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Box 35, Finland +3)Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, +Tennessee 37831 +4)Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250 +(Dated: 30 January 2023) +Previous works have controversially claimed near-room temperature ferromagnetism in two-dimensional (2D) VSe2, +with conflicting results throughout the literature. These discrepancies in magnetic properties between both phases (T +and H phase) of 2D VSe2 are most likely due to the structural parameters being coupled to the magnetic properties. +Specifically, both phases have a close lattice match and similar total energies, which makes it difficult to determine +which phase is being observed experimentally. In this study, we used a combination of density functional theory +(DFT), highly accurate diffusion Monte Carlo (DMC) and a surrogate Hessian line-search optimization technique to +resolve the previously reported discrepancy in structural parameters and relative phase stability. With DMC accuracy, +we determined the freestanding geometry of both phases and constructed a phase diagram. Our findings demonstrate +the successes of the DMC method coupled with the surrogate Hessian structural optimization technique when applied +to a 2D magnetic system. +I. +INTRODUCTION +One of the most promising two-dimensional (2D) mag- +netic materials that has been extensively studied experimen- +tally and theoretically is 2D VSe2. Similar to other 2D tran- +sition metal dichalcogenides (such as MoS2)1, VSe2 exists +in two phases, the T (octahedral phase (1T)-centered honey- +combs) phase which is metallic and the H (the trigonal pris- +matic phase (2H)-hexagonal honeycombs, see Fig. 1) phase +which is semiconducting. Several experimental and theoret- +ical studies have controversially claimed near-room tempera- +ture ferromagnetism in VSe2, with conflicting results through- +out the literature. Density functional theory (DFT) along with +classical Monte Carlo simulations have been used to obtain +a) +b) +1T-VSe2 +2H-VSe2 +FIG. 1. Top and side view of the atomic structure of monolayer VSe2 +in the a) 1T and b) 2H phase. +a)Electronic mail: daniel.wines@nist.gov +b)Electronic mail: ataca@umbc.edu +an estimate of the Curie temperature of H-VSe2 (291 K)2, but +the model Ising Hamiltonian used did not take into account the +magnetic anisotropy energies, which are essential for an accu- +rate estimation of the Curie temperature of a 2D lattice. The +Curie temperature of multilayered 2D H-VSe2 has been ex- +perimentally measured to be 425 K, with the ferromagnetism +softening as the thickness of the sample increases3. Addi- +tionally, the experimental Curie temperature for monolayer T- +VSe2 has ranged from 300 K to 470 K4,5 depending on which +substrate is used (MoS2, graphite, SiO2-coated silicon). The +experimental magnetization of T-VSe2 has also been met with +controversy, with values of 15 µB and 5 µB (per formula unit) +being reported from two separate studies4,6. Insight has also +been reported with regards to how the ferromagnetism is en- +hanced with defects, molecular adsorption and the choice of +substrate for VSe24,5,7. A wide range of values have also been +reported for the charge density wave (CDW) transition tem- +perature for T-VSe2, ranging from 120 K to 350 K3,6,8–10. +These discrepancies in the electronic and magnetic proper- +ties of either phase of 2D VSe2 arise from the structural pa- +rameters of each phase being coupled closely to the magnetic +and electronic properties and the external factors (substrates, +defects) of the individual samples. One example of this has +been a reported discrepancy on which phase (T or H) is en- +ergetically more favorable. Both the T and H phases have a +close lattice match and similar total energies, which makes it +difficult to determine which phase is being observed experi- +mentally. Recently, it has been reported experimentally that +the T phase is favored for bulk VSe2, but with dimension- +ality decrease, the H phase is favored3,11. It has also been +reported that a T-to-H phase transition can be realized by ther- +mal annealing11. This same structural phase transition has +even been reported by applying a biaxial strain of ≈ 3 % (from +calculated results)7,11,12. Researchers have proposed that this +lattice strain can be induced by the mismatch that occurs from +arXiv:2301.11404v1 [cond-mat.str-el] 26 Jan 2023 + +b +C +aC +a +bb2 +putting 2D VSe2 on a substrate7,12. +From a computational perspective, results for VSe2 depend +heavily on which methodology is employed. In most cases, +DFT with an empirical Hubbard correction (+U) for corre- +lated electrons is used13. For example, if the U correction is +applied for T and H-VSe2, the T phase is more energetically +favorable, while if no U correction is applied, the H phase +is more favorable14. In addition to the discrepancies in re- +sults calculated with DFT+U, results between van der Waals +(vdW) corrected functionals and hybrid functionals are also +inconclusive14 in terms of predicting the relative phase stabil- +ity. In order to alleviate the uncertainty in DFT methods, more +sophisticated methods can be used such as Diffusion Monte +Carlo (DMC)15. DMC is a correlated, many-body electronic +structure method that has demonstrated success for the elec- +tronic and magnetic properties of a variety of bulk and 2D +systems16–24. This method has a weaker dependence on the +starting density functional and U parameter and can success- +fully achieve results with an accuracy beyond the DFT+U15. +Due to the fact that T and H-VSe2 have structural parame- +ters that are coupled to their electronic and magnetic proper- +ties, it makes it difficult to produce conclusive results that rely +solely on DFT or DFT+U. For this reason, we employed our +recently developed energy-based surrogate Hessian method +for structural optimization with stochastic electronic structure +theories (such as DMC)22 to obtain the geometry of T and +H-VSe2 with DMC accuracy, resulting in high-accuracy bond +lengths that resolve previous functional dependent structural +discrepancies. After obtaining an accurate geometry for both +structures, we constructed a phase diagram between T and H- +VSe2 using DMC calculated energies and obtained accurate +magnetic properties of each structure. The accurate estimates +for lattice geometry, relative phase energy and the DMC phase +diagram assist in clarifying previously inconclusive theoreti- +cal and experimental results regarding T and H phase VSe2. +For full details of the computational methods used, see the +Supporting Information (SI). +As an initial starting point for our study, we performed +benchmarking DFT and DFT+U calculations using a variety +of density functionals (local density approximation (LDA)25, +Perdew-Burke-Ernzerhof (PBE)26, and strongly constrained +and appropriately normed (SCAN)27 meta-GGA functionals, +see SI for more details) and the Vienna Ab initio Simulation +Package (VASP) code for monolayer T-VSe2 and H-VSe2. +The goal of these simulations were to assess how sensitive +the relative energy between the T and H phase is with re- +spect to functional and material geometry. Another goal of +these simulations was to benchmark the structural parameters +of each material with respect to several density functionals. It +is advantageous to perform these reference calculations with +VASP and PAW pseudopotentials as a precursor to the more +expensive DMC calculations due to the fact that they require +a much smaller cutoff energy and are more cost effective for +a large number of simulations. It is important to note that +for all DFT and DMC simulations, we assumed a ferromag- +netic ground state for both T and H-VSe2. Although recent +reports have suggested that T-VSe2 could be experimentally +paramagnetic3, we infer that this paramagnetism can be in- +duced by magnetic anisotropy. In addition, the modeling of +paramagnetism with computational methods imposes a great +challenge, which is why we focus on the freestanding ferro- +magnetic ground states of both phases. A more robust treat- +ment of the magnetic structure can be explored in future work, +but is beyond the scope of this work which primarily focuses +on determining the geometric structure and phase stability of +2D T and H-VSe2. +In Fig. 2 we present a comprehensive look at the difference +in total energy between T-VSe2 and H-VSe2, using several +DFT functionals under different geometric constraints. We +performed these calculations for a variety of U values in three +different ways: fully relaxing the structure at each value of U +(Fig. 2 a) ), fixing the lattice and atomic positions to the U += 0 eV relaxed geometry of that particular functional and cal- +culating the static energy at each value of U (Fig 2 b)), fixing +the lattice to the U = 0 eV relaxed geometry of that particular +functional and relaxing just the atomic positions at each value +of U (Fig. 2 c)). The results in Fig. 2 indicate that there is +a significant disagreement between DFT functionals, U value +used, and material geometries, with all three factors playing +a significant role in the energy difference between T and H +phase. Specifically, regardless of relaxation method, all bare +(no U correction) SCAN, PBE, and PBEsol functionals pre- +dict H favorable, while bare LDA predicts T favorable. For +all functionals, there is a critical value of U that reverses the +relative phase stability, which is dependent on functional and +relaxation method. The SCAN functional with a U correction +predicts T phase favorable, with larger energy differences. As +seen in Fig. 2, the trends in the relative phase stability be- +tween Fig. 2 b) and c) are nearly identical, but significantly +vary from Fig. a). This implies that the density functional is +strongly coupled to material geometry, but the lattice constant +change has more of an effect on phase stability than atomic +positions and bond distances. This is most prevalent for higher +U values (> 2 eV), where the relaxed geometry changes more +drastically with U. The interrelated nature of the material’s +geometry, density functional, and value of U are reasons to +seek out higher levels of theory beyond DFT/DFT+U such as +DMC to accurately determine the optimal geometry and rela- +tive energy between the phases of 2D VSe2. +The relaxed lattice constants, V-Se distances, and T - H en- +ergies from Fig. 2 a) are presented in Table I and Fig. 3, +along with additional VASP reference calculations performed +with the vdW corrected functionals (PBE-D228, PBE-D329, +SCAN+rvv1030). The DMC computed parameters are also +given for comparison in Table I and Fig. 3 (more discussion +to follow). We observe a ≈ 7 % variability in lattice constant +across the different methods for T-VSe2 and a ≈ 4 % variabil- +ity in lattice constant across the different methods for H-VSe2. +Between both phases, we observe a ≈ 3 % variability in V-Se +distance (dV−Se). Most strikingly, the energy difference be- +tween the T and H phases (ET−H) drastically varies depend- +ing on the material geometry and computational methodology, +ranging from -0.2 eV/f.u. to 0.06 eV/f.u.. Due to the fact +that a strain-induced phase transition has been reported be- +tween T- and H-VSe27,11,12, we decided to perform additional +VASP benchmarking calculations that involved the applica- + +3 +0 +1 +2 +3 +4 +U (eV) +0 +1 +2 +3 +4 +U (eV) +0 +1 +2 +3 +4 +U (eV) +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +T - H Energy (eV/f.u.) +-0.3 +-0.2 +-0.1 +0 +0.1 +-0.05 + 0.05 +-0.15 +-0.25 +-0.35 +T - H Energy (eV/f.u.) +-0.3 +-0.2 +-0.1 +0 +0.1 +-0.05 + 0.05 +-0.15 +-0.25 +-0.35 +T - H Energy (eV/f.u.) +PBE +LDA +SCAN +PBESOL +Full relaxation +Fixed lattice/relaxed positions +Fixed lattice/positions +a) +b) +c) +FIG. 2. Relative (T - H) energy between T and H phase 2D VSe2 as a function of U parameter for several density functionals and methods of +atomic relaxation: a) fully relaxing the structure, b) fixing the lattice and atomic positions to the U = 0 eV relaxed geometry of that particular +functional and calculating the static energy, c) fixing the lattice to the U = 0 eV relaxed geometry of that particular functional and relaxing just +the atomic positions. The dotted line indicates 0 eV. +TABLE I. Tabulated results for lattice constant, V-Se distance, and relative energy (T - H) for both T and H phase 2D VSe2 for several +computational methods. DMC error bars (standard error about the mean) are included in parenthesis. +T-VSe2 +H-VSe2 +Method +a (Å) +dV−Se (Å) a (Å) +dV−Se (Å) ET−H (eV/f.u.) +PBE +3.336 +2.489 +3.333 +2.502 +0.045 +PBE+U=2 +3.435 +2.526 +3.364 +2.520 +-0.008 +LDA +3.228 +2.438 +3.229 +2.445 +-0.026 +LDA+U=2 +3.277 +2.455 +3.266 +2.464 +0.045 +SCAN +3.387 +2.486 +3.329 +2.486 +0.045 +SCAN+U=2 +3.462 +2.524 +3.353 +2.502 +-0.202 +PBEsol +3.262 +2.458 +3.272 +2.471 +0.013 +PBEsol+U=2 3.323 +2.483 +3.301 +2.487 +0.025 +PBE-D2 +3.323 +2.484 +3.318 +2.496 +0.010 +PBE-D3 +3.315 +2.485 +3.319 +2.497 +0.042 +SCAN+rvv10 3.379 +2.481 +3.319 +2.482 +0.051 +DMC +3.414(12) 2.505(7) +3.335(8) 2.503(5) +0.06(2) +tion of tensile and compressive strain for each monolayer. We +performed these calculations for PBE, SCAN, and LDA (with +U = 0 eV and U = 2 eV), starting from the U = 0 eV geom- +etry for each functional. The resulting equations of state are +depicted in Fig. S3. As seen in the figure, the equation of +state and resulting strain-induced phase transition is entirely +dependent on the functional and U value, with no consistent +trend. +The strong sensitivity of each monolayer with respect to +geometry and functional are grounds for using a higher-order +method such as DMC to obtain a statistically accurate estimate +of the lattice parameters and relative energy between phases. +Prior to performing the DMC/line-search calculations, we op- +timized our nodal surface (orbitals selected for DFT wave- +function generation). Since DMC has the zero-variance prop- +erty, it means that as the trial wave function approaches the + +4 +ET-H - ET-H +(DMC)(eV/f.u.) +dV-Se - dV-Se +(DMC) (Å) +dV-Se - dV-Se +(DMC) (Å) +a - aDMC (Å) +a - aDMC (Å) +T-VSe2 +H-VSe2 +a) +b) +c) +FIG. 3. A summary of the deviation of the geometric properties rel- +ative to the DMC calculated geometric properties for a) T-VSe2 and +b) H-VSe2 and c) the the deviation of T - H energy relative to the +DMC calculated T - H energy for a variety of DFT functionals (U = +2 eV), where the DMC error bar (standard error about the mean) is +represented by the red bars. +exact ground state, the statistical fluctuations in the energy +reduce to zero15. Although there have been instances where +various sophisticated methods have been used to optimize the +nodal surface31–34, we employed the PBE+U approach, where +the Hubbard (U) value was used as a variational parameter +to optimize the nodal surface using DMC (similar to other +successful DMC studies of magnetic materials16,20,21,24,35–37). +We performed these calculations for both T and H-VSe2 (24 +atom supercells), where we tuned the U value from (1 to 4) eV +while creating the trial wavefunction and computed the DMC +energy. The results of these calculations are depicted in Fig. +S4, where we observe that U = 2 eV yields the lowest energy +for both phases. It is important to note that for the H phase, +the DMC energies for U = 1 and U = 2 eV are statistically +identical. Based on this, we created the trial wavefunction us- +ing PBE+U (U = 2 eV) for all subsequent DMC calculations +within the surrogate Hessian line-search for both phases (all +52 DMC energy evaluations). Since we obtained an optimal +U value of 2 eV for both materials, we focused our DFT+U +benchmarking efforts more on U = 2 eV (Fig. 3, Fig 5, Table +I, Fig. 2, Fig. S3). +Based on the DMC line-search results, we determined ac- +curate bounds on the lattice parameter (a) and off-plane dis- +placement of Se (z), within an error tolerance of 0.018 Å or +lower for both parameters. This translates to within ≈ 0.5% +accuracy in a parameter set of a and dV−Se with 95% con- +fidence. Convergence (absence of significant displacements +2.45 +2.50 +2.55 +2.60 +V-Se distance (˚A) +T-phase +H-phase +Fit eqm. (T) +Fit eqm. (H) +LS eqm. (T) +LS eqm. (H) +3.2 +3.3 +3.4 +3.5 +3.6 +3.7 +3.8 +Lattice constant (˚A) +−2459.5 +−2459.0 +−2458.5 +Energy/f.u. (eV) +PES (T) +PES (H) +FIG. 4. (Top) The phase diagram of 2D VSe2 in terms of a and +dV−Se. The phase boundary (solid line, black) is estimated from +bicubic fits. To assure quality of the fits, the estimated ±0.01 eV +error contours (dotted line) and the minima from the fits (’x’) and the +line-search (’o’) are all well separated. (Bottom) Slices of the PES at +dV−Se = 2.505 Å. +outside of the error tolerance) was achieved after two parallel +line-search iterations for both phases. This convergence is il- +lustrated in Fig. S5, where the convergence of the parameter +offsets of a and z and the convergence of the total energy per +f.u. are depicted for both T and H phase 2D VSe2 for the ini- +tial DFT relaxed structure (1) and both subsequent iterations +of DMC (2 - 3). In addition, the final energy of both of the +fitted structures (square points) are given. +The final geometric parameters and relative phase energies +determined with DMC are given in Table I and Fig. 3. For +T-VSe2, we determined a lattice constant of 3.414(12) Å and +a V-Se distance of 2.505(7) Å . For H-VSe2, we determined a +lattice constant of 3.335(8) Å and a V-Se distance of 2.503(5) +Å . The DMC finite-size extrapolated energy difference (T +- H) between the two phases was determined to be 0.06(2) +eV/f.u., indicating that in freestanding form at the equilibrium +geometry, H-VSe2 is favored over T-VSe2. When comparing +these DMC results to the other DFT functionals in Table I and +Fig. 3, it is clear that very few DFT functionals can repro- +duce the DMC results for lattice constant, V-Se distance and +relative energy difference. The SCAN functional comes the +closest to reproducing all three simultaneous DMC values, but +still falls slightly short for the V-Se distances of both phases +and the lattice constant of T-VSe2. The fact that SCAN+U +successfully predicts the structural properties (for H-VSe2) +and the fact that SCAN+rvv10 produces an energy difference +closest to the average DMC energy difference for both phases +loosely implies that a simultaneous description of correlated +magnetism and vdW interactions are both needed to correctly +represent the physics of VSe2. Experimental measurements of + +5 +the lattice constant and V-Se distance of freestanding mono- +layer VSe2 are scarce and often times dependent on external +factors such as the substrate (more discussion to follow) and +sample preparation technique4,5,38,39. However, Chen et al.38 +have recently reported a lattice constant of 3.4 Å for thin films +of T-VSe2 and Liu et al.39 have recently reported a lattice +constant of 3.3 Å for epitaxially grown monolayer H-VSe2. +Both of these measured values are in excellent agreement with +our DMC computed lattice constants. Additionally, we deter- +mined the near-equilibrium PES of both T and H 2D VSe2 +with DMC accuracy, which are both depicted in Fig. S6. +The phase diagram presented in Fig. 4 is based on similar +fits to data, where the z displacement has been remapped to +dV−Se. This DMC phase diagram can directly be compared to +the energy vs. strain DFT benchmarking calculations in Fig. +S3, which emphasizes the need for an accurate representation +of the phase boundary between the two phases. The freestand- +ing geometries of both T and H lie in the energetic H phase, +but a slice of the phase diagram along dV−Se = 2.505 Å in- +dicates that the T phase becomes favorable over H at biaxial +strain of a ≳ 3.5 Å. This implies that in freestanding form, +once T-VSe2 is positively strained at least ≈ 2.5 %, T phase is +favored over H. Alternatively, if freestanding H-VSe2 is pos- +itively strained at least ≈ 5 %, T phase is also favored over +H This strain can easily be accomplished by placing mono- +layer VSe2 on a substrate with significant lattice mismatch. In +fact, this type of mismatch has been reported to alter the mate- +rial properties4,5,40,41, significantly contributing to the contro- +versies of T and H-VSe2 (for energetic favorability, magnetic +properties). Whether or not the changes in energetic favorabil- +ity or magnetic properties with respect to the substrate are due +to lattice mismatch or more complicated interactions between +the substrate and the monolayer remains to be answered and +is beyond the scope of this work, which has focused solely on +the freestanding forms of T and H-VSe2. However, such cal- +culations can be employed for future work using higher order +methods such as DMC. The proximity of the phase boundary +between T and H phase (Fig. 4) is emphasized by the small en- +ergy difference between the two phases (0.06(2) eV/f.u., at the +equilibrium geometry) between the two curves. Since this en- +ergy difference is so close to room temperature (≈ 0.024 eV), +this implies that a process such as thermal annealing can eas- +ily induce a phase transition. In fact, recently it was demon- +strated that a structural phase transition of multilayer VSe2 +from T to H occurs through annealing at 650 K, along with a +metal-insulator transition11. +To gain a deeper understanding of the magnetic properties +of 2D T and H-VSe2, we extracted the spin densities (using a +trial wavefunction at U = 2 eV and 24 atom supercell at the +final equilibrium geometry predicted by DMC/line-search). +The spin density isosurfaces of each phase (ρup - ρdown) are +depicted in the insets of Fig. 5 a) and c) for T-VSe2 and H- +VSe2 respectively. For both phases, we observe the V atoms +are highly spin-polarized, while the Se atoms are slightly an- +tiparallel with respect to the V atoms. For more calculation +details regarding spin density, see SI. +We went on to plot the radial averaged spin densities as a +function of distance, separately for V and Se for T and H-VSe2 +(depicted in Fig. 5 a) - d)). This allows us to view the spa- +tial variations in spin density. Additionally, we benchmarked +these V and Se radially averaged densities with PBE+U (U += 2 eV) using NC pseudopotentials at the equilibrium geom- +etry (the calculation required to create the trial WF for the +subsequent DMC runs). As seen in Fig. 5 a) and c), there is +a substantial difference in the V spin density between DMC +and PBE+U (U = 2 eV) for both T and H phase. This same +substantial difference between DMC and PBE+U also occurs +for the total charge density. This discrepancy is most preva- +lent near the radial density peak (peak of d orbital) and can +be attributed to the fact that DFT functionals (even with the +added Hubbard correction) tend to delocalize and unsuccess- +fully capture 3d orbitals. This large discrepancy in the spin +densities highlights the need for more accurate, many-body +computational methodologies for correlated materials such as +VSe2, where DFT fails. In contrast, there is closer agreement +between the DMC and PBE+U spin densities for Se in T and +H-VSe2 (see Fig. 5 b) and d). +Finally, we estimated the site-averaged atomic magnetic +moments per V and Se for both T and H phase by integrating +the DMC and PBE+U spin densities depicted in Fig. 5. At the +DMC level, we estimated a magnetic moment of 1.06(2) µB +for V and -0.09(2) µB for Se in T-VSe2 and a magnetic mo- +ment of 1.02(1) µB for V and -0.14(1) µB for Se in H-VSe2. +At the PBE+U (U = 2 eV) level, we estimated a magnetic mo- +ment of 1.30 µB for V and -0.12 µB for Se in T-VSe2 and a +magnetic moment of 1.40 µB for V and -0.15 µB for Se in H- +VSe2. Consistent with the radial spin density results in Fig. +5, we find that the DMC and PBE+U magnetic moments for +Se are in much closer agreement than for V (for both T and +H phase). By analyzing the spin densities and obtaining the +on-site magnetic moments, we obtain a clear picture of how +the magnetization of each ion depends on the computational +method used, serving as a benchmark for the magnetic prop- +erties of 2D VSe2. +In this work, we used a combination of DFT, DMC and +a recently developed surrogate Hessian line-search optimiza- +tion technique to resolve the previously reported discrepancy +in structural parameters and relative phase stability of mono- +layer T-VSe2 and H-VSe2. Using these methods, we deter- +mined the lattice constant and V-Se distance (with DMC ac- +curacy) to be 3.414(12) Å and 2.505(7) Å respectively for T- +VSe2 and 3.335(8) Å and 2.503(5) respectively for H-VSe2. +In addition, we find the relative energy between the phases (T +- H) to be 0.06(2) eV/f.u. at the DMC level, indicating that +in freestanding form, H-VSe2 is more energetically favorable +than T-VSe2. We went on to obtain a phase diagram between +T and H phase from the PES and determined that a phase tran- +sition can be induced by strain or mechanisms such as ther- +mal annealing. Additionally, we benchmarked the magnetic +properties such as spin density and on-site magnetic moment +for both phases and find substantial differences between DMC +and DFT. The results of this study demonstrate the successes +of the DMC method coupled with the surrogate Hessian line- +search structural optimization technique when applied to a 2D +magnetic system. +The estimates for lattice constant, bond +distance, relative phase energy and the extracted structural- + +6 +a) +b) +c) +d) +4̟r2[ρup - ρdown ] (Ne/Å) +4̟r2[ρup - ρdown ] (Ne/Å) +4̟r2[ρup - ρdown ] (Ne/Å) +4̟r2[ρup - ρdown ] (Ne/Å) +T-VSe2 (V) +T-VSe2 (Se) +H-VSe2 (V) +H-VSe2 (Se) +r (Å) +r (Å) +r (Å) +r (Å) +MV= 1.30 µB +MV= 1.06(2) µB +MV= 1.40 µB +MV= 1.02(1) µB +MSe= -0.12 µB +MSe= -0.09(2) µB +MSe= -0.15 µB +MSe= -0.14(1) µB +FIG. 5. The radially averaged spin density (ρup - ρdown) as a function of distance, calculated with DMC and PBE+U (U = 2 eV) of a) V and +b) Se for 2D T-VSe2 and c) V and d) Se for 2D H-VSe2. The inset of a) and c) depicts the spin isosurface density of T-VSe2 and H-VSe2 +respectively, where the isosurface value was set to 6 x 10−3 e/Å3. The standard error about the mean for DMC is indicated by error bars in +blue. +dependent phase diagram assist in clarifying previously incon- +clusive theoretical and experimental results regarding T and H +phase VSe2. +II. +CODE AVAILABILITY STATEMENT +Software packages mentioned in the article can be found at +https://github.com/usnistgov/jarvis. Please note that the use of +commercial software (VASP) does not imply recommendation +by the National Institute of Standards and Technology. +III. +COMPETING INTERESTS +The authors declare no competing interests. +IV. +ACKNOWLEDGMENTS +The authors thank the National Institute of Standards +and Technology for funding, +computational, +and data- +management resources. +The authors thank Dr. +Kamal +Choudhary and Dr. +Francesca Tavazza for fruitful discus- +sions. +We acknowledge grants of computer capacity from +the Finnish Grid and Cloud Infrastructure (persistent identi- +fier urn:nbn:fi:research-infras-2016072533). +REFERENCES +1C. Ataca, H. ¸Sahin, and S. Ciraci, “Stable, single-layer MX2 transition- +metal oxides and dichalcogenides in a honeycomb-like structure,” The Jour- +nal of Physical Chemistry C 116, 8983–8999 (2012). +2H.-R. Fuh, C.-R. Chang, Y.-K. Wang, R. F. L. Evans, R. W. Chantrell, and +H.-T. 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Li, “Structural and transport +properties of 1t-vse2 single crystal under high pressures,” Frontiers in Ma- +terials 8 (2021), 10.3389/fmats.2021.710849. + +Supporting Information: A Quantum Monte +Carlo study of the structural, energetic, and +magnetic properties of two-dimensional (2D) H +and T phase VSe2 +Daniel Wines,∗,† Juha Tiihonen,‡ Kayahan Saritas,¶ Jaron Krogel,§ and Can +Ataca∗,∥ +†Materials Science and Engineering Division, National Institute of Standards and +Technology (NIST), Gaithersburg, MD 20899 +‡Department of Physics, Nanoscience Center, University of Jyv¨askyl¨a, P.O. Box 35, +Finland +¶ Department of Applied Physics, Yale University, New Haven CT 06520 +§ Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, +Tennessee 37831 +∥Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250 +E-mail: daniel.wines@nist.gov; ataca@umbc.edu +Computational Methods +Density functional theory (DFT) benchmarks for the T and H phase of 2D VSe2 were per- +formed using the Vienna Ab initio Simulation Package (VASP) code with projector aug- +mented wave (PAW) pseudopotentials.1,2 For these calculations, the local density approxi- +S1 +arXiv:2301.11404v1 [cond-mat.str-el] 26 Jan 2023 + +mation (LDA),3 Perdew-Burke-Ernzerhof (PBE),4 and strongly constrained and appropri- +ately normed (SCAN)5 meta-GGA functionals were used with the added Hubbard correction +(U)6 to treat the on-site Coulomb interaction of the 3d orbitals of the V atoms. At least 20 +˚A of vacuum was given between periodic layers of VSe2 in the c-direction. In addition, we +used a reciprocal grid of 24x24x1 and a kinetic energy cutoff of 400 eV. +Our Quantum Monte Carlo (QMC) simulations used DFT-PBE to generate the trial +wavefunction for fixed-node diffusion Monte Carlo (DMC) calculations. +The Quantum +Espresso (QE)7 code was used for our DFT calculations to create the trial wavefunction. +This trial wavefunction was created for the ferromagnetic configuration of 2D VSe2 using +different U values with the goal of variationally determining the optimal nodal surface (U +value that yields the lowest total energy). For V, we used norm-conserving (NC) RRKJ +(OPT) pseudopotentials8 and for Se, we used NC Burkatzki-Fillipi-Dolg (BFD) pseudopo- +tentials.9 After testing at the DFT level, a kinetic energy cutoff of 4,080 eV (300 Ry) and +a k-grid of 6x6x1 was used (see Fig. S1 and S2) to generate trial wavefunctions for DMC. +To accelerate the line-search method convergence for the metallic T phase, we increased the +k-grid to 12x12x1. +After the trial wavefunction was generated with DFT, Variational Monte Carlo (VMC) +and DMC10,11 calculations were performed using the QMCPACK12,13 code. The single de- +terminant DFT wavefunction is converted into a many-body wavefunction by use of the Jas- +trow parameters,14,15 which assist in modeling electron correlation with the goal of reducing +the statistical uncertainty in DMC calculations.16,17 Up to two-body Jastrow18 correlation +functions were included, where the linear method19 was used to minimize the variance and +energy of the VMC energies. The cost function of the variance optimization is 100 % vari- +ance minimization and the cost function of the energy optimization is split as 95 % energy +minimization and 5 % variance minimization, which has been proven to reduce the uncer- +tainty of DMC calculated results.16 The Nexus20 software suite was used to automate the +DFT-VMC-DMC workflow. The locality approximation17 was used to evaluate the nonlocal +S2 + +part of the pseudopotentials in DMC and an optimal timestep of 0.01 Ha−1 was determined +for DMC simulations due to the fact that it yielded an acceptance ratio greater than 99 % +(see Table S1). A full summary of the VMC and DMC methods can be found in reference.10 +The total charge density and spin density was extracted from our DMC calculations. +The spin density is defined as the difference between the spin-up contribution to the total +charge density and the spin-down contribution to the total charge density (ρup − ρdown). We +used an extrapolation scheme on the DMC charge densities with the goal of eliminating the +bias that occurs from using a mixed estimator. Since the charge density estimator does not +commute with the fixed-node Hamiltonian, the DMC charge density was obtained from a +mixed estimator between the pure fixed-node DMC and VMC densities. The extrapolation +formula takes the form:10 +ρ1 = 2ρDMC − ρVMC + O[(Φ − ΨT)2] +(1) +where ρDMC and ρVMC are the DMC and VMC charge densities respectively. Φ is the trial +wavefunction from the DMC Hamiltonian and ΨT is the trial wavefunction from VMC. +In addition, we integrated the DFT+U and DMC spin densities up to a cutoff radius +rcut (which we define as 1.34 ˚A , due to the fact that it is approximately half of the V-Se +bond distance in 2D T and H-VSe2) in order to estimate the site-averaged atomic magnetic +moment per V and Se. To obtain these magnetic moments per atom (MA), we sum over the +spherically interpolated spin densities: +MA = 4π +� rcut +0 +r2ρs(r)dr ≈ 4π +rcut/∆r +� +i=0 +r2 +i ρs(ri)∆r +(2) +where ri is the distance from the center of the atom to a given point on the grid and ∆r is +the radial grid size. +To optimize the structural parameters of both T and H-VSe2 according to the DMC po- +tential energy surface (PES), we use a surrogate Hessian accelerated optimization method.21 +S3 + +In the method, we consider the PES around equilibrium as the second-order expansion in +Wyckoff parameter space, p: +E(p) = E0 + 1 +2(p − p0)THp(p − p0), +(3) +where Hp is the Hessian, or the force-constant matrix, E0 is the energy minimum and p0 +the energy-minimizing parameters. Diagonalizing the parameter Hessian, i.e., Hp = U TΛU, +forms an optimal basis for a conjugate line-search in the parameter space, namely the eigen- +vectors U. The line-searches along U can be conducted in parallel, and ideally, they locate +the minimum in just one parallel iteration within the quadratic region. Here, we conduct +the line-search according to a set of 2 parameters: the lattice constant a and the Wyckoff +parameter z, which is the unsigned displacement of the Se atoms along the z axis (see Fig. +1). For reporting purposes, the line-search parameters a and z are remapped to a and d, +where d is the V-Se distance. +In the surrogate Hessian scheme, we obtain a cheap but relatively accurate Hessian from +DFT, and use it to the inform line-search on the DMC PES, in particular by providing the +search directions. We also resample the DFT PES to predict fitting errors. Thus, we may +minimize the computational cost of the DMC runs, while maintaining an error tolerance. +The surrogate DFT PES was based on QE with a 4,080 eV (300 Ry) cutoff using PBE with +no DFT+U correction. The DMC PES was based on DFT-PBE with U = 2 eV orbitals +and finite-size extrapolation through supercell sizes of 9 and 24 atoms. Each line-search was +based on a 3rd order polynomial fit and set to contain 7 points, or displaced geometries, +totaling 13 energy evaluations per phase, per iteration. However, alternative techniques, +including (bi)polynomial fitting, were used in some parts to incorporate auxiliary DMC +data and ensure convergence to the quadratic region. Effectively, two parallel line-search +iterations for both phases were carried out, and the convergence was claimed in the absence +of significant displacements. +S4 + +a) +b) +Figure S1: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b) +H-VSe2 as a function of plane wave cutoff energy for the norm-conserving pseudopotentials +calculated with DFT using the PBE functional at a k-point grid of 6x6x1. The results show +a converged cutoff energy of 4,080 eV (300 Ry) for both phases. +a) +b) +Figure S2: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b) +H-VSe2 as a function of K-point grid for the norm-conserving pseudopotentials calculated +with DFT (PBE) at the converged cutoff energy (see Fig. S1). The results show a converged +k-point grid of 6x6x1 (36) for both monolayers. The number of K-points was scaled appro- +priately to obtain the converged grid depending on the supercell size and shape for all DFT +and DMC calculations. +S5 + +PBE (U = 0) +PBE (U = 2) +SCAN (U = 0) +SCAN (U = 2) +3.15 +3.25 +3.35 +3.45 +3.55 +3.15 +3.25 +3.35 +3.45 +3.55 +3.15 +3.25 +3.35 +3.45 +3.55 +3.15 +3.25 +3.35 +3.45 +3.55 +-18.04 +-18.00 +-17.96 +-17.92 +-17.88 +-17.84 +-16.00 +-15.95 +-15.90 +-15.85 +-15.80 +-15.65 +-15.75 +-15.70 +-59.90 +-59.85 +-59.80 +-59.75 +-58.70 +-59.65 +-58.10 +-58.00 +-57.90 +-57.80 +-57.70 +-57.60 +Total Energy (eV) +Total Energy (eV) +Total Energy (eV) +Total Energy (eV) +Lattice Constant (Å) +Lattice Constant (Å) +Lattice Constant (Å) +Lattice Constant (Å) +3.05 +3.15 +3.25 +3.35 +-18.04 +-18.00 +-17.90 +-17.80 +Total Energy (eV) +Lattice Constant (Å) +LDA (U = 0) +3.05 +3.15 +3.25 +3.35 +Lattice Constant (Å) +LDA (U = 2) +-20.30 +-20.20 +-20.15 +-20.10 +Total Energy (eV) +-20.25 +-20.05 +T +H +Figure S3: Total energy as a function of lattice strain for T (blue) and H (red) phase 2D +VSe2, calculated with various functionals and U values. Density functionals include LDA, +PBE, and SCAN. +S6 + +Table S1: Tabulated results for the DMC timestep convergence of a 12 atom cell of 2D +T-VSe2 and H-VSe2. The acceptance ratio of 0.99 indicates that 0.01 Ha−1 is an appropriate +timestep to use for all subsequent DMC simulations. +T-VSe2 +Timestep (Ha−1) +DMC Total Energy (Ha) +Error (Ha) +Acceptance Ratio +0.02 +-361.730 +0.001 +0.985 +0.01 +-361.709 +0.002 +0.994 +0.005 +-361.709 +0.003 +0.997 +0.002 +-361.702 +0.002 +0.999 +H-VSe2 +Timestep (Ha−1) +DMC Total Energy (Ha) +Error (Ha) +Acceptance Ratio +0.02 +-361.673 +0.001 +0.985 +0.01 +-361.657 +0.002 +0.994 +0.005 +-361.654 +0.002 +0.998 +0.002 +-361.657 +0.003 +0.999 +1 +2 +3 +4 +U (eV) +-2460.30 +-2460.25 +-2460.20 +-2460.15 +-2460.10 +-2460.05 +-2460.00 +-2459.95 +Total Energy (eV/f.u.) +T +H +Figure S4: DMC calculated total energies of a 24-atom supercell (normalized per formula +unit (f.u.)) +of 2D T (blue) and H (red) phase VSe2 calculated as a function of the U +parameter used to variationally determine the optimal trial wave function. The DMC error +bars represent the standard error about the mean. +S7 + +1.0 +1.5 +2.0 +2.5 +3.0 +−0.02 +−0.01 +0.00 +0.01 +a (˚A ) +1.0 +1.5 +2.0 +2.5 +3.0 +−0.010 +−0.005 +0.000 +0.005 +0.010 +z (˚A ) +1 +2 +3 +Iteration +-2459.55 +-2459.6 +-2459.65 +-2459.7 +E/f.u. (eV ) +T +H +Figure S5: The convergence of the a and z parameters and DMC energies per f.u. for both +T (blue) and H (red) phase of 2D VSe2 based on parallel line-search iterations along the +DMC PES. The starting parameters (iteration 1) are from DFT, the zero offset is the mean +over iterations 2 and 3, and dotted lines indicate the error tolerances for each case (95 % +confidence). The DMC energies from respective equilibrium geometries are plotted with +1SEM (one standard error of the mean) uncertainties, with extra squares marking energies +from the predicted minimum geometry. +S8 + +3.00 +3.25 +3.50 +3.75 +4.00 +Lattice constant (˚A ) +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Z-offset (˚A ) +LS eqm . +Fit eqm . +LS # 0 +LS # 1 +3.00 +3.25 +3.50 +3.75 +4.00 +Lattice constant (˚A ) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Figure S6: Contour reconstructions of the DMC PESs (eV) of T (left) and H (right) phases of +2D VSe2 with respect to a and z parameters. The contours are based on bicubic fits to sparse +data, and thus, subject to biases and statistical uncertainties not indicated in the figures. +The markers (’x’ and ’+’) indicate data points from two parallel line-search iterations. +References +(1) Kresse, G.; Furthm¨uller, J. Efficient iterative schemes for ab initio total-energy calcu- +lations using a plane-wave basis set. Phys. Rev. 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QMCPACK: Advances in the development, efficiency, and applica- +tion of auxiliary field and real-space Variational and Diffusion Quantum Monte Carlo. +The Journal of Chemical Physics 2020, 152, 174105. +(14) Slater, J. C. The Theory of Complex Spectra. Phys. Rev. 1929, 34, 1293–1322. +(15) Jastrow, R. Many-Body Problem with Strong Forces. Phys. Rev. 1955, 98, 1479–1484. +(16) Umrigar, C. J.; Filippi, C. Energy and Variance Optimization of Many-Body Wave +Functions. Phys. Rev. Lett. 2005, 94, 150201. +S10 + +(17) Mitas, L.; Shirley, E. L.; Ceperley, D. M. Nonlocal pseudopotentials and Diffusion +Monte Carlo. The Journal of Chemical Physics 1991, 95, 3467–3475. +(18) Drummond, N. D.; Towler, M. D.; Needs, R. J. Jastrow correlation factor for atoms, +molecules, and solids. Phys. Rev. B 2004, 70, 235119. +(19) Umrigar, C. J.; Toulouse, J.; Filippi, C.; Sorella, S.; Hennig, R. G. Alleviation of the +Fermion-Sign Problem by Optimization of Many-Body Wave Functions. Phys. 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The Journal of Chemical Physics +2022, 156, 054104. +S11 + diff --git a/-dFIT4oBgHgl3EQf9CsF/content/tmp_files/load_file.txt b/-dFIT4oBgHgl3EQf9CsF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4acf3d48ca3d785f177209adead8eafeeab49e02 --- /dev/null +++ b/-dFIT4oBgHgl3EQf9CsF/content/tmp_files/load_file.txt @@ -0,0 +1,1258 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf,len=1257 +page_content='A Quantum Monte Carlo study of the structural, energetic, and magnetic properties of two-dimensional (2D) H and T phase VSe2 Daniel Wines,1, a) Juha Tiihonen,2 Kayahan Saritas,3 Jaron Krogel,3 and Can Ataca4, b) 1)Materials Science and Engineering Division, National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899 2)Department of Physics, Nanoscience Center, University of Jyväskylä, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Box 35, Finland 3)Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 4)Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250 (Dated: 30 January 2023) Previous works have controversially claimed near-room temperature ferromagnetism in two-dimensional (2D) VSe2, with conflicting results throughout the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' These discrepancies in magnetic properties between both phases (T and H phase) of 2D VSe2 are most likely due to the structural parameters being coupled to the magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Specifically, both phases have a close lattice match and similar total energies, which makes it difficult to determine which phase is being observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In this study, we used a combination of density functional theory (DFT), highly accurate diffusion Monte Carlo (DMC) and a surrogate Hessian line-search optimization technique to resolve the previously reported discrepancy in structural parameters and relative phase stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' With DMC accuracy, we determined the freestanding geometry of both phases and constructed a phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Our findings demonstrate the successes of the DMC method coupled with the surrogate Hessian structural optimization technique when applied to a 2D magnetic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' INTRODUCTION One of the most promising two-dimensional (2D) mag- netic materials that has been extensively studied experimen- tally and theoretically is 2D VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Similar to other 2D tran- sition metal dichalcogenides (such as MoS2)1, VSe2 exists in two phases, the T (octahedral phase (1T)-centered honey- combs) phase which is metallic and the H (the trigonal pris- matic phase (2H)-hexagonal honeycombs, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 1) phase which is semiconducting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Several experimental and theoret- ical studies have controversially claimed near-room tempera- ture ferromagnetism in VSe2, with conflicting results through- out the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Density functional theory (DFT) along with classical Monte Carlo simulations have been used to obtain a) b) 1T-VSe2 2H-VSe2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Top and side view of the atomic structure of monolayer VSe2 in the a) 1T and b) 2H phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' a)Electronic mail: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='wines@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='gov b)Electronic mail: ataca@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='edu an estimate of the Curie temperature of H-VSe2 (291 K)2, but the model Ising Hamiltonian used did not take into account the magnetic anisotropy energies, which are essential for an accu- rate estimation of the Curie temperature of a 2D lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The Curie temperature of multilayered 2D H-VSe2 has been ex- perimentally measured to be 425 K, with the ferromagnetism softening as the thickness of the sample increases3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Addi- tionally, the experimental Curie temperature for monolayer T- VSe2 has ranged from 300 K to 470 K4,5 depending on which substrate is used (MoS2, graphite, SiO2-coated silicon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The experimental magnetization of T-VSe2 has also been met with controversy, with values of 15 µB and 5 µB (per formula unit) being reported from two separate studies4,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Insight has also been reported with regards to how the ferromagnetism is en- hanced with defects, molecular adsorption and the choice of substrate for VSe24,5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' A wide range of values have also been reported for the charge density wave (CDW) transition tem- perature for T-VSe2, ranging from 120 K to 350 K3,6,8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' These discrepancies in the electronic and magnetic proper- ties of either phase of 2D VSe2 arise from the structural pa- rameters of each phase being coupled closely to the magnetic and electronic properties and the external factors (substrates, defects) of the individual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' One example of this has been a reported discrepancy on which phase (T or H) is en- ergetically more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Both the T and H phases have a close lattice match and similar total energies, which makes it difficult to determine which phase is being observed experi- mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Recently, it has been reported experimentally that the T phase is favored for bulk VSe2, but with dimension- ality decrease, the H phase is favored3,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' It has also been reported that a T-to-H phase transition can be realized by ther- mal annealing11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This same structural phase transition has even been reported by applying a biaxial strain of ≈ 3 % (from calculated results)7,11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Researchers have proposed that this lattice strain can be induced by the mismatch that occurs from arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='11404v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='str-el] 26 Jan 2023 b C aC a bb2 putting 2D VSe2 on a substrate7,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' From a computational perspective, results for VSe2 depend heavily on which methodology is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In most cases, DFT with an empirical Hubbard correction (+U) for corre- lated electrons is used13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For example, if the U correction is applied for T and H-VSe2, the T phase is more energetically favorable, while if no U correction is applied, the H phase is more favorable14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition to the discrepancies in re- sults calculated with DFT+U, results between van der Waals (vdW) corrected functionals and hybrid functionals are also inconclusive14 in terms of predicting the relative phase stabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In order to alleviate the uncertainty in DFT methods, more sophisticated methods can be used such as Diffusion Monte Carlo (DMC)15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' DMC is a correlated, many-body electronic structure method that has demonstrated success for the elec- tronic and magnetic properties of a variety of bulk and 2D systems16–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This method has a weaker dependence on the starting density functional and U parameter and can success- fully achieve results with an accuracy beyond the DFT+U15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Due to the fact that T and H-VSe2 have structural parame- ters that are coupled to their electronic and magnetic proper- ties, it makes it difficult to produce conclusive results that rely solely on DFT or DFT+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For this reason, we employed our recently developed energy-based surrogate Hessian method for structural optimization with stochastic electronic structure theories (such as DMC)22 to obtain the geometry of T and H-VSe2 with DMC accuracy, resulting in high-accuracy bond lengths that resolve previous functional dependent structural discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' After obtaining an accurate geometry for both structures, we constructed a phase diagram between T and H- VSe2 using DMC calculated energies and obtained accurate magnetic properties of each structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The accurate estimates for lattice geometry, relative phase energy and the DMC phase diagram assist in clarifying previously inconclusive theoreti- cal and experimental results regarding T and H phase VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For full details of the computational methods used, see the Supporting Information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' As an initial starting point for our study, we performed benchmarking DFT and DFT+U calculations using a variety of density functionals (local density approximation (LDA)25, Perdew-Burke-Ernzerhof (PBE)26, and strongly constrained and appropriately normed (SCAN)27 meta-GGA functionals, see SI for more details) and the Vienna Ab initio Simulation Package (VASP) code for monolayer T-VSe2 and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The goal of these simulations were to assess how sensitive the relative energy between the T and H phase is with re- spect to functional and material geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Another goal of these simulations was to benchmark the structural parameters of each material with respect to several density functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' It is advantageous to perform these reference calculations with VASP and PAW pseudopotentials as a precursor to the more expensive DMC calculations due to the fact that they require a much smaller cutoff energy and are more cost effective for a large number of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' It is important to note that for all DFT and DMC simulations, we assumed a ferromag- netic ground state for both T and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Although recent reports have suggested that T-VSe2 could be experimentally paramagnetic3, we infer that this paramagnetism can be in- duced by magnetic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition, the modeling of paramagnetism with computational methods imposes a great challenge, which is why we focus on the freestanding ferro- magnetic ground states of both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' A more robust treat- ment of the magnetic structure can be explored in future work, but is beyond the scope of this work which primarily focuses on determining the geometric structure and phase stability of 2D T and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 we present a comprehensive look at the difference in total energy between T-VSe2 and H-VSe2, using several DFT functionals under different geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We performed these calculations for a variety of U values in three different ways: fully relaxing the structure at each value of U (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 a) ), fixing the lattice and atomic positions to the U = 0 eV relaxed geometry of that particular functional and cal- culating the static energy at each value of U (Fig 2 b)), fixing the lattice to the U = 0 eV relaxed geometry of that particular functional and relaxing just the atomic positions at each value of U (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 indicate that there is a significant disagreement between DFT functionals, U value used, and material geometries, with all three factors playing a significant role in the energy difference between T and H phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Specifically, regardless of relaxation method, all bare (no U correction) SCAN, PBE, and PBEsol functionals pre- dict H favorable, while bare LDA predicts T favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For all functionals, there is a critical value of U that reverses the relative phase stability, which is dependent on functional and relaxation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The SCAN functional with a U correction predicts T phase favorable, with larger energy differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2, the trends in the relative phase stability be- tween Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 b) and c) are nearly identical, but significantly vary from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This implies that the density functional is strongly coupled to material geometry, but the lattice constant change has more of an effect on phase stability than atomic positions and bond distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This is most prevalent for higher U values (> 2 eV), where the relaxed geometry changes more drastically with U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The interrelated nature of the material’s geometry, density functional, and value of U are reasons to seek out higher levels of theory beyond DFT/DFT+U such as DMC to accurately determine the optimal geometry and rela- tive energy between the phases of 2D VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The relaxed lattice constants, V-Se distances, and T - H en- ergies from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2 a) are presented in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3, along with additional VASP reference calculations performed with the vdW corrected functionals (PBE-D228, PBE-D329, SCAN+rvv1030).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The DMC computed parameters are also given for comparison in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3 (more discussion to follow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We observe a ≈ 7 % variability in lattice constant across the different methods for T-VSe2 and a ≈ 4 % variabil- ity in lattice constant across the different methods for H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Between both phases, we observe a ≈ 3 % variability in V-Se distance (dV−Se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Most strikingly, the energy difference be- tween the T and H phases (ET−H) drastically varies depend- ing on the material geometry and computational methodology, ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06 eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='. Due to the fact that a strain-induced phase transition has been reported be- tween T- and H-VSe27,11,12, we decided to perform additional VASP benchmarking calculations that involved the applica- 3 0 1 2 3 4 U (eV) 0 1 2 3 4 U (eV) 0 1 2 3 4 U (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 T - H Energy (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 T - H Energy (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 T - H Energy (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') PBE LDA SCAN PBESOL Full relaxation Fixed lattice/relaxed positions Fixed lattice/positions a) b) c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Relative (T - H) energy between T and H phase 2D VSe2 as a function of U parameter for several density functionals and methods of atomic relaxation: a) fully relaxing the structure, b) fixing the lattice and atomic positions to the U = 0 eV relaxed geometry of that particular functional and calculating the static energy, c) fixing the lattice to the U = 0 eV relaxed geometry of that particular functional and relaxing just the atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The dotted line indicates 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Tabulated results for lattice constant, V-Se distance, and relative energy (T - H) for both T and H phase 2D VSe2 for several computational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' DMC error bars (standard error about the mean) are included in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' T-VSe2 H-VSe2 Method a (Å) dV−Se (Å) a (Å) dV−Se (Å) ET−H (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') PBE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='336 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='489 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='333 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='045 PBE+U=2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='435 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='526 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='364 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='008 LDA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='228 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='438 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='229 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='026 LDA+U=2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='277 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='455 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='266 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='045 SCAN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='387 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='486 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='329 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='045 SCAN+U=2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='462 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='524 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='353 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='202 PBEsol 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='262 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='458 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='272 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='013 PBEsol+U=2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='323 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='483 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='301 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='025 PBE-D2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='323 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='484 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='318 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='010 PBE-D3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='315 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='485 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='319 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='042 SCAN+rvv10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='379 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='481 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='319 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='051 DMC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='414(12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='505(7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='335(8) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='503(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) tion of tensile and compressive strain for each monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We performed these calculations for PBE, SCAN, and LDA (with U = 0 eV and U = 2 eV), starting from the U = 0 eV geom- etry for each functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The resulting equations of state are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' As seen in the figure, the equation of state and resulting strain-induced phase transition is entirely dependent on the functional and U value, with no consistent trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The strong sensitivity of each monolayer with respect to geometry and functional are grounds for using a higher-order method such as DMC to obtain a statistically accurate estimate of the lattice parameters and relative energy between phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Prior to performing the DMC/line-search calculations, we op- timized our nodal surface (orbitals selected for DFT wave- function generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Since DMC has the zero-variance prop- erty, it means that as the trial wave function approaches the 4 ET-H - ET-H (DMC)(eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') dV-Se - dV-Se (DMC) (Å) dV-Se - dV-Se (DMC) (Å) a - aDMC (Å) a - aDMC (Å) T-VSe2 H-VSe2 a) b) c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' A summary of the deviation of the geometric properties rel- ative to the DMC calculated geometric properties for a) T-VSe2 and b) H-VSe2 and c) the the deviation of T - H energy relative to the DMC calculated T - H energy for a variety of DFT functionals (U = 2 eV), where the DMC error bar (standard error about the mean) is represented by the red bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' exact ground state, the statistical fluctuations in the energy reduce to zero15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Although there have been instances where various sophisticated methods have been used to optimize the nodal surface31–34, we employed the PBE+U approach, where the Hubbard (U) value was used as a variational parameter to optimize the nodal surface using DMC (similar to other successful DMC studies of magnetic materials16,20,21,24,35–37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We performed these calculations for both T and H-VSe2 (24 atom supercells), where we tuned the U value from (1 to 4) eV while creating the trial wavefunction and computed the DMC energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The results of these calculations are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S4, where we observe that U = 2 eV yields the lowest energy for both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' It is important to note that for the H phase, the DMC energies for U = 1 and U = 2 eV are statistically identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Based on this, we created the trial wavefunction us- ing PBE+U (U = 2 eV) for all subsequent DMC calculations within the surrogate Hessian line-search for both phases (all 52 DMC energy evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Since we obtained an optimal U value of 2 eV for both materials, we focused our DFT+U benchmarking efforts more on U = 2 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3, Fig 5, Table I, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Based on the DMC line-search results, we determined ac- curate bounds on the lattice parameter (a) and off-plane dis- placement of Se (z), within an error tolerance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='018 Å or lower for both parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This translates to within ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5% accuracy in a parameter set of a and dV−Se with 95% con- fidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Convergence (absence of significant displacements 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='60 V-Se distance (˚A) T-phase H-phase Fit eqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (T) Fit eqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (H) LS eqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (T) LS eqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (H) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='8 Lattice constant (˚A) −2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 −2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 −2458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 Energy/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (eV) PES (T) PES (H) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (Top) The phase diagram of 2D VSe2 in terms of a and dV−Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The phase boundary (solid line, black) is estimated from bicubic fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' To assure quality of the fits, the estimated ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 eV error contours (dotted line) and the minima from the fits (’x’) and the line-search (’o’) are all well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (Bottom) Slices of the PES at dV−Se = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='505 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' outside of the error tolerance) was achieved after two parallel line-search iterations for both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This convergence is il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S5, where the convergence of the parameter offsets of a and z and the convergence of the total energy per f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' are depicted for both T and H phase 2D VSe2 for the ini- tial DFT relaxed structure (1) and both subsequent iterations of DMC (2 - 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition, the final energy of both of the fitted structures (square points) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The final geometric parameters and relative phase energies determined with DMC are given in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For T-VSe2, we determined a lattice constant of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='414(12) Å and a V-Se distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='505(7) Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For H-VSe2, we determined a lattice constant of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='335(8) Å and a V-Se distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='503(5) Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The DMC finite-size extrapolated energy difference (T H) between the two phases was determined to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=', indicating that in freestanding form at the equilibrium geometry, H-VSe2 is favored over T-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' When comparing these DMC results to the other DFT functionals in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 3, it is clear that very few DFT functionals can repro- duce the DMC results for lattice constant, V-Se distance and relative energy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The SCAN functional comes the closest to reproducing all three simultaneous DMC values, but still falls slightly short for the V-Se distances of both phases and the lattice constant of T-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The fact that SCAN+U successfully predicts the structural properties (for H-VSe2) and the fact that SCAN+rvv10 produces an energy difference closest to the average DMC energy difference for both phases loosely implies that a simultaneous description of correlated magnetism and vdW interactions are both needed to correctly represent the physics of VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Experimental measurements of 5 the lattice constant and V-Se distance of freestanding mono- layer VSe2 are scarce and often times dependent on external factors such as the substrate (more discussion to follow) and sample preparation technique4,5,38,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' However, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='38 have recently reported a lattice constant of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='4 Å for thin films of T-VSe2 and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='39 have recently reported a lattice constant of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3 Å for epitaxially grown monolayer H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Both of these measured values are in excellent agreement with our DMC computed lattice constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Additionally, we deter- mined the near-equilibrium PES of both T and H 2D VSe2 with DMC accuracy, which are both depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The phase diagram presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 4 is based on similar fits to data, where the z displacement has been remapped to dV−Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This DMC phase diagram can directly be compared to the energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' strain DFT benchmarking calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S3, which emphasizes the need for an accurate representation of the phase boundary between the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The freestand- ing geometries of both T and H lie in the energetic H phase, but a slice of the phase diagram along dV−Se = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='505 Å in- dicates that the T phase becomes favorable over H at biaxial strain of a ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This implies that in freestanding form, once T-VSe2 is positively strained at least ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 %, T phase is favored over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Alternatively, if freestanding H-VSe2 is pos- itively strained at least ≈ 5 %, T phase is also favored over H This strain can easily be accomplished by placing mono- layer VSe2 on a substrate with significant lattice mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In fact, this type of mismatch has been reported to alter the mate- rial properties4,5,40,41, significantly contributing to the contro- versies of T and H-VSe2 (for energetic favorability, magnetic properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Whether or not the changes in energetic favorabil- ity or magnetic properties with respect to the substrate are due to lattice mismatch or more complicated interactions between the substrate and the monolayer remains to be answered and is beyond the scope of this work, which has focused solely on the freestanding forms of T and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' However, such cal- culations can be employed for future work using higher order methods such as DMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The proximity of the phase boundary between T and H phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 4) is emphasized by the small en- ergy difference between the two phases (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=', at the equilibrium geometry) between the two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Since this en- ergy difference is so close to room temperature (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='024 eV), this implies that a process such as thermal annealing can eas- ily induce a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In fact, recently it was demon- strated that a structural phase transition of multilayer VSe2 from T to H occurs through annealing at 650 K, along with a metal-insulator transition11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' To gain a deeper understanding of the magnetic properties of 2D T and H-VSe2, we extracted the spin densities (using a trial wavefunction at U = 2 eV and 24 atom supercell at the final equilibrium geometry predicted by DMC/line-search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The spin density isosurfaces of each phase (ρup - ρdown) are depicted in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5 a) and c) for T-VSe2 and H- VSe2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For both phases, we observe the V atoms are highly spin-polarized, while the Se atoms are slightly an- tiparallel with respect to the V atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For more calculation details regarding spin density, see SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We went on to plot the radial averaged spin densities as a function of distance, separately for V and Se for T and H-VSe2 (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5 a) - d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This allows us to view the spa- tial variations in spin density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Additionally, we benchmarked these V and Se radially averaged densities with PBE+U (U = 2 eV) using NC pseudopotentials at the equilibrium geom- etry (the calculation required to create the trial WF for the subsequent DMC runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5 a) and c), there is a substantial difference in the V spin density between DMC and PBE+U (U = 2 eV) for both T and H phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This same substantial difference between DMC and PBE+U also occurs for the total charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This discrepancy is most preva- lent near the radial density peak (peak of d orbital) and can be attributed to the fact that DFT functionals (even with the added Hubbard correction) tend to delocalize and unsuccess- fully capture 3d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This large discrepancy in the spin densities highlights the need for more accurate, many-body computational methodologies for correlated materials such as VSe2, where DFT fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In contrast, there is closer agreement between the DMC and PBE+U spin densities for Se in T and H-VSe2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5 b) and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Finally, we estimated the site-averaged atomic magnetic moments per V and Se for both T and H phase by integrating the DMC and PBE+U spin densities depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' At the DMC level, we estimated a magnetic moment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) µB for V and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='09(2) µB for Se in T-VSe2 and a magnetic mo- ment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02(1) µB for V and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='14(1) µB for Se in H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' At the PBE+U (U = 2 eV) level, we estimated a magnetic mo- ment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='30 µB for V and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='12 µB for Se in T-VSe2 and a magnetic moment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='40 µB for V and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 µB for Se in H- VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Consistent with the radial spin density results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5, we find that the DMC and PBE+U magnetic moments for Se are in much closer agreement than for V (for both T and H phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' By analyzing the spin densities and obtaining the on-site magnetic moments, we obtain a clear picture of how the magnetization of each ion depends on the computational method used, serving as a benchmark for the magnetic prop- erties of 2D VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In this work, we used a combination of DFT, DMC and a recently developed surrogate Hessian line-search optimiza- tion technique to resolve the previously reported discrepancy in structural parameters and relative phase stability of mono- layer T-VSe2 and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Using these methods, we deter- mined the lattice constant and V-Se distance (with DMC ac- curacy) to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='414(12) Å and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='505(7) Å respectively for T- VSe2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='335(8) Å and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='503(5) respectively for H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition, we find the relative energy between the phases (T H) to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' at the DMC level, indicating that in freestanding form, H-VSe2 is more energetically favorable than T-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We went on to obtain a phase diagram between T and H phase from the PES and determined that a phase tran- sition can be induced by strain or mechanisms such as ther- mal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Additionally, we benchmarked the magnetic properties such as spin density and on-site magnetic moment for both phases and find substantial differences between DMC and DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The results of this study demonstrate the successes of the DMC method coupled with the surrogate Hessian line- search structural optimization technique when applied to a 2D magnetic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The estimates for lattice constant, bond distance, relative phase energy and the extracted structural- 6 a) b) c) d) 4̟r2[ρup - ρdown ] (Ne/Å) 4̟r2[ρup - ρdown ] (Ne/Å) 4̟r2[ρup - ρdown ] (Ne/Å) 4̟r2[ρup - ρdown ] (Ne/Å) T-VSe2 (V) T-VSe2 (Se) H-VSe2 (V) H-VSe2 (Se) r (Å) r (Å) r (Å) r (Å) MV= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='30 µB MV= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06(2) µB MV= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='40 µB MV= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02(1) µB MSe= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='12 µB MSe= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='09(2) µB MSe= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 µB MSe= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='14(1) µB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The radially averaged spin density (ρup - ρdown) as a function of distance, calculated with DMC and PBE+U (U = 2 eV) of a) V and b) Se for 2D T-VSe2 and c) V and d) Se for 2D H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The inset of a) and c) depicts the spin isosurface density of T-VSe2 and H-VSe2 respectively, where the isosurface value was set to 6 x 10−3 e/Å3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The standard error about the mean for DMC is indicated by error bars in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' dependent phase diagram assist in clarifying previously incon- clusive theoretical and experimental results regarding T and H phase VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' CODE AVAILABILITY STATEMENT Software packages mentioned in the article can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='com/usnistgov/jarvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Please note that the use of commercial software (VASP) does not imply recommendation by the National Institute of Standards and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' COMPETING INTERESTS The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank the National Institute of Standards and Technology for funding, computational, and data- management resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Kamal Choudhary and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Francesca Tavazza for fruitful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We acknowledge grants of computer capacity from the Finnish Grid and Cloud Infrastructure (persistent identi- fier urn:nbn:fi:research-infras-2016072533).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' REFERENCES 1C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Zhu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Ibrahim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Gao, “Epitaxially grown monolayer vse2: an air-stable magnetic two-dimensional material with low work function at edges,” Science Bulletin 63, 419–425 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 40A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Karn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Chan, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Chazarin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Chen, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Pai, “Modification of monolayer 1t-vse2 by selective deposition of vanadium and tellurium,” AIP Advances 12, 035240 (2022), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1063/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0001402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 41D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' He, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Li, “Structural and transport properties of 1t-vse2 single crystal under high pressures,” Frontiers in Ma- terials 8 (2021), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='3389/fmats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='710849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Supporting Information: A Quantum Monte Carlo study of the structural, energetic, and magnetic properties of two-dimensional (2D) H and T phase VSe2 Daniel Wines,∗,† Juha Tiihonen,‡ Kayahan Saritas,¶ Jaron Krogel,§ and Can Ataca∗,∥ †Materials Science and Engineering Division, National Institute of Standards and Technology (NIST), Gaithersburg, MD 20899 ‡Department of Physics, Nanoscience Center, University of Jyv¨askyl¨a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Box 35, Finland ¶ Department of Applied Physics, Yale University, New Haven CT 06520 § Material Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 ∥Department of Physics, University of Maryland Baltimore County, Baltimore MD 21250 E-mail: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='wines@nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' ataca@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='edu Computational Methods Density functional theory (DFT) benchmarks for the T and H phase of 2D VSe2 were per- formed using the Vienna Ab initio Simulation Package (VASP) code with projector aug- mented wave (PAW) pseudopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='1,2 For these calculations, the local density approxi- S1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='11404v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='str-el] 26 Jan 2023 mation (LDA),3 Perdew-Burke-Ernzerhof (PBE),4 and strongly constrained and appropri- ately normed (SCAN)5 meta-GGA functionals were used with the added Hubbard correction (U)6 to treat the on-site Coulomb interaction of the 3d orbitals of the V atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' At least 20 ˚A of vacuum was given between periodic layers of VSe2 in the c-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition, we used a reciprocal grid of 24x24x1 and a kinetic energy cutoff of 400 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Our Quantum Monte Carlo (QMC) simulations used DFT-PBE to generate the trial wavefunction for fixed-node diffusion Monte Carlo (DMC) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The Quantum Espresso (QE)7 code was used for our DFT calculations to create the trial wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' This trial wavefunction was created for the ferromagnetic configuration of 2D VSe2 using different U values with the goal of variationally determining the optimal nodal surface (U value that yields the lowest total energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For V, we used norm-conserving (NC) RRKJ (OPT) pseudopotentials8 and for Se, we used NC Burkatzki-Fillipi-Dolg (BFD) pseudopo- tentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='9 After testing at the DFT level, a kinetic energy cutoff of 4,080 eV (300 Ry) and a k-grid of 6x6x1 was used (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S1 and S2) to generate trial wavefunctions for DMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' To accelerate the line-search method convergence for the metallic T phase, we increased the k-grid to 12x12x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' After the trial wavefunction was generated with DFT, Variational Monte Carlo (VMC) and DMC10,11 calculations were performed using the QMCPACK12,13 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The single de- terminant DFT wavefunction is converted into a many-body wavefunction by use of the Jas- trow parameters,14,15 which assist in modeling electron correlation with the goal of reducing the statistical uncertainty in DMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='16,17 Up to two-body Jastrow18 correlation functions were included, where the linear method19 was used to minimize the variance and energy of the VMC energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The cost function of the variance optimization is 100 % vari- ance minimization and the cost function of the energy optimization is split as 95 % energy minimization and 5 % variance minimization, which has been proven to reduce the uncer- tainty of DMC calculated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='16 The Nexus20 software suite was used to automate the DFT-VMC-DMC workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The locality approximation17 was used to evaluate the nonlocal S2 part of the pseudopotentials in DMC and an optimal timestep of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 Ha−1 was determined for DMC simulations due to the fact that it yielded an acceptance ratio greater than 99 % (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' A full summary of the VMC and DMC methods can be found in reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='10 The total charge density and spin density was extracted from our DMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The spin density is defined as the difference between the spin-up contribution to the total charge density and the spin-down contribution to the total charge density (ρup − ρdown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We used an extrapolation scheme on the DMC charge densities with the goal of eliminating the bias that occurs from using a mixed estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Since the charge density estimator does not commute with the fixed-node Hamiltonian, the DMC charge density was obtained from a mixed estimator between the pure fixed-node DMC and VMC densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The extrapolation formula takes the form:10 ρ1 = 2ρDMC − ρVMC + O[(Φ − ΨT)2] (1) where ρDMC and ρVMC are the DMC and VMC charge densities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Φ is the trial wavefunction from the DMC Hamiltonian and ΨT is the trial wavefunction from VMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In addition, we integrated the DFT+U and DMC spin densities up to a cutoff radius rcut (which we define as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='34 ˚A , due to the fact that it is approximately half of the V-Se bond distance in 2D T and H-VSe2) in order to estimate the site-averaged atomic magnetic moment per V and Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' To obtain these magnetic moments per atom (MA), we sum over the spherically interpolated spin densities: MA = 4π � rcut 0 r2ρs(r)dr ≈ 4π rcut/∆r � i=0 r2 i ρs(ri)∆r (2) where ri is the distance from the center of the atom to a given point on the grid and ∆r is the radial grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' To optimize the structural parameters of both T and H-VSe2 according to the DMC po- tential energy surface (PES), we use a surrogate Hessian accelerated optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='21 S3 In the method, we consider the PES around equilibrium as the second-order expansion in Wyckoff parameter space, p: E(p) = E0 + 1 2(p − p0)THp(p − p0), (3) where Hp is the Hessian, or the force-constant matrix, E0 is the energy minimum and p0 the energy-minimizing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Diagonalizing the parameter Hessian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=', Hp = U TΛU, forms an optimal basis for a conjugate line-search in the parameter space, namely the eigen- vectors U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The line-searches along U can be conducted in parallel, and ideally, they locate the minimum in just one parallel iteration within the quadratic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Here, we conduct the line-search according to a set of 2 parameters: the lattice constant a and the Wyckoff parameter z, which is the unsigned displacement of the Se atoms along the z axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' For reporting purposes, the line-search parameters a and z are remapped to a and d, where d is the V-Se distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' In the surrogate Hessian scheme, we obtain a cheap but relatively accurate Hessian from DFT, and use it to the inform line-search on the DMC PES, in particular by providing the search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' We also resample the DFT PES to predict fitting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Thus, we may minimize the computational cost of the DMC runs, while maintaining an error tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The surrogate DFT PES was based on QE with a 4,080 eV (300 Ry) cutoff using PBE with no DFT+U correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The DMC PES was based on DFT-PBE with U = 2 eV orbitals and finite-size extrapolation through supercell sizes of 9 and 24 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Each line-search was based on a 3rd order polynomial fit and set to contain 7 points, or displaced geometries, totaling 13 energy evaluations per phase, per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' However, alternative techniques, including (bi)polynomial fitting, were used in some parts to incorporate auxiliary DMC data and ensure convergence to the quadratic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Effectively, two parallel line-search iterations for both phases were carried out, and the convergence was claimed in the absence of significant displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S4 a) b) Figure S1: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b) H-VSe2 as a function of plane wave cutoff energy for the norm-conserving pseudopotentials calculated with DFT using the PBE functional at a k-point grid of 6x6x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The results show a converged cutoff energy of 4,080 eV (300 Ry) for both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' a) b) Figure S2: The total energy per atom of the unit cell (3 atoms) of 2D a) T-VSe2 and b) H-VSe2 as a function of K-point grid for the norm-conserving pseudopotentials calculated with DFT (PBE) at the converged cutoff energy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The results show a converged k-point grid of 6x6x1 (36) for both monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The number of K-points was scaled appro- priately to obtain the converged grid depending on the supercell size and shape for all DFT and DMC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S5 PBE (U = 0) PBE (U = 2) SCAN (U = 0) SCAN (U = 2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='04 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='96 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='92 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='88 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='84 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='95 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='90 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='85 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='80 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='65 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='75 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='70 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='90 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='85 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='80 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='75 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='70 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='65 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='10 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='90 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='80 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='70 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='60 Total Energy (eV) Total Energy (eV) Total Energy (eV) Total Energy (eV) Lattice Constant (Å) Lattice Constant (Å) Lattice Constant (Å) Lattice Constant (Å) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='04 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='90 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='80 Total Energy (eV) Lattice Constant (Å) LDA (U = 0) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='35 Lattice Constant (Å) LDA (U = 2) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='30 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='10 Total Energy (eV) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 T H Figure S3: Total energy as a function of lattice strain for T (blue) and H (red) phase 2D VSe2, calculated with various functionals and U values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Density functionals include LDA, PBE, and SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S6 Table S1: Tabulated results for the DMC timestep convergence of a 12 atom cell of 2D T-VSe2 and H-VSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The acceptance ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='99 indicates that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 Ha−1 is an appropriate timestep to use for all subsequent DMC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' T-VSe2 Timestep (Ha−1) DMC Total Energy (Ha) Error (Ha) Acceptance Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='005 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='999 H-VSe2 Timestep (Ha−1) DMC Total Energy (Ha) Error (Ha) Acceptance Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='005 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='002 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='999 1 2 3 4 U (eV) 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='30 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='20 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='15 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='10 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='05 2460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='95 Total Energy (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=') T H Figure S4: DMC calculated total energies of a 24-atom supercell (normalized per formula unit (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=')) of 2D T (blue) and H (red) phase VSe2 calculated as a function of the U parameter used to variationally determine the optimal trial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The DMC error bars represent the standard error about the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='01 a (˚A ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='010 z (˚A ) 1 2 3 Iteration 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='55 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='6 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='65 2459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='7 E/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' (eV ) T H Figure S5: The convergence of the a and z parameters and DMC energies per f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' for both T (blue) and H (red) phase of 2D VSe2 based on parallel line-search iterations along the DMC PES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The starting parameters (iteration 1) are from DFT, the zero offset is the mean over iterations 2 and 3, and dotted lines indicate the error tolerances for each case (95 % confidence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The DMC energies from respective equilibrium geometries are plotted with 1SEM (one standard error of the mean) uncertainties, with extra squares marking energies from the predicted minimum geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 Lattice constant (˚A ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='0 Z-offset (˚A ) LS eqm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Fit eqm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' LS # 0 LS # 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 Lattice constant (˚A ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content='10 Figure S6: Contour reconstructions of the DMC PESs (eV) of T (left) and H (right) phases of 2D VSe2 with respect to a and z parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The contours are based on bicubic fits to sparse data, and thus, subject to biases and statistical uncertainties not indicated in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The markers (’x’ and ’+’) indicate data points from two parallel line-search iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' References (1) Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Furthm¨uller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Efficient iterative schemes for ab initio total-energy calcu- lations using a plane-wave basis set.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' Surrogate Hessian accelerated structural op- timization for stochastic electronic structure theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' The Journal of Chemical Physics 2022, 156, 054104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} +page_content=' S11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFIT4oBgHgl3EQf9CsF/content/2301.11404v1.pdf'} diff --git a/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf b/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cd8ec2ae97d34eb009b9406ff6336b910650f30c --- /dev/null +++ b/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 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+1,894 @@ +AI based approach to Trailer Generation for Online +Educational Courses +1st Prakhar Mishra +IIIT +Bangalore, India +prakhar.mishra@iiitb.ac.in +2nd Chaitali Diwan +IIIT +Bangalore, India +chaitali.diwan@iiitb.ac.in +3rd Srinath Srinivasa +IIIT +Bangalore, India +sri@iiitb.ac.in +4th G. Srinivasaraghavan +IIIT +Bangalore, India +gsr@iiitb.ac.in +Abstract—In this paper, we propose an AI based approach +to Trailer Generation in the form of short videos for online +educational courses. Trailers give an overview of the course to +the learners and help them make an informed choice about the +courses they want to learn. It also helps to generate curiosity +and interest among the learners and encourages them to pursue +a course. While it is possible to manually generate the trailers, it +requires extensive human efforts and skills over a broad spectrum +of design, span selection, video editing, domain knowledge, etc., +thus making it time-consuming and expensive, especially in an +academic setting. The framework we propose in this work is a +template based method for video trailer generation, where most of +the textual content of the trailer is auto-generated and the trailer +video is automatically generated, by leveraging Machine Learning +and Natural Language Processing techniques. The proposed +trailer is in the form of a timeline consisting of various frag- +ments created by selecting, para-phrasing or generating content +using various proposed techniques. The fragments are further +enhanced by adding voice-over text, subtitles, animations, etc., to +create a holistic experience. Finally, we perform user evaluation +with 63 human evaluators for evaluating the trailers generated +by our system and the results obtained were encouraging. +Index Terms—Video Trailer Generation, Machine Learning, +Natural Language Processing +I. INTRODUCTION +The growth of the internet has significantly increased the +amount of free instructional content. These resources are +offered not only by big institutions but also by individual +content creators over various platforms such as Coursera, +Udemy, YouTube, etc. This increase in content production rate +has resulted in the creation of redundant courses and tutoring +videos for many topics over time. In spite of advantages +like on-demand accessibility, the abundance of options has +increased confusion and made it more challenging to select +a course that might be in line with learner’s interests. And +often, enrolling to a course that doesn’t meet the learner’s +expectations for a course’s curriculum and other aspects such +as expected level of commitment, the availability of support, +etc., causes the learner to lose motivation and eventually drop +the course. [1], [2]. +This problem can be tackled to a certain extent by presenting +a video trailer to the learners before the start of the course +(learning pathway) to help them quickly glance through the +pathway and get an overall idea of the course content and its +format [3]–[5]. +The idea of Trailers is not brand-new, and the film industry +has been using them extensively for a while. Trailers, in +context of movies are mostly about advertising. They notify +viewers about an upcoming movie while generating interest +among them. Often the effectiveness of a trailer affects the +perception of the movie, even before it is released publicly. +The course trailers serve a greater purpose in the educational +context than simple course promotion. Before beginning the +learning journey, they aid in helping learners set realistic +expectations for their learning outcomes and competency +mastery. +Concept of trailers might resemble with that of summariza- +tion [6]–[8], but apart from incorporating a few elements of +summarization like shortening and abstracting out information +from substantial sized input source, trailers are different in +terms of their motivation, purpose and the impact they cre- +ate on the end users. Unlike summaries, trailers need not +be complete in their coverage. Also, they are designed to +give glimpses of a few interesting segments of the narrative +without revealing the main plot or climax of the underlying +narrative [9]. Although there is no clear demarcation of what +a climax is in academic narratives, based on our analysis of +many academic course trailers in popular MOOCs (Massive +Open Online Courses) such as Udemy1 and Coursera2, we +see prevalence of a common pattern in trailer timelines. The +timeline starts with an introduction about the course and the +instructor and ends with a call-to-action (CTA) which offers +opportunity to the learners to take action or start the course. +In between, there are several elements and factoids about the +course and its contents, that aim to arouse viewer interest. +The current approach of generating trailers is manual, +cumbersome and time-consuming, it requires someone with +relevant skills like designing, video editing, and a subject +matter expert to help in curating the trailer content. Although, +there are software products like Apple iMovie3, Windows +Movie Maker4 and others that people can use for generating +trailers by performing basic editing like cuts, merging frames, +1https://www.udemy.com +2https://www.coursera.org +3https://www.apple.com/in/imovie +4https://www.microsoft.com/en-us/p/movie-maker-video-editor/ +9mvfq4lmz6c9 +arXiv:2301.03957v1 [cs.CL] 10 Jan 2023 + +Fig. 1. Trailer Structure +etc. Yet the content to be placed in the trailer has to be curated +entirely by a human expert. +In our work, we propose a semi-automatic template based +framework for generating video trailers for learning pathways, +which are a sequence of related educational documents of +various forms [10]–[12]. Here, most of the content that is +placed in the trailer is auto-generated with a scope for taking +inputs from the creator. The framework for trailer generation +consists of various essential trailer fragments arranged as +a timeline of the trailer. Each fragment is composed of a +sequence of frames that are coherent within themselves in +terms of the topical information they present. And inherently, +each frame is composed of various types of elements and their +properties like font size, text styling, image size, etc. Fig. 1 +shows the illustration for the same. +Once all the elements are generated and placed at their +respective positions within a frame of a trailer fragment, +a template is applied to it. The template consists of the +multi-modal experiences such as voice-over, subtitles, sounds, +animations, etc. It also determines the elements of the trailer +design such as number and ordering of fragments, frames and +elements. Fig. 2 shows the visual view of some of the frames +for one of the templates with it’s corresponding elements and +their positioning in the frames. +II. RELATED WORK +There are studies that discuss the idea, use and motivation +of having trailers for academic courses [3]–[5]. Also, there are +online educational platforms like Coursera and Udemy which +have course trailers. However, we could not find literature on +approaches to generating trailers for academic courses. Hence, +in the following paragraphs we discuss some of the pioneering +works of trailer generation in-general across other domains. +Trailer generation can also be seen as special case of larger +research interest of adding an element of surprise to the engage +receiver’s attention in midst of information overload [13], [14]. +Authors in [15]–[18] present an approach for automatic +trailer generation from movies as input. Hermes et al. [16] +create trailers for action movies by analyzing audio and +video signals present in movies and automatically detecting +features like faces, scene cuts, sound-volume, etc and use +ontology of the corresponding domain for producing trailers. +Irie et al. [17] propose a movie trailer generation method +which extracts symbols like title logo, main theme music and +selects impressive shot or speech segments based on clustering +methods and EM algorithm. Brachmann et al. [15] propose an +approach of generating action movie trailers using the concept +of trailer grammar, knowledge base and various ML techniques +for analyzing audio and images present in the movie. Smith +et al. [18] propose a system that understands and encodes +the patterns and emotions present in horror movies using +Convolution Neural Networks(CNN). +All the above methods use visual and audio cues to derive +the trailer frames, whereas we use raw text data and build +the necessary discriminative and generative Neural Network +models to create frames and its elements to be placed in the +trailer. +Hesham et al. in [19] explore the idea of creating movie +trailers from their subtitles. They first classify the movie by +genre, identify important keywords and then rank important +subtitles. The trailer is then generated by stacking the movie +time-frames corresponding to the important subtitles. Gaikwad +et al. in [20] propose a technique to create previews of movies +by utilizing subtitles and finding the most representative scenes +by matching them with the plot summaries. Chi et al. [21] +propose an approach to automatically create marketing-style +short videos for a given product page url by extracting +elements and their styles present in the product html page +under specified tags. +Unlike the aforementioned works which primarily focus +on generating trailers based on an extractive strategies, in +our work we develop various modules that comprehend in- +put document and generate content for the trailer either by +paraphrasing or by using Natural Language Generator based +model. +As far as we know, automatic/semi-automatic generation +of video trailers for learning pathways is unexplored. Our +proposed approach of video trailer generation using Machine +Learning, Natural Language Processing and Generation tech- +niques is also unique. +III. PROPOSED SYSTEM +We propose a framework for trailer generation consisting of +different trailer fragments that form a trailer timeline, genera- +tion of the trailer fragments and finally applying templates that +determine the look and feel of the trailer. Based on our analysis +of multiple trailers presented for various online courses offered +on various educational platforms like Coursera and Udemy, we +designed and structured our trailer elements, fragments and +overall flow of the trailer. +We propose a trailer timeline consisting of 7 trailer frag- +ments namely, Splash, Trailer Title, Author Details, Outline, +Meta-Information, Social Proof and finally the Call-to-Action. +Figure 3 shows the timeline of all the above-mentioned frag- +ments in the trailer. Each of these fragments define a specific +part of the trailer, their purpose and their importance in the +trailer. We define the fragments in detail in further part of +this section. As discussed earlier, fragments are composed of + +Trailer + Fragment 1 +Fragment 2 +Fragment t +Frame? 1 +Frame? 2 +Frame?2 f +Element 1 Element 1 +Element eFig. 2. Illustration of Frames +Fig. 3. Trailer Timeline +a sequence of frames and each frame is composed of various +types of elements and their properties. +The overall approach for trailer generation is illustrated in +Fig. 4. All the resources mapped to a learning pathway form +the input to our Fragment Data Generator (FDG) module. +Template constraints that define the elements, fragments and +frames also form the input to FDG. The FDG along with other +sources like creator’s input, any images or information from +the web or knowledge bases, etc., can be incorporated into +the frames or the fragments. Once the elements for all the +frames across all the fragments are generated, we pass it to +the composition module for adding in other important aspects +of the trailer like voice-over, subtitles, sounds, etc., to add to +its multi-modal experience. +A. Fragment Data Generation +Following are the proposed trailer fragments arranged in the +order of their appearance in the trailer timeline- +Splash Fragment: The idea of splash fragment is to +display any introductory information related to the trailer such +as credits, software logo, etc., mostly obtained from creator’s +input. This optional fragment could also be the last fragment +in the trailer depending on the creator’s preference. +Trailer Title Fragment: In this fragment we generate a +short yet representative title for the entire trailer, hence giving +a quick idea about the topic that summarizes the underlying +pathway or the set of resources. We apply Hierarchical Title +Generation model [22] over the resources mapped to the +learning pathway to get the list of trailer titles. We select a title +among them based on their Term Frequency. In case, none of +the titles are above a threshold, we fall back on the fact that the +first resource in the pathway is the proxy to the introductory +resource, and we generate the trailer title for it by applying +Single Document Title Generator [23], [24]. Figure 5 shows +the trailer title fragment generation flow. +Author Details Fragment: A quick introduction about +the author or the instructor of the learning pathway could +help the learners build an implicit connect and trust. Majority +of the elements in the Author Details Fragment like author +names, affiliations and author’s image are expected from the +creator while creating the trailer. Template constraints such +as addressing multiple authors with different frame elements, +handling and getting relevant images to be put in this fragment +etc are also obtained from trailer creator. These inputs and +template constraints are plugged in the automation system +to fill the overall author frame. Additionally, we crawl the +web to get relevant images, for example: we crawl the web +and get relevant affiliation images and place it in the desired +coordinates as defined by the template. Also for the templates +that allow for having only the frontal face of author, we make +use of an open-sourced face recognition model5 to crop the +face from the uploaded author image. In case no author image +is provided to the system by the creator, we place a dummy +caricatured relevant sized image. Similarly, we have defined +defaults for the features, frames and templates in case there is +no input from the trailer creator. For example, when multiple +authors exists, we display information w.r.t to the the first +author entered by the creator and treat him/her as the primary +instructor and rest all the authors are abstracted by placing +them under the “and others” category. +Outline Fragment: This fragment gives an idea about +the specific topics that would be covered in the learning +pathway. This could help in setting learners’ expectation in +terms of the topics covered and in deciding whether the content +aligns to his/her end goals. For this we use Single Document +5https://docs.opencv.org/3.4/db/d28/tutorial cascade classifier.html + +AddTextHere +Add Text Here +What you will learn .. +AddTextHere +Add Text Here +Add Text Here +1 +Add TextHere +② +3 +4 +Add Text +Add Text +Add Text +Here +Here +Here +Frame 1 +Frame 2 +Frame 3Meta- +Splash +Title +Author +Outline +Information +Social Proof +CTA +Introduction +Introduction +Overview of +Course +Building +Credits/Logo +Defining +to the Course +about the +topics covered +Structure and +Validation +Next Steps +Instructor +other details +and TrustFig. 4. Trailer Generation Flow +Fig. 5. Trailer Title Fragment Generation Flow +Title Generator [23], [24] model to generate titles for all the +resources in the learning pathway which represents the outline +of the learning pathway. +Every template under the outline fragment limits the number +of text elements to be listed on the screen with the aim to +balance aesthetics and information at the same time. To adhere +to this prior constraint, we design a multi-step process to select +diverse, yet impactful set of elements from a relatively larger +list of outlines generated in the previous step. Fig. 6 shows +the entire pipeline of Outline Text Selection. +Let K be the number of text elements that the frame requires +and N be the total number of resources we have as input +and let K < N. We start with all the resources (N) given +by the user and remove any instance of assessments and +short documents under the assumption that such documents +won’t hold much informational content. After this we remove +any occurrence of exact duplicates and near duplicates in the +remaining set and pass the remaining resource list to the title +generator system to generate title for every resource. +Post this, we fix the first and the last position of the outline +with the first and last resource title. We specifically do this +action because of the inherent ordering present in the input +resource as a part of learning pathway. Also intuitively, picking +first and last sets a bound over the topic space to be covered +under a particular course. +Finally on this reduced set, we divide the space into bins of +equal size from which we randomly sample one outline ele- +ment from each bin to remaining K−2 positions in the outline +list. We use threshold based Jaccard and cosine similarity for +filtering syntactic and semantic duplicates respectively. The +Jaccard similarity between any two documents is calculated as +an intersection over union of word sets for both documents. It +helps us get sense of syntactic similarity between documents. +For calculating cosine similarity, we vectorise our inputs using + +Learning Pathway +R1 +RR3 +R4 +R5 +Ra +R7 +Rs +Template +Constraints +Fragment Data Generator +OtherSources +Creator +Input +Fragment Data +Splash +Trailer Title +Outline +Meta- +Social Proof +Call-to- +Knowledge +AuthorDetails +Information +Action +Base +Fragment Data +Web +Composition +Voice-over +Text-to- +Frame +Subtitle +ixal +Speech +Generation +Generation +Duration +Generation +Trailer +Music +ArchiveTask: Generate +No +Hierarchical Title +Titles List +/Winning +No +KUserInput +Pick 1st Resource +Trailer Title +Generation +Title ? +from Input +Yes +Yes + Single Document Title Generator +Trailer Title +4pre-trained Sentence Transformers [25] and then measure the +semantic closeness between them using cosine similarity. +Algorithm 1 Duplicates Filter +1: resources = Array(1, 2, . . . , N − 1, N) +2: remaining resources = Array(1, N) +3: for i ← 2 to N − 1 do +4: +scores = Array() +5: +for r ← remaining resources do +6: +scores ← calculate similarity(i, r) +7: +end for +8: +if max(scores) < threshold then +9: +remaining resources ← i +10: +end if +11: end for +12: return remaining resources +Since every pathway is composed of different resources of +various properties like length, style, etc., having one threshold +that fits all does not work. Hence, our threshold is adaptable in +a way that guarantees at-least K items are selected post any of +the syntactic or semantic pruning steps. The threshold search +space is between 0 to 1 where for efficiency and tractability we +quantize it at 0.1. Then for each threshold we get remaining +resources as defined in Algorithm 1. Finally the threshold that +guarantees at-least K items and possibly reduces the input set +by maximum is chosen as the final threshold. +Meta-Information Fragment: The idea of having Meta- +Information Fragment is to inform learners about other impor- +tant aspects of the course like course structure, total reading +time, total number of resources, etc. We believe this would +help learners understand more about the learning pathway or +resources apart from just knowing the topics that would be +covered. Also, such information can be used by learners in +charting out their learning hours and estimating the efforts +it would take for the successful completion of the course. +Some of the elements that we generate automatically as part +of this fragment are: generating topical word clouds 6 bases on +word frequencies after pre-processing like stop-word removal, +estimating total reading time based on average reading speed +statistics and other pathway level derived statistics like total +resources, availability of discussion forum, etc. +Social Proof Fragment: Social Proof is one of the most +prominent ways of social influence and is based on the +heuristic that the users follow others similar to them when +uncertain [26]. We collect these statistics from the deployed +learning environments. This information is added to the video +trailer over time when different learners take this course and +the analytical data is available. +Call-to-Action Fragment: CTA is a marketing term which +is designed to push the audience in taking the desired actions. +It is an important aspect of any trailer because all of the +enthusiasm that is built in a learner while watching the trailer is +of no use if the learner is not clear on the next actionable [27], +6https://pypi.org/project/wordcloud/ +[28] item. In our system, we randomly select phrases from a +set of pre-defined list of potential key-phrases to be placed on +the screen at a pre-defined location under this fragment. Some +of the phrases we use are ‘Start your learning today’, ‘Let’s +get started’, ‘Are you ready?’, etc., along with the action that +will take the learner on the learning pathway. +B. Additional Elements +In this subsection, we discuss two other interesting elements +that we propose to be added to the trailers, namely, Definition +Extractor and Paraphraser. These are shown as suggestions +to the trailer creator and it’s up to the creator to include them +and decide their placement in the trailer. +Definition Extractor: Definitions are descriptive elements +that we believe can help in introduction of concepts. To +select the definition from the learning resource, we propose a +discriminative model that classifies a given piece of text into +Definition or Non-Definition class. For building the classifier +model, we use a dataset7 that contains positive and negative +definition candidates extracted from Wikipedia for various +topics. Our best performing model is a fine-tuned DistilBERT- +base-uncased8 model with a Definition class F1-score of 0.96 +and Non-Definition class F1-score of 0.97 on the test set. +Paraphraser: We believe that this is an useful utility that +can be used in the Outline and Trailer title fragments. This +gives the creator an ability to re-write concisely any substan- +tially larger textual content present in any frame. We use a +publicly available pre-trained model9 for this task which fine- +tunes a large sized T5 (Text-to-Text Transfer Transformer) [7] +model on a parallel corpus of sentence and it’s corresponding +paraphrase. +C. Video Composition +Video Composition module is responsible for stitching +together all the elements that need to be part of the trailer, such +as the Frame data, Voice-over text, Text-to-Speech (TTS), etc., +into a trailer video. Fig. 4 pictorially shows the overall flow +of the various components that are a part of the video compo- +sition. We use Python’s MoviePy library10 as our choice for +video editing and composition of the templates as it provides +us with all the necessary editing functions like inserting text, +concatenations and cuts, which we use to draft our templates. +After the frame-level data elements are in-place, the next +step is to generate voice-over text for each of the frames. +Voice-over text is defined as the spoken-text that the narrator +speaks while a frame is displayed on the screen. For this, +we select grammar from a pre-defined set of slot based +text grammars which we define per frame. The slots in the +grammar are nothing but the screen’s text elements. Finally, +once the Voice-over Text is generated for every frame, we +pass them through the IBM Watson’s Text-to-speech (TTS) +7http://nlp.uniroma1.it/wcl/ +8https://huggingface.co/distilbert-base-uncased +9https://github.com/ramsrigouthamg/Questgen.ai +10 https://zulko.github.io/moviepy + +Fig. 6. Outline Text Selection +API11 with relevant parameters such as voice-type, gender, +etc., by choosing from a list of speaker profiles to get the +audio files for every frame. Fig. 7 illustrates the flow from +grammar selection to voice generation for the Trailer Title +Fragment. We then derive the frame duration accordingly to +make sure that the visual and audio aspects of the frames are +in sync and minimize any kind of lag on either ends. Finally, +along with all the above details, we input template constraints +like positioning of elements, and styles, user preferences, and +some basic animations like fade-in and fade-out settings to +come up with the final trailer. +IV. EXPERIMENTS +In this section, we describe the dataset, evaluation strategy +and results obtained for the trailers generated by our proposed +system. +Dataset: Apart from the datasets which we have used for +training and evaluating specific modules that are responsible +for generating fragment relevant data. We created three dif- +ferent learning pathways for our experiments and evaluation +of the generated trailers. Each learning pathway differs with +each other in the number of resources and stylometry. Two of +the pathways are based on text book chapters with difference +in number of resources mapped, and one pathway is video +lectures. We tried to take different pathways to evaluate our +model’s flexibility on different types of learning pathways. +First one was created by sampling some chapters sequentially +from a freely available Machine Learning textbook [29]. For +second, we chose the speech-to-text transcription of a week’s +video lectures from an academic course on NLP. Our third +learning pathway is the entire ML textbook [29]12. All the +three corpus are analogous to learning pathways as they are all +semantically coherent, progressive and share the same global +topic. +Evaluation and Results: Trailers can be seen as gen- +erative tasks with an inherent notion of creativity. Here the +objective evaluation is not straight-forward because the ef- +fectiveness of a trailer is highly subjective and relies on the +human perception. However, we think that human evaluation +11https://cloud.ibm.com/catalog/services/speech-to-text +12Datasets can be found at: https://bit.ly/3ro3JLO +1 +The first trailer looked more catchy compared to the second +one. Being generated by an AI agent, both seems to be good. +2 +Looks amazing. Great work! +3 +You guys have truly done a remarkable work! +4 +Good job, keep it up! +5 +Great! +TABLE I +POSITIVE COMMENTS +1 +Maybe I just felt that he was conveying info too fast +2 +As of now, it sounds a bit robotic. Some improvements w.r.t +the TTS can help make it better. +3 +Slowing the video when the information that is being conveyed +is relatively dense would be helpful. For example, when going +through the list of topics, speaking slowly helps. When giving +instructor names, one can be fast. +4 +Also, if there’s some way to bring viewer’s attention to the +part of the slide that’s being mentioned, that would be better +where the content is not sequential. +5 +Remove the date from the frame. Add something about what +can they do once they learn the course(what type of problems +can they solve) +TABLE II +IMPROVEMENTS SUGGESTED BY USERS +on various trailers generated can give us a good perspective +on the quality of the trailers. We had 63 human evaluators +consisting of Engineering graduates, Post-graduates and PhD +students well versed in the technical domain that represent our +dataset. +We evaluate 6 trailers13 in total that were generated from 3 +different learning pathways as discussed above, i.e., 2 trailer +per learning pathway. These two trailers are based on two +templates T1, T2 created by us. Both the templates differ in +aesthetics and level-of-detail(LOD). The evaluation for each +trailer was done on a set of 8 questions on Likert-scale from +1 to 5, where 1 would mean very poor and 5 would mean very +good. +There were three separate groups of evaluators. Each group +was provided with 2 trailers based on 2 templates for the +same pathway. We thoughtfully perform this diversification to +simulate the cluster sampling procedure, since showing all 6 +trailers to the same evaluators would have created boredom, +resulting in not so accurate evaluation. +13Sample Trailers: https://bit.ly/3Hscie9 + +Filtering Less-informative Documents +Syntactic Filters over Document Text +All Input +Filter Assessments +Filter Short +Filter Exact +Filter Near +Generate Title for +Resources +Documents +Duplicates +Duplicates +every Resource +R = [1, 2, ... N-1, N] +Select 1st and pth +Randomly Select 1 +resource and add in +Outline Elements +resource from each +Outline then Divide +Filter Semantic +Filter Near +Filter Exact +bin +P-2 resources into K- +Duplicates +Duplicates +Duplicates +O = [1, 2, ... K-1, K] +2 equal spaced bins +Semantic Filter over Titles +R = [1, 2, .. P-1, P] +Syntactic Filters over Titles +where, K<=P<=NFig. 7. Flow of Grammar selection to Voice-over generation +We also encouraged the evaluators to give free comments +for the trailers they evaluated, as this would help us improve +our system in future iterations. Table. I and II lists down some +of the positive comments and improvements suggested by the +users. Fig. 8 shows some of the trailer fragments generated by +our proposed system14. +Following is the list of 8 questions that were asked to the +evaluator during the evaluation. The text in italics highlights +the broader aspect of the evaluation feature. +Q1. Did you find the trailer to be self-contained? +Q2. How were the fonts and styles used in the trailer in +terms of readability? +Q3. How did you find the length and pace of the trailer? +Q4. As a user, how impressed are you with this trailer +overall? +Q5. Could this trailer evoke interest in someone taking +this course? (Ignoring any prior inclination to the topic) +Q6. How was the average duration of each frame? +Q7. Based on the trailer you just saw, do you think you +have a good impression of the course now? +Q8. How did you find the sync between the audio and +visuals you saw? +As can be seen in Fig. 9, the scores obtained for each of the +survey questions are good and far above the average(score of +3) for almost all the trailers generated by our approach. Also, +in our study, we found both the templates performed equally +good. However, for Q5, the average scores is relatively lower +compared to other questions. On digging deeper we found +some of the comments of total 24 comments we received +mentioned about the difficulty of the course for not getting +interested in the course. This could mean that this question +(Q5) is more subjective. +14Detailed +demo +walk-through: +https://www.youtube.com/watch?v= +06VVuAlFhTk +V. CONCLUSIONS AND FUTURE WORK +In this paper, we presented a novel framework for au- +tomatically generating video trailers for a learning pathway +using ML and NLP techniques. We validated our trailers on +multiple corpus of varied granularity with human evaluation +and the results obtained were encouraging. This approach can +be adapted to different domains given enough data to train +the models involved in the entire process. We believe that +this approach can lay foundation to building more advanced +versions of trailer. +In future, we plan to improve the existing system by +incorporating suggestions obtained in the user evaluation and +adding more interesting themes like automatically detecting +learning outcomes given the resources. We also intend to create +an interactive dashboard to take inputs from the creator and +allow the creator to make edits to the auto-generated content. +ACKNOWLEDGMENT +We thank the Center of Excellence on Cognitive Com- +puting, funded by Mphasis F1 Foundation for funding this +research. We also thank Dr. Prasad Ram and Gooru team +(https://gooru.org) for the topical discussions and encourage- +ment. +REFERENCES +[1] O. Simpson, “Student retention in distance education: are we failing +our students?” Open Learning: The Journal of Open, Distance and e- +Learning, vol. 28, no. 2, pp. 105–119, 2013. +[2] M. Hartnett, A. St George, and J. 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Srinivasaraghavan, “Automatic +title generation for learning resources and pathways with pre-trained +transformer models,” International Journal of Semantic Computing, +vol. 15, no. 04, pp. 487–510, 2021. +[23] ——, “Automatic title generation for text with pre-trained transformer +language model,” in 2021 IEEE 15th International Conference on +Semantic Computing (ICSC). +IEEE, 2021, pp. 17–24. +[24] J. Tan, X. Wan, and J. Xiao, “From neural sentence summarization to +headline generation: A coarse-to-fine approach.” in IJCAI, vol. 17, 2017, +pp. 4109–4115. +[25] N. Reimers and I. Gurevych, “Sentence-bert: Sentence embeddings using +siamese bert-networks,” arXiv preprint arXiv:1908.10084, 2019. +[26] R. B. Cialdini and L. James, Influence: Science and practice. +Pearson +education Boston, MA, 2009, vol. 4. +[27] “Call-to-action (cta),” https://bit.ly/3DDUBp4, accessed: 2021-12-08. +[28] “3 reasons a call to action is important,” https://bit.ly/33c7WbO, ac- +cessed: 2021-12-08. +[29] J. Gareth, W. Daniela, H. Trevor, and T. Robert, An introduction to +statistical learning: with applications in R. +Spinger, 2013. + +What you will learn ... +October 6, 20: +October 6, 2021 +Readtimo +~8hr +model +Resources +oneet al +Let's get started! +Regularization in +Convolutional Neural +function +Deep Learning +Networks +1 +2 +3 +4 +6 +Text +layergradient +Regular +Resources +neuralnetwork +Assessments +Are you +rward Networks +Optimization Techniques +Recurrent Noural +ready? +Networks +Training +Active +Discussion +Forum +Ihis course, In this specaly curated course +ol the curriculum you wll go througn +startyourjourney +Outline Frame +Meta-Information Frame +CTA Frame5 +4 +Likert Value +3 +2 +1 +0 +Q1 +Q2 +Q3 +Q4 +Q5 +Q6 +Q7 +Q8 +SurveyQuestion +P1-T1 +P1-T2 +P2-T1 +P2-T2 +P3-T1 +P3-T2 \ No newline at end of file diff --git a/0tE2T4oBgHgl3EQfiQfL/content/tmp_files/load_file.txt b/0tE2T4oBgHgl3EQfiQfL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13dd36d45531237a5da3122a67a86e8a09097fd6 --- /dev/null +++ b/0tE2T4oBgHgl3EQfiQfL/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf,len=564 +page_content='AI based approach to Trailer Generation for Online Educational Courses 1st Prakhar Mishra IIIT Bangalore, India prakhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='mishra@iiitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='in 2nd Chaitali Diwan IIIT Bangalore, India chaitali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='diwan@iiitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='in 3rd Srinath Srinivasa IIIT Bangalore, India sri@iiitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='in 4th G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Srinivasaraghavan IIIT Bangalore, India gsr@iiitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='in Abstract—In this paper, we propose an AI based approach to Trailer Generation in the form of short videos for online educational courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailers give an overview of the course to the learners and help them make an informed choice about the courses they want to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' It also helps to generate curiosity and interest among the learners and encourages them to pursue a course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' While it is possible to manually generate the trailers, it requires extensive human efforts and skills over a broad spectrum of design, span selection, video editing, domain knowledge, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', thus making it time-consuming and expensive, especially in an academic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The framework we propose in this work is a template based method for video trailer generation, where most of the textual content of the trailer is auto-generated and the trailer video is automatically generated, by leveraging Machine Learning and Natural Language Processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The proposed trailer is in the form of a timeline consisting of various frag- ments created by selecting, para-phrasing or generating content using various proposed techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The fragments are further enhanced by adding voice-over text, subtitles, animations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', to create a holistic experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Finally, we perform user evaluation with 63 human evaluators for evaluating the trailers generated by our system and the results obtained were encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Index Terms—Video Trailer Generation, Machine Learning, Natural Language Processing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' INTRODUCTION The growth of the internet has significantly increased the amount of free instructional content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' These resources are offered not only by big institutions but also by individual content creators over various platforms such as Coursera, Udemy, YouTube, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This increase in content production rate has resulted in the creation of redundant courses and tutoring videos for many topics over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In spite of advantages like on-demand accessibility, the abundance of options has increased confusion and made it more challenging to select a course that might be in line with learner’s interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' And often, enrolling to a course that doesn’t meet the learner’s expectations for a course’s curriculum and other aspects such as expected level of commitment, the availability of support, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', causes the learner to lose motivation and eventually drop the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This problem can be tackled to a certain extent by presenting a video trailer to the learners before the start of the course (learning pathway) to help them quickly glance through the pathway and get an overall idea of the course content and its format [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The idea of Trailers is not brand-new, and the film industry has been using them extensively for a while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailers, in context of movies are mostly about advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' They notify viewers about an upcoming movie while generating interest among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Often the effectiveness of a trailer affects the perception of the movie, even before it is released publicly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The course trailers serve a greater purpose in the educational context than simple course promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Before beginning the learning journey, they aid in helping learners set realistic expectations for their learning outcomes and competency mastery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Concept of trailers might resemble with that of summariza- tion [6]–[8], but apart from incorporating a few elements of summarization like shortening and abstracting out information from substantial sized input source, trailers are different in terms of their motivation, purpose and the impact they cre- ate on the end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Unlike summaries, trailers need not be complete in their coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also, they are designed to give glimpses of a few interesting segments of the narrative without revealing the main plot or climax of the underlying narrative [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Although there is no clear demarcation of what a climax is in academic narratives, based on our analysis of many academic course trailers in popular MOOCs (Massive Open Online Courses) such as Udemy1 and Coursera2, we see prevalence of a common pattern in trailer timelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The timeline starts with an introduction about the course and the instructor and ends with a call-to-action (CTA) which offers opportunity to the learners to take action or start the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In between, there are several elements and factoids about the course and its contents, that aim to arouse viewer interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The current approach of generating trailers is manual, cumbersome and time-consuming, it requires someone with relevant skills like designing, video editing, and a subject matter expert to help in curating the trailer content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Although, there are software products like Apple iMovie3, Windows Movie Maker4 and others that people can use for generating trailers by performing basic editing like cuts, merging frames, 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='udemy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='coursera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='org 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com/in/imovie 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com/en-us/p/movie-maker-video-editor/ 9mvfq4lmz6c9 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='03957v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='CL] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer Structure etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Yet the content to be placed in the trailer has to be curated entirely by a human expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In our work, we propose a semi-automatic template based framework for generating video trailers for learning pathways, which are a sequence of related educational documents of various forms [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Here, most of the content that is placed in the trailer is auto-generated with a scope for taking inputs from the creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The framework for trailer generation consists of various essential trailer fragments arranged as a timeline of the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Each fragment is composed of a sequence of frames that are coherent within themselves in terms of the topical information they present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' And inherently, each frame is composed of various types of elements and their properties like font size, text styling, image size, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 1 shows the illustration for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Once all the elements are generated and placed at their respective positions within a frame of a trailer fragment, a template is applied to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The template consists of the multi-modal experiences such as voice-over, subtitles, sounds, animations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' It also determines the elements of the trailer design such as number and ordering of fragments, frames and elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 2 shows the visual view of some of the frames for one of the templates with it’s corresponding elements and their positioning in the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' RELATED WORK There are studies that discuss the idea, use and motivation of having trailers for academic courses [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also, there are online educational platforms like Coursera and Udemy which have course trailers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' However, we could not find literature on approaches to generating trailers for academic courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Hence, in the following paragraphs we discuss some of the pioneering works of trailer generation in-general across other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer generation can also be seen as special case of larger research interest of adding an element of surprise to the engage receiver’s attention in midst of information overload [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Authors in [15]–[18] present an approach for automatic trailer generation from movies as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Hermes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [16] create trailers for action movies by analyzing audio and video signals present in movies and automatically detecting features like faces, scene cuts, sound-volume, etc and use ontology of the corresponding domain for producing trailers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Irie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [17] propose a movie trailer generation method which extracts symbols like title logo, main theme music and selects impressive shot or speech segments based on clustering methods and EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Brachmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [15] propose an approach of generating action movie trailers using the concept of trailer grammar, knowledge base and various ML techniques for analyzing audio and images present in the movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [18] propose a system that understands and encodes the patterns and emotions present in horror movies using Convolution Neural Networks(CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' All the above methods use visual and audio cues to derive the trailer frames, whereas we use raw text data and build the necessary discriminative and generative Neural Network models to create frames and its elements to be placed in the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Hesham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' in [19] explore the idea of creating movie trailers from their subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' They first classify the movie by genre, identify important keywords and then rank important subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The trailer is then generated by stacking the movie time-frames corresponding to the important subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Gaikwad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' in [20] propose a technique to create previews of movies by utilizing subtitles and finding the most representative scenes by matching them with the plot summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' [21] propose an approach to automatically create marketing-style short videos for a given product page url by extracting elements and their styles present in the product html page under specified tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Unlike the aforementioned works which primarily focus on generating trailers based on an extractive strategies, in our work we develop various modules that comprehend in- put document and generate content for the trailer either by paraphrasing or by using Natural Language Generator based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' As far as we know, automatic/semi-automatic generation of video trailers for learning pathways is unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Our proposed approach of video trailer generation using Machine Learning, Natural Language Processing and Generation tech- niques is also unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' PROPOSED SYSTEM We propose a framework for trailer generation consisting of different trailer fragments that form a trailer timeline, genera- tion of the trailer fragments and finally applying templates that determine the look and feel of the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Based on our analysis of multiple trailers presented for various online courses offered on various educational platforms like Coursera and Udemy, we designed and structured our trailer elements, fragments and overall flow of the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We propose a trailer timeline consisting of 7 trailer frag- ments namely, Splash, Trailer Title, Author Details, Outline, Meta-Information, Social Proof and finally the Call-to-Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Figure 3 shows the timeline of all the above-mentioned frag- ments in the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Each of these fragments define a specific part of the trailer, their purpose and their importance in the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We define the fragments in detail in further part of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' As discussed earlier, fragments are composed of Trailer Fragment 1 Fragment 2 Fragment t Frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 1 Frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 2 Frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='2 f Element 1 Element 1 Element eFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Illustration of Frames Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer Timeline a sequence of frames and each frame is composed of various types of elements and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The overall approach for trailer generation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' All the resources mapped to a learning pathway form the input to our Fragment Data Generator (FDG) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Template constraints that define the elements, fragments and frames also form the input to FDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The FDG along with other sources like creator’s input, any images or information from the web or knowledge bases, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', can be incorporated into the frames or the fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Once the elements for all the frames across all the fragments are generated, we pass it to the composition module for adding in other important aspects of the trailer like voice-over, subtitles, sounds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', to add to its multi-modal experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fragment Data Generation Following are the proposed trailer fragments arranged in the order of their appearance in the trailer timeline- Splash Fragment: The idea of splash fragment is to display any introductory information related to the trailer such as credits, software logo, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', mostly obtained from creator’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This optional fragment could also be the last fragment in the trailer depending on the creator’s preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer Title Fragment: In this fragment we generate a short yet representative title for the entire trailer, hence giving a quick idea about the topic that summarizes the underlying pathway or the set of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We apply Hierarchical Title Generation model [22] over the resources mapped to the learning pathway to get the list of trailer titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We select a title among them based on their Term Frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In case, none of the titles are above a threshold, we fall back on the fact that the first resource in the pathway is the proxy to the introductory resource, and we generate the trailer title for it by applying Single Document Title Generator [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Figure 5 shows the trailer title fragment generation flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Author Details Fragment: A quick introduction about the author or the instructor of the learning pathway could help the learners build an implicit connect and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Majority of the elements in the Author Details Fragment like author names, affiliations and author’s image are expected from the creator while creating the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Template constraints such as addressing multiple authors with different frame elements, handling and getting relevant images to be put in this fragment etc are also obtained from trailer creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' These inputs and template constraints are plugged in the automation system to fill the overall author frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Additionally, we crawl the web to get relevant images, for example: we crawl the web and get relevant affiliation images and place it in the desired coordinates as defined by the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also for the templates that allow for having only the frontal face of author, we make use of an open-sourced face recognition model5 to crop the face from the uploaded author image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In case no author image is provided to the system by the creator, we place a dummy caricatured relevant sized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Similarly, we have defined defaults for the features, frames and templates in case there is no input from the trailer creator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For example, when multiple authors exists, we display information w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='t to the the first author entered by the creator and treat him/her as the primary instructor and rest all the authors are abstracted by placing them under the “and others” category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Outline Fragment: This fragment gives an idea about the specific topics that would be covered in the learning pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This could help in setting learners’ expectation in terms of the topics covered and in deciding whether the content aligns to his/her end goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For this we use Single Document 5https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='opencv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='org/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='4/db/d28/tutorial cascade classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='html AddTextHere Add Text Here What you will learn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='. AddTextHere Add Text Here Add Text Here 1 Add TextHere ② 3 4 Add Text Add Text Add Text Here Here Here Frame 1 Frame 2 Frame 3Meta- Splash Title Author Outline Information Social Proof CTA Introduction Introduction Overview of Course Building Credits/Logo Defining to the Course about the topics covered Structure and Validation Next Steps Instructor other details and TrustFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer Generation Flow Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Trailer Title Fragment Generation Flow Title Generator [23], [24] model to generate titles for all the resources in the learning pathway which represents the outline of the learning pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Every template under the outline fragment limits the number of text elements to be listed on the screen with the aim to balance aesthetics and information at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' To adhere to this prior constraint, we design a multi-step process to select diverse, yet impactful set of elements from a relatively larger list of outlines generated in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 6 shows the entire pipeline of Outline Text Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Let K be the number of text elements that the frame requires and N be the total number of resources we have as input and let K < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We start with all the resources (N) given by the user and remove any instance of assessments and short documents under the assumption that such documents won’t hold much informational content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' After this we remove any occurrence of exact duplicates and near duplicates in the remaining set and pass the remaining resource list to the title generator system to generate title for every resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Post this, we fix the first and the last position of the outline with the first and last resource title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We specifically do this action because of the inherent ordering present in the input resource as a part of learning pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also intuitively, picking first and last sets a bound over the topic space to be covered under a particular course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Finally on this reduced set, we divide the space into bins of equal size from which we randomly sample one outline ele- ment from each bin to remaining K−2 positions in the outline list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We use threshold based Jaccard and cosine similarity for filtering syntactic and semantic duplicates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The Jaccard similarity between any two documents is calculated as an intersection over union of word sets for both documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' It helps us get sense of syntactic similarity between documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For calculating cosine similarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' we vectorise our inputs using ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Learning Pathway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='R1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='RR3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='R4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='R5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Ra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='R7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Rs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Fragment Data Generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='OtherSources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Creator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Fragment Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Splash ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Trailer Title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Outline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Meta- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Social Proof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Call-to- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='AuthorDetails ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Fragment Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Composition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Voice-over ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Text-to- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Frame ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Subtitle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ixal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Speech ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Duration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Trailer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ArchiveTask: Generate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Hierarchical Title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Titles List ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='/Winning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='KUserInput ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Pick 1st Resource ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Trailer Title ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='Title ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' from Input Yes Yes Single Document Title Generator Trailer Title 4pre-trained Sentence Transformers [25] and then measure the semantic closeness between them using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Algorithm 1 Duplicates Filter 1: resources = Array(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' , N − 1, N) 2: remaining resources = Array(1, N) 3: for i ← 2 to N − 1 do 4: scores = Array() 5: for r ← remaining resources do 6: scores ← calculate similarity(i, r) 7: end for 8: if max(scores) < threshold then 9: remaining resources ← i 10: end if 11: end for 12: return remaining resources Since every pathway is composed of different resources of various properties like length, style, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', having one threshold that fits all does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Hence, our threshold is adaptable in a way that guarantees at-least K items are selected post any of the syntactic or semantic pruning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The threshold search space is between 0 to 1 where for efficiency and tractability we quantize it at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Then for each threshold we get remaining resources as defined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Finally the threshold that guarantees at-least K items and possibly reduces the input set by maximum is chosen as the final threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Meta-Information Fragment: The idea of having Meta- Information Fragment is to inform learners about other impor- tant aspects of the course like course structure, total reading time, total number of resources, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We believe this would help learners understand more about the learning pathway or resources apart from just knowing the topics that would be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also, such information can be used by learners in charting out their learning hours and estimating the efforts it would take for the successful completion of the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Some of the elements that we generate automatically as part of this fragment are: generating topical word clouds 6 bases on word frequencies after pre-processing like stop-word removal, estimating total reading time based on average reading speed statistics and other pathway level derived statistics like total resources, availability of discussion forum, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Social Proof Fragment: Social Proof is one of the most prominent ways of social influence and is based on the heuristic that the users follow others similar to them when uncertain [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We collect these statistics from the deployed learning environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This information is added to the video trailer over time when different learners take this course and the analytical data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Call-to-Action Fragment: CTA is a marketing term which is designed to push the audience in taking the desired actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' It is an important aspect of any trailer because all of the enthusiasm that is built in a learner while watching the trailer is of no use if the learner is not clear on the next actionable [27], 6https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='org/project/wordcloud/ [28] item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In our system, we randomly select phrases from a set of pre-defined list of potential key-phrases to be placed on the screen at a pre-defined location under this fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Some of the phrases we use are ‘Start your learning today’, ‘Let’s get started’, ‘Are you ready?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', along with the action that will take the learner on the learning pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Additional Elements In this subsection, we discuss two other interesting elements that we propose to be added to the trailers, namely, Definition Extractor and Paraphraser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' These are shown as suggestions to the trailer creator and it’s up to the creator to include them and decide their placement in the trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Definition Extractor: Definitions are descriptive elements that we believe can help in introduction of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' To select the definition from the learning resource, we propose a discriminative model that classifies a given piece of text into Definition or Non-Definition class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For building the classifier model, we use a dataset7 that contains positive and negative definition candidates extracted from Wikipedia for various topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Our best performing model is a fine-tuned DistilBERT- base-uncased8 model with a Definition class F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='96 and Non-Definition class F1-score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='97 on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Paraphraser: We believe that this is an useful utility that can be used in the Outline and Trailer title fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This gives the creator an ability to re-write concisely any substan- tially larger textual content present in any frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We use a publicly available pre-trained model9 for this task which fine- tunes a large sized T5 (Text-to-Text Transfer Transformer) [7] model on a parallel corpus of sentence and it’s corresponding paraphrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Video Composition Video Composition module is responsible for stitching together all the elements that need to be part of the trailer, such as the Frame data, Voice-over text, Text-to-Speech (TTS), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', into a trailer video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 4 pictorially shows the overall flow of the various components that are a part of the video compo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We use Python’s MoviePy library10 as our choice for video editing and composition of the templates as it provides us with all the necessary editing functions like inserting text, concatenations and cuts, which we use to draft our templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' After the frame-level data elements are in-place, the next step is to generate voice-over text for each of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Voice-over text is defined as the spoken-text that the narrator speaks while a frame is displayed on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For this, we select grammar from a pre-defined set of slot based text grammars which we define per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The slots in the grammar are nothing but the screen’s text elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Finally, once the Voice-over Text is generated for every frame, we pass them through the IBM Watson’s Text-to-speech (TTS) 7http://nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='it/wcl/ 8https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='co/distilbert-base-uncased 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com/ramsrigouthamg/Questgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ai 10 https://zulko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='io/moviepy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Outline Text Selection API11 with relevant parameters such as voice-type, gender, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', by choosing from a list of speaker profiles to get the audio files for every frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 7 illustrates the flow from grammar selection to voice generation for the Trailer Title Fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We then derive the frame duration accordingly to make sure that the visual and audio aspects of the frames are in sync and minimize any kind of lag on either ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Finally, along with all the above details, we input template constraints like positioning of elements, and styles, user preferences, and some basic animations like fade-in and fade-out settings to come up with the final trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' EXPERIMENTS In this section, we describe the dataset, evaluation strategy and results obtained for the trailers generated by our proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Dataset: Apart from the datasets which we have used for training and evaluating specific modules that are responsible for generating fragment relevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We created three dif- ferent learning pathways for our experiments and evaluation of the generated trailers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Each learning pathway differs with each other in the number of resources and stylometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Two of the pathways are based on text book chapters with difference in number of resources mapped, and one pathway is video lectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We tried to take different pathways to evaluate our model’s flexibility on different types of learning pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' First one was created by sampling some chapters sequentially from a freely available Machine Learning textbook [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For second, we chose the speech-to-text transcription of a week’s video lectures from an academic course on NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Our third learning pathway is the entire ML textbook [29]12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' All the three corpus are analogous to learning pathways as they are all semantically coherent, progressive and share the same global topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Evaluation and Results: Trailers can be seen as gen- erative tasks with an inherent notion of creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Here the objective evaluation is not straight-forward because the ef- fectiveness of a trailer is highly subjective and relies on the human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' However, we think that human evaluation 11https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com/catalog/services/speech-to-text 12Datasets can be found at: https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ly/3ro3JLO 1 The first trailer looked more catchy compared to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Being generated by an AI agent, both seems to be good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 2 Looks amazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Great work!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 3 You guys have truly done a remarkable work!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 4 Good job, keep it up!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 5 Great!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' TABLE I POSITIVE COMMENTS 1 Maybe I just felt that he was conveying info too fast 2 As of now, it sounds a bit robotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Some improvements w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='t the TTS can help make it better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 3 Slowing the video when the information that is being conveyed is relatively dense would be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' For example, when going through the list of topics, speaking slowly helps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' When giving instructor names, one can be fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 4 Also, if there’s some way to bring viewer’s attention to the part of the slide that’s being mentioned, that would be better where the content is not sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 5 Remove the date from the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Add something about what can they do once they learn the course(what type of problems can they solve) TABLE II IMPROVEMENTS SUGGESTED BY USERS on various trailers generated can give us a good perspective on the quality of the trailers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We had 63 human evaluators consisting of Engineering graduates, Post-graduates and PhD students well versed in the technical domain that represent our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We evaluate 6 trailers13 in total that were generated from 3 different learning pathways as discussed above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=', 2 trailer per learning pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' These two trailers are based on two templates T1, T2 created by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Both the templates differ in aesthetics and level-of-detail(LOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The evaluation for each trailer was done on a set of 8 questions on Likert-scale from 1 to 5, where 1 would mean very poor and 5 would mean very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' There were three separate groups of evaluators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Each group was provided with 2 trailers based on 2 templates for the same pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We thoughtfully perform this diversification to simulate the cluster sampling procedure, since showing all 6 trailers to the same evaluators would have created boredom, resulting in not so accurate evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 13Sample Trailers: https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='ly/3Hscie9 Filtering Less-informative Documents Syntactic Filters over Document Text All Input Filter Assessments Filter Short Filter Exact Filter Near Generate Title for Resources Documents Duplicates Duplicates every Resource R = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' N-1, N] Select 1st and pth Randomly Select 1 resource and add in Outline Elements resource from each Outline then Divide Filter Semantic Filter Near Filter Exact bin P-2 resources into K- Duplicates Duplicates Duplicates O = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' K-1, K] 2 equal spaced bins Semantic Filter over Titles R = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='. P-1, P] Syntactic Filters over Titles where, K<=P<=NFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Flow of Grammar selection to Voice-over generation We also encouraged the evaluators to give free comments for the trailers they evaluated, as this would help us improve our system in future iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' I and II lists down some of the positive comments and improvements suggested by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 8 shows some of the trailer fragments generated by our proposed system14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Following is the list of 8 questions that were asked to the evaluator during the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' The text in italics highlights the broader aspect of the evaluation feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Did you find the trailer to be self-contained?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' How were the fonts and styles used in the trailer in terms of readability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' How did you find the length and pace of the trailer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' As a user, how impressed are you with this trailer overall?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Could this trailer evoke interest in someone taking this course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' (Ignoring any prior inclination to the topic) Q6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' How was the average duration of each frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Based on the trailer you just saw, do you think you have a good impression of the course now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Q8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' How did you find the sync between the audio and visuals you saw?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 9, the scores obtained for each of the survey questions are good and far above the average(score of 3) for almost all the trailers generated by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Also, in our study, we found both the templates performed equally good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' However, for Q5, the average scores is relatively lower compared to other questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' On digging deeper we found some of the comments of total 24 comments we received mentioned about the difficulty of the course for not getting interested in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This could mean that this question (Q5) is more subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' 14Detailed demo walk-through: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content='v= 06VVuAlFhTk V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this paper, we presented a novel framework for au- tomatically generating video trailers for a learning pathway using ML and NLP techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We validated our trailers on multiple corpus of varied granularity with human evaluation and the results obtained were encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' This approach can be adapted to different domains given enough data to train the models involved in the entire process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We believe that this approach can lay foundation to building more advanced versions of trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' In future, we plan to improve the existing system by incorporating suggestions obtained in the user evaluation and adding more interesting themes like automatically detecting learning outcomes given the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We also intend to create an interactive dashboard to take inputs from the creator and allow the creator to make edits to the auto-generated content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank the Center of Excellence on Cognitive Com- puting, funded by Mphasis F1 Foundation for funding this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' We also thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Prasad Ram and Gooru team (https://gooru.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} +page_content=' Networks Training Active Discussion Forum Ihis course, In this specaly curated course ol the curriculum you wll go througn startyourjourney Outline Frame Meta-Information Frame CTA Frame5 4 Likert Value 3 2 1 0 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 SurveyQuestion P1-T1 P1-T2 P2-T1 P2-T2 P3-T1 P3-T2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfiQfL/content/2301.03957v1.pdf'} diff --git a/1dFLT4oBgHgl3EQfpy-5/vector_store/index.pkl b/1dFLT4oBgHgl3EQfpy-5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2632d49f0e3386a9d07cfc0e0808b2bac568a719 --- /dev/null +++ b/1dFLT4oBgHgl3EQfpy-5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e3ebd44b889223021ca073749402a0626fc467147d97a1e2c2d358cec0551f38 +size 235245 diff --git a/1tE3T4oBgHgl3EQfngpH/content/tmp_files/load_file.txt b/1tE3T4oBgHgl3EQfngpH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..663560cf1bb67e2b86d64ddf4bba0dac46096cb6 --- /dev/null +++ b/1tE3T4oBgHgl3EQfngpH/content/tmp_files/load_file.txt @@ -0,0 +1,746 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf,len=745 +page_content='Multivariate Regression via Enhanced Response Envelope: Envelope Regularization and Double Descent Oh-Ran Kwon and Hui Zou School of Statistics, University of Minnesota Abstract The envelope model provides substantial efficiency gains over the standard multi- variate linear regression by identifying the material part of the response to the model and by excluding the immaterial part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In this paper, we propose the enhanced response envelope by incorporating a novel envelope regularization term in its formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' It is shown that the enhanced response envelope can yield better prediction risk than the original envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The enhanced response envelope naturally handles high- dimensional data for which the original response envelope is not serviceable without necessary remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In an asymptotic high-dimensional regime where the ratio of the number of predictors over the number of samples converges to a non-zero constant, we characterize the risk function and reveal an interesting double descent phenomenon for the first time for the envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' A simulation study confirms our main theoret- ical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Simulations and real data applications demonstrate that the enhanced response envelope does have significantly improved prediction performance over the original envelope method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Keywords: Double descent, Envelope model, High-dimension asymptotics, Prediction, Reg- ularization 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04625v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='ME] 11 Jan 2023 1 Introduction The envelope model first introduced by Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2010) is a modern approach to estimat- ing an unknown regression coefficient matrix β ∈ Rr×p in multivariate linear regression of the response vector y ∈ Rr on the predictors x ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' It was shown by Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2010) that the envelope estimator of β results in substantial efficiency gains relative to the standard maximum likelihood estimator of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The gains arise by identifying the part of the response vector that is material to the regression and by excluding the immaterial part in the estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The original envelope model has been later extended to the envelope model based on excluding immaterial parts of the predictors to the regression by Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2013) then established the connection between the latter envelope model and partial least squares, providing a statistical understanding of partial least squares algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The success of the envelope models and their theories motivated some authors to propose new envelope models by applying or extending the core idea of envelope modeling to various statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The two most common are the response envelope models and the predictor envelope models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The response envelope models (predictor envelope models) achieve estima- tion and prediction gains by eliminating the variability arising from the immaterial part of the responses (predictors) that is invariant to the changes in the predictors (responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Pa- pers on response envelope models include the original envelope model (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2010), the partial envelope model (Su and Cook, 2011), the scaled response envelope model (Cook and Su, 2013), the reduced-rank envelope model (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2015), the sparse envelope model (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2016), the Bayesian envelope model (Khare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2017), the tensor response enve- lope model (Li and Zhang, 2017), the envelope model for matrix variate regression (Ding and Cook, 2018), and the spatial envelope model for spatially correlated data (Rekabdarkolaee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Papers on predictor envelope models include the envelope model for predictor reduction (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2013), the envelope model for generalized linear models and Cox’s proportional hazard model (Cook and Zhang, 2015a), the scaled predictor envelope model (Cook and Su, 2016), the envelope quantile regression model (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2020), the envelope model for the censored quantile regression (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022), tensor envelope partial least squares regression (Zhang and Li, 2017), and envelope-based sparse partial least squares regression (Zhu and Su, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For a comprehensive review of the envelope models, readers 2 are referred to Cook (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' High-dimensional data have become common in many fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' It is only natural to consider the performance of the envelope model under high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The likelihood-based method to estimate β under both the response/predictor envelope model is not serviceable for high- dimensional data because the likelihood-based method requires the inversion of the sample covariance matrix of predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hence, one has to find effective ways to mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For the predictor envelope model, its connection to partial least squares provides one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Partial least squares (De Jong, 1993) can be used for estimating β for the predictor envelope model (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The partial least squares algorithm is an iterative moment-based algorithm involving the sample covariance of predictors and the sample covariance between the response vector and predictors, which does not require inversion of the sample covariance matrix of predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In addition, the algorithm provides the root-n consistent estimator of β in the predictor envelope model with the number of predictors fixed (Chun and Kele¸s, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2013) and can yield accurate prediction in the asymptotic high-dimensional regime when the response is univariate (Cook and Forzani, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Motivated by this, Zhu and Su (2020) introduced envelope-based sparse partial least squares and showed the consistency of the estimator for the sparse predictor envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Zhang and Li (2017) proposed a tensor envelope partial least squares algorithm, which provides the consistent estimator for the tensor predictor envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Another way to apply predictor envelope models for high-dimensional data is by selecting the principal components of predictors and then using likelihood-based estimation on the principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This simple remedy is adapted by Rimal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2019) to compare the prediction performance of the likelihood-based predictor envelope method, principal component regression, and partial least squares regression for high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Their extensive numerical study showed that this simple remedy produced better prediction performance than principal component regression and partial least squares regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The impact of high dimensions is more severe for the response envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' There is far less work on making the response envelope model serviceable for high- dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The Bayesian approach for the response envelope model (Khare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2017) can handle high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The sparse envelope model (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2016) which performs variable selection on the responses can handle data with the sample size smaller 3 than the number of responses, but still requires the number of predictors smaller than the number of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In this paper, we propose the enhanced response envelope for high-dimensional data by incorporating a novel envelope regularization term in its formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope regu- larization term respects the fundamental idea of the original envelope model by considering the presence of the material and immaterial parts of the response in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The en- hancements are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' First, our enhanced response envelope estimator can handle both low- and high-dimensional data, while the original envelope estimator can only handle low- dimensional data where the sample size n is smaller than the number of predictors p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' From the connection between the original envelope estimator and the enhanced response envelope estimator in low-dimension, we extend the definition of the original envelope estimator to high-dimensional data by considering the limiting case of the enhanced response envelope es- timator with a vanishing regularization parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' see the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Second, we prove that the enhanced response envelope can reduce the prediction risk relative to the original envelope for all values of n and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Moreover, we study the asymptotics of the predic- tion risk for the original envelope estimator and the enhanced response envelope estimator when both n, p → ∞ and their ratio converges to a nonzero constant p/n → γ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This kind of asymptotic regime has been considered in high-dimensional machine learning theory (El Karoui, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Dobriban and Wager, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Liang and Rakhlin, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022) for analyzing the behavior of prediction risk of certain predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We derive an interesting asymptotic prediction risk curve for the envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The risk increases as γ increases, and then decreases after γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This phenomenon is known as the double descent phenomenon in the machine learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Although the double descent phenomenon has been observed for neural networks and ridgeless regression (Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022), this is the first time that such a phenomenon is shown for the envelope models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We review the original envelope model and the corresponding envelope estimator in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2, we introduce a new regularization term called the envelope regularization based on which we propose the enhanced response envelope in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The enhanced response envelope estimator nat- urally provides a definition for the envelope estimator when p > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='4 describes 4 how to implement this new method in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1, we prove that the enhanced response envelope can yield better prediction risk than the original envelope for any (n, p) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Considering n, p → ∞ and p/n → γ ∈ (0, ∞), we derive the limiting prediction risk result of the original envelope and the enhanced response envelope in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This result along with our simulation study in Section 4 verify the double descent phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Real data analyses are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Proofs of theorems are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 2 Enhanced response envelope 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 Review of envelope model Envelope model Let us begin with the classical multivariate linear regression model of a response vector y ∈ Rr given a predictor vector x ∈ Rp: y = βx + ε, ε ∼ N(0, Σ), (1) where ε is the error vector with a positive definite Σ and independent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' β ∈ Rr×p is an unknown matrix of regression coefficients and x ∼ Px where Px is a distribution on Rp such that E(x) = 0 and Cov(x) = Σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We omit an intercept by assuming E(y) = 0 for easy communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope model allows for the possibility that there is a part of the response vector that is unaffected by changes in the predictor vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Specifically, let E ⊆ Rr be a subspace such that for all x1 and x2, (i) QEy|(x = x1) ∼ QEy|(x = x2) and (ii) PEy ⊥⊥ QEy|x, (2) where PE is the projection onto E and QE = I − PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Condition (i) states that the marginal distribution of QEy is invariant to changes in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Condition (ii) says that QEy does not affect PEy if x is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Conditions together imply that PE includes the relevant depen- dency information of y on x (the material part) while QE is the irrelevant information (the immaterial part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let B = span(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The conditions in (2) hold if and only if span(β) = B ⊆ E and Σ = PEΣPE + QEΣQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (3) 5 The definition of an envelope introduced by Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2007, 2010) formalizes the smallest subspace satisfying the conditions in (2) using the equivalence relation of (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope is defined as the intersection of all subspaces E satisfying (3) and is denoted by EΣ,B, Σ-envelope of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope model arises by parameterizing the multivariate linear model in terms of the envelope EΣ,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The parameterization is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let u = dim(EΣ,B), Γ ∈ Rr×u be any semi-orthogonal basis matrix for EΣ,B, and Γ0 ∈ Rr×(r−u) is any semi-orthogonal basis matrix for the orthogonal complement of EΣ,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Then the multivariate linear model can be written as y = Γηx + ε, ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT 0 ), (4) where β = Γη with η ∈ Ru×p, and Ω ∈ Rr×r and Ω0 ∈ R(r−u)×(r−u) are symmetric positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Model (4) is called the envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Envelope estimator The parameters in the envelope model are estimated by maximizing the likelihood function from model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Assume that p+r < n and u is the dimension u of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, and SY|X = SY−SY,XS−1 X SX,Y, where Y ∈ Rn×r has rows yT i and X ∈ Rn×p has rows xT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope estimator of β is determined as ˆEΣ,B = span{arg min G∈Gr(r,u)(log |GTSY|XG| + log |GTS−1 Y G|)}, (5) where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Define ˆΓ as any semi- orthogonal basis matrix for ˆEΣ,B and let ˆΓ0 be any semi-orthogonal basis matrix for the orthogonal complement of ˆEΣ,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The estimator of β is given by ˆβ = ˆΓˆΓTSY,XS−1 X , (6) and Σ is estimated by ˆΣ = ˆΓ ˆΩˆΓ + ˆΓT 0 ˆΩ0ˆΓ0 where ˆΩ = ˆΓTSY|XˆΓ, ˆΩ0 = ˆΓT 0 SY ˆΓ0, (7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2 Envelope regularization In this section, we introduce the envelope regularization term that respects the fundamental idea in the envelope model by considering the presence of material and immaterial parts, 6 PEΣ,By and QEΣ,By, in the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We define the envelope regularization term as ρ(η, Ω) = tr(ηTΩ−1η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (8) The envelope model distinguishes between PEΣ,By and QEΣ,By in the estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The log-likelihood function of the envelope model is decomposed into two log-likelihood functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' One is the log-likelihood function for the multivariate regression of ΓTy on x, ΓTy = ηx+ΓTε where ΓTε ∼ N(0, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The other is the log-likelihood function for the zero- mean model of ΓT 0 y, ΓT 0 y = ΓT 0 ε where ΓT 0 ε ∼ N(0, Ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope regularization term (8) is the function of η and Ω, the parameters in the likelihood for the material part of the envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope regularization term (8) can be seen as imposing the Frobenius norm regularization on the coefficient after standardizing the material part of the regression to have uncorrelated errors, Ω−1/2ΓTy = Ω−1/2ηx + Ω−1/2ΓTε where Ω−1/2ΓTε ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We emphasize that the envelope regularization is different from the ridge regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' While the ridge regularization ∥β∥2 F is the quadratic function of β, the envelope regulariza- tion is not because the components of Ω are not fixed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The envelope regularization is the function of both η and Ω, and thus is optimized over η and Ω simultaneously, as shown in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3 The proposed estimator We only assume that r ≤ n but p is allowed to be bigger than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The log-likelihood function under the envelope model (4) is Lu(η, EΣ,B, Ω, Ω0) = − (nr/2) log(2π) − (n/2) log |ΓΩΓT + Γ0Ω0ΓT 0 | − (1/2) n � i=1 (yi − Γηxi)T(ΓΩΓT + Γ0Ω0ΓT 0 )−1(yi − Γηxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' By incorporating the envelope regularization term ρ given in the last subsection, we propose the following enhanced response envelope estimator via penalized maximum likelihood: arg max{Lu(η, EΣ,B, Ω, Ω0) − n 2λ · ρ(η, Ω)}, (9) where λ > 0 serves as a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 7 Let SX = n−1XTX, SY = n−1YTY, SY,X = n−1YTX, Sλ X = SX + λI and Sλ Y|X = SY − SY,X(Sλ X)−1SX,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' After some basic calculations, (9) can be expressed as ˆEΣ,B(λ) = span{arg min G∈Gr(r,u)(log |GTSλ Y|XG| + log |GTS−1 Y G|)}, (10) where Gr(r, u) = {G ∈ Rr×u : G is a semi-orthogonal matrix}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let ˆΓλ be any semi-orthogonal basis matrix for ˆEΣ,B(λ) and ˆΓ0,λ be any semi-orthogonal basis matrix for the orthogonal complement of ˆEΣ,B(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The enhanced envelope estimator of β is given by ˆβ(λ) = ˆΓλˆΓT λSY,X(Sλ X)−1 (11) and Σ is estimated by ˆΣ(λ) = ˆΓλ ˆΩ(λ)ˆΓλ + ˆΓT 0,λ ˆΩ0(λ)ˆΓ0,λ where ˆΩ(λ) = ˆΓT λSλ Y|XˆΓλ, ˆΩ0(λ) = ˆΓT 0,λSY ˆΓ0,λ, (12) The enhanced response envelope estimator can naturally handle the case where p ≥ n−r, while the original envelope estimator (5) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Motivated by the definition of ridgeless regression (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022), we can consider taking the limit of the enhanced response envelope estimator with λ → 0+: ˆEΣ,B = span{arg min G∈Gr(r,u)( lim λ→0+ log |GTSλ Y|XG| + log |GTS−1 Y G|)}, ˆβ = lim λ→0+ ˆβ(λ) (13) We take (13) as the definition of envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Obviously, when p < n−r, this extended definition recovers the original envelope estimator (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This definition enables the use of the envelope estimator when p ≥ n−r, without altering the definition of the original envelope estimator (5) when p < n−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In practice, we implement (13) by computing the enhanced response envelope estimator (10) with a very small value of λ such as 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' As the enhanced response envelope estimator (9) has flexibility on λ, the enhanced re- sponse envelope estimator with an appropriate choice of λ can yield better prediction risk compared to the envelope estimator, which is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We discuss the Grass- mannian manifold optimization required in (10) in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='4 Implementation Suppose that the dimension u is specified and λ is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Our proposed estimator ˆEΣ,B(λ) for EΣ(B) requires the optimization over the Grassmannian G(u, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Such a computation 8 problem exists for the original envelope model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' So far, the best-known algorithm for solving envelope models is the algorithm introduced by Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Thus, we employ their algorithm to compute ˆEΣ,B(λ) in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Note that we standardize X so that each column has a mean of 0 and a standard deviation of 1 before fitting any model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In practice, the tuning parameter λ and the dimension u of the envelope are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We use the cross-validation method to choose (u, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For the original envelope, u can be selected by using AIC, BIC, LRT or cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' BIC and LRT may be preferred as shown by simulations in Su and Cook (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Because the enhanced response envelope model has an additional tuning parameter λ, we propose to use cross-validation to find the best tuning parameter combination of u and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We have implemented the enhanced response envelope method in R and the code is available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 3 Theory In this section, we show that the enhanced response envelope can reduce the prediction risk over the envelope for any (n, p) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We then consider the asymptotic setting when n, p → ∞ p/n → γ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This asymptotic regime has been considered in the literature (El Karoui, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Dobriban and Wager, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Liang and Rakhlin, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022) for analyzing the behavior of prediction risk of certain predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In our discussion, we consider the case where EΣ(B) is known, which has been assumed in the existing envelope papers to understand the core mechanism of envelope methodologies (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Cook and Zhang, 2015a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 Reduction in prediction risk Consider a test point xnew ∼ Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For an estimator ˆβ, we define the prediction risk as R( ˆβ|X) = E[∥ ˆβxnew − βxnew∥2|X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Note that this definition has the bias-variance decomposition, R( ˆβ|X) = ∥bias(vec( ˆβ)|X)∥2 + tr{Var(vec( ˆβ)|X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 9 Let Γ be a semi-orthogonal basis matrix for EΣ,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Following the discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3, we take (13) as the definition of the envelope estimator ˆβΓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The prediction risk of ˆβΓ is R( ˆβΓ|X) = vecT(β)[ΠXΣxΠX ⊗ Ir]vec(β) � �� � bias2 + tr(Ω) n tr(S+ XΣx) � �� � var , where ΠX = Ip − S+ XSX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The prediction risk of the enhanced response envelope estimator ˆβΓ(λ) is R( ˆβΓ(λ)|X) = E[∥ ˆβΓ(λ)xnew − βxnew∥2|X] = λ2vecT(β)[(SX + λI)−1Σx(SX + λI)−1 ⊗ Ir]vec(β) � �� � bias2 + tr(Ω) n tr(ΣxSX(SX + λI)−2) � �� � var .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (14) Theorem 1 shows that using the envelope regularization always improves the prediction risk of the envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' There always exists a λ > 0 such that R( ˆβΓ(λ)|X) < R( ˆβΓ|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2 Limiting prediction risk and double descent phenomenon The asymptotics of the envelope model are well-established in the case where n diverges while p is fixed (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2010), while not in a high-dimensional asymptotic setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In this section, we examine the limiting risk of both the enhanced response envelope estimator and the envelope estimator in the high-dimensional asymptotic regime where n, p → ∞ with p/n → γ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The number of response variables r is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This kind of asymptotic regime has been considered in high-dimensional machine learning theory (El Karoui, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Dobriban and Wager, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Liang and Rakhlin, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022) for analyzing the behavior of prediction risk of certain predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let x = Σ1/2 x x∗, where E(x∗) = 0 and Cov(x∗) = Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Then the envelope model (4) of y on x can be expressed as the envelope model of y on x∗: y = Γηx + ε = Γη∗x∗ + ε, where η∗ = ηΣ1/2 and ε ∼ N(0, ΓΩΓT + Γ0Ω0ΓT 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We take advantage of the invariance property of the envelope model in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Considering the envelope on (y, x∗) amounts to assuming the covariance of the predictor is Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 10 0 2 4 6 8 0 5 10 15 20 25 γ Limiting prediction risk Envelope Enhanced response envelope Figure 1: The limiting prediction risks of the enhanced response envelope with λ∗ = tr(Ω)γ/c2 (gray solid line) and the envelope (black solid line), illustrating Theorem 2 when tr(Ω) = 10 and tr(βTβ) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Assume that x has a bounded 4th moment and that tr(ηTη) = c2 for all n, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Then as n, p → ∞, such that p/n → γ ∈ (0, ∞), almost surely, R( ˆβΓ|X) → � � � � � tr(Ω) γ 1−γ for γ < 1 c2(1 − 1 γ) + tr(Ω) 1 γ−1 for γ > 1, and R( ˆβΓ(λ∗)|X) → tr(Ω)γm(−λ∗), where λ∗ = tr(Ω)γ/c2 and m(z) = 1−γ−z−√ (1−γ−z)2−4γz (2γz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Figure 1 visualizes the limiting prediction risk curves in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' It plots the limiting risks of envelope (black solid line) and the enhanced response envelope with λ∗ = tr(Ω)γ/c2 (dark-gray solid line), when tr(Ω) = 10 and tr(ηTη) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We have four remarks from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The limiting risk of envelope increases before γ = 1 and then decreases after γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The double descent phenomenon has been observed in popular methods such as neural networks, kernel machines and ridgeless regression (Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2022), but this is the first time that such a result is established 11 in the envelope literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Second, the enhanced response envelope estimator always has a better asymptotic prediction risk than the envelope estimator (for any c2, tr(Ω), and γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Third, in Theorem 1, we show the existence of a λ that gives a smaller prediction risk of the enhanced response envelope than the envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In an asymptotic regime, we specify such a λ value: λ∗ = tr(Ω)γ/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Lastly, the gap between two limiting prediction risks, limn,p→∞ R( ˆβΓ|X) and n,p→∞R( ˆβΓ(λ∗)|X), increases as γ increases from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' It is easy to see as 1 1−γ > m(−λ∗), 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 4 Simulation In this section, we use simulations to compare the performance of the enhanced response envelope estimator and the envelope estimator in terms of the prediction risk, E[∥ ˆβxnew − βxnew∥2|X] = tr[( ˆβ − β)Cov(xnew)( ˆβ − β)T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In addition, we use simulations to have a numeric illustration of the double descent phenomenon to confirm the asymptotic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We consider a setting where yi ∈ R3 is generated from the model yi = βxi + εi, εi ∼ N(0, Σ), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , n, and xi ∈ Rp is generated independently from xi ∼ N(0, Σx(ρ)) where (i, j)th element of Σx(ρ) ∈ Rp×p is ρ|i−j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The covariance matrix Σ is set using three orthonormal vectors and has eigenvalues 10, 8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The columns of Γ are the second and third eigenvectors of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Each component of ˜η ∈ R2×p is generated from the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We then set η = √ 10 · ˜η/∥˜η∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In this setting, tr(ηTη) = 10, tr(Ω) = 10, and tr(Ω0) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We assume that dim(EΣ,B) = 2 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Prediction risk comparison In this simulation, we try different combinations of n, p and ρ where n ∈ {50, 90, 200, 500}, p/n ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2} and ρ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We compare the prediction risk of the enhanced response envelope estimator to three different estimators: the envelope estimator, multivariate linear regression, and multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For the enhanced response envelope and the multivariate ridge regression, we perform ten-fold cross-validation on simulated data to select λ among equally spaced 100 candidate λ-values in the scale of logarithm base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We compute the envelope estimator for data 12 with n ≤ p−r by taking a very small value of λ = 10−8 in the enhanced response envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We fit multivariate regression model to n < p data by taking a tiny value of λ = 10−8 in the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We then calculate the prediction risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This process is repeated 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In Table 1, we report the prediction risk averaged over 100 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' First, we see that the prediction risks from the enhanced response envelope are consistently smaller than the envelope, as indicated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Second, the enhanced response envelope consistently gives smaller prediction risks compared to the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' When u = r, the enhanced response envelope model reduces to the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Therefore, the prediction risk of the enhanced envelope model can be smaller than that of multivariate ridge regression as long as tr(Ω0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Double descent confirmation This simulation is designed to support Theorem 2 and to illustrate the double descent phenomenon in the envelope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We set n ∈ {200, 2000} and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' p/n varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We compute the envelope and the enhanced response envelope with setting λ∗ = tr(Ω)p/(nc2) = p/n on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We then calculate the prediction risk for each estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Again, we fit n ≤ p−r data to the envelope estimator by taking a very small value of λ = 10−8 in the enhanced response envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Figure 2 displays the prediction risks from n = 2000 with various p values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The gray rectangle points denote the prediction risk for the enhanced response envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The black triangle points are the prediction risk for the envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We see a fascinating double descent prediction risk curve for the envelope model, as discussed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Also, the enhanced response envelope gives a smaller prediction risk across the entire range of p/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Figure 3 plots the prediction risk curves from n = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We see that Figure 3 exhibits the same messages for the much smaller sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Although Theorem 2 is established when considering EΣ,B is known, we did not use this information in the actual estimation in the simulation study, yet the core message of Theorem 2 is confirmed by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 13 n p Enhanced envelope Envelope Multivariate linear reg Multivariate ridge reg Example 1: p/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1, ρ = 0 50 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='40 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='39 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='12) 90 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='08) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='41 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='10) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='92 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='09) 200 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='26 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='93 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 500 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='85 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) Example 2: p/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8, ρ = 0 50 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='73 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='18) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='89 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='80) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='45 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='09) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 90 72 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='44 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='14) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='10 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='93) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='81 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='24) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='08) 200 160 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='10) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='99) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='63) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='06) 500 400 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='61 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='85) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='89 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) Example 3: p/n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2, ρ = 0 50 60 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='23) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='70 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='79 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='83) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='08 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 90 108 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='58 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='13) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='01 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='36) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='38 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='60) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='07) 200 240 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='07) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='91 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='18) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='94 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='83) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 500 600 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='43 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='91) 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='55) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) Example 4: p/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8 50 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='19) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='39 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='84 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='07) 90 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='40 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='08) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='06) 200 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='90 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='30 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 500 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='04) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='02) Example 5: p/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8 50 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='50 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='34) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='45 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='09) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='76 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='12) 90 72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='15) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='14 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='85) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='81 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='24) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='10) 200 160 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='40 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='99) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='33 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='63) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05) 500 400 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='69 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='87) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) Example 6: p/n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='8 50 60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='24 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='23) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='43 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='12) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='37) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='80 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='14) 90 108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='41 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='12) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='84 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='68) 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='01 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='09) 200 240 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='07) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='46 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='30) 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='34 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='17) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='98 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='06) 500 600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='98) 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='62 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='71) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='03) Table 1: Prediction risk, averaged over 100 replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The standard error is given in paren- theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For n ≤ p−r data, we compute the envelope by taking a very small value of λ = 10−8 in the enhanced response envelope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' see the definition of the envelope estimator (13) in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For n < p data, we fit the multivariate regression model by taking a tiny value of λ = 10−8 in the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 14 0 2 4 6 8 0 5 10 15 20 25 p/n Prediction risk Envelope Enhanced response envelope Figure 2: Prediction risk of the envelope and the enhanced response envelope with λ∗ = tr(Ω)p/(nc2), when n = 2000 and p varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For n ≤ p−r data, we fit the envelope by taking a very small value of λ = 10−8 in the enhanced response envelope estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' see the definition of the envelope estimator (13) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 5 Real data In this section, we use two real datasets to illustrate the enhanced response envelope esti- mator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We use air pollution data in which the number of samples is bigger than the number of predictors (n > p) in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2, we analyze near-infrared spectroscopy data in which the number of predictors is much bigger than the number of predictors (p ≫ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We compare the prediction performance of the enhanced response envelope estimator to the envelope estimator, multivariate regression, and multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 Air pollution data The air pollution data are available and obtained directly from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='5 of Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The response vector y ∈ R5 consists of atmospheric concentrations of CO, NO, NO2, O3, and HC, recorded at noon in the Los Angeles area on 42 different days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The two predictors 15 0 2 4 6 8 0 5 10 15 20 25 p/n Prediction risk Envelope Enhanced response envelope Figure 3: Prediction risk of the envelope and the enhanced response envelope with λ∗ = tr(Ω)p/(nc2), when n = 200 and p varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For n ≤ p−r data, we fit the envelope by taking a very small value of λ = 10−8 in the enhanced response envelope estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' see the definition of the envelope estimator (13) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' are wind speed and solar radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This data were analyzed in Cook (2018) to illustrate the effectiveness of the original envelope model compared to the standard multivariate regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' They showed that the asymptotic standard errors of estimated components of β from the envelope model are significantly reduced compared to those from the standard multivariate regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We use the data to predict atmospheric concentrations from wind speed and solar radiation and compare the prediction performance of the enhanced response envelope estimator to the envelope estimator, the standard multivariate regression, and multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' To compare the prediction performance, we borrow the nested cross validation idea (Wang and Zou, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', 2021), in which an inner cross-validation is performed to tune a model and an outer cross-validation is performed to provide a prediction error of the tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We adopt the leave-one-out cross-validation (LOOCV) procedure for the outer loop because the LOOCV error is an unbiased estimator of the generalization error of the tuned model and is shown to have nice performance compared to other methods for 16 Enhanced envelope Envelope Multivariate linear reg Multivariate ridge reg Error 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='859 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='951 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='192 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='124 Table 2: Air pollution data: prediction error of the enhanced response envelope method, the original envelope method, the multivariate linear regression, and the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' estimating generalization errors (Wang and Zou, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We take the ith observation out from the data and set the remaining n−1 observations as the training set to fit and tune models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We standardize X of the training set so that each column has a mean of 0 and a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We perform ten-fold cross-validation to select (u, λ) from a fine grid of u ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , 5} and 20 equally spaced candidate λ-values in the scale of logarithm base 10 for the enhanced response envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For the envelope, we perform ten-fold cross-validation to choose u from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For the multivariate ridge model, ten-fold cross-validation is performed to select λ from 20 equally spaced λ-values in the scale of logarithm base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The ith observation we take out at the beginning is set as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We standardize xi of the test set using the mean and standard deviation of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We then calculate the squared prediction error, ∥yi − ˆβ(−i)xi∥2 2/r, where ˆβ(−i) is the estimated regression coefficient derived from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We repeat this process for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , n and report �n i=1 ∥yi − ˆβ(−i)xi∥2 2/(nr) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We see that the enhanced response envelope estimator gives the smallest prediction error among all competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2 Near-infrared spectroscopy data of fresh cattle manure Near-infrared spectroscopy data of cattle manure were collected by Gog´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The data are available in the Data INRAE Repository at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='15454/JIGO8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This data contain 73 cattle manure samples that were analyzed by near-infrared spectroscopy using a NIRFlex device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Near-infrared spectra were recorded every 2 nm from 1100 to 2498 nm on fresh homogenized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In addition, the cattle manure samples were analyzed for three chemical properties: the amount of dry matter, magnesium oxide, and potassium 17 Enhanced envelope Envelope Multivariate linear reg Multivariate ridge reg Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='492 Table 3: Near-infrared spectroscopy data: prediction error from the enhanced response enve- lope method, the envelope method, the multivariate linear regression, and the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We compute the envelope estimator by taking a very small value of λ = 10−8 in the enhanced response envelope estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' see the definition of the envelope estimator (13) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We fit the multivariate regression model by taking a very small value of λ = 10−8 in the multivariate ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We use the data of 62 cattle manure samples which have no missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We standardize each chemical property to have a sample mean of 0 and a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In our analysis, we consider the multivariate linear model, where xi ∈ R700 is the vector of near-infrared spectroscopy measurements and yi ∈ R3 is the vector of three chemical measurements to predict the three chemical properties from the absorbance spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In Table 3, we report the prediction error which is calculated using the same procedure described in the previous subsection, except that u is chosen from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Again, We see that the enhanced response envelope estimator has the smallest prediction error among all competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 6 Discussion In this paper, we have developed a novel envelope regularization function which is used to define the enhanced envelope estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We have shown that the enhanced envelope estimator is indeed better than the un-regularized envelope estimator in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The asymptotic analysis of the risk function of envelope reveals, for the first time in the envelope literature, an interesting double descent phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The numeric examples in this work also suggest that the enhanced response envelope estimator is a promising new tool for multivariate regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 18 Although this paper is focused on the case where the number of responses (r) is less than the number of samples and the number of predictors, it is interesting to consider the case when r → ∞ in ultrahigh-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2016) studied the response envelope for r → ∞ but p is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' When both p, r > n and diverge, there are additional technical issues to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' For example, we may need another penalty term to handle the issues caused by the large r in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' This direction of research will be investigated in a separate paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' References Bai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', Miao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', and Pan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2007), “On asymptotics of eigenvectors of large sample covariance matrix,” The Annals of Probability, 35, 1532–1572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2017), “Tensor envelope partial least-squares regression,” Technomet- rics, 59, 426–436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', Van Keilegom, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', and Ding, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2022), “Envelopes for censored quantile regression,” Scandinavian Journal of Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Zhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' and Su, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2020), “Envelope-based sparse partial least squares,” The Annals of Statistics, 48, 161–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' A Proofs of Theorems A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 Proof of Theorem 1 Note that R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−1Σx(SX + λI)−1βT) + tr(Ω) n tr(ΣxSX(SX + λI)−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Therefore, we have ∂ ∂λR( ˆβΓ(λ)|X) = 2λ · tr(βSX(SX + λI)−2Σx(SX + λI)−1βT) − 2tr(Ω) n tr(ΣxSX(SX + λI)−3) ≤ p � i=1 � 2λ · σi(βTβ) − 2tr(Ω) n � σi(ΣxSX(SX + λI)−3), where σi(M) denotes the i-th largest eigenvalue of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The inequality above comes from Von Neumann’s trace inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 22 Since ∂ ∂λR( ˆβΓ(λ)|X) < 0 if λ < tr(Ω)/(nσ1 � βTβ) � , R( ˆβΓ(λ)|X) is a monotonically decreasing function if 0 ≤ λ ≤ tr(Ω)/(nσ1 � βTβ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Therefore, we have R( ˆβΓ(λ)|X) < tr(Ω) n tr(ΣxS+ X), when 0 < λ < tr(Ω)/(nσ1 � βTβ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Since tr(Ω) n tr(ΣxS+ X) ≤ R( ˆβΓ|X), we prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2 Proof of Theorem 2 Our analyses of limiting prediction risk follow that of Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' As Σx = I, R( ˆβΓ|X) = vecT(β)[ΠX ⊗ Ir]vec(β) + tr(Ω) n tr(S+ X), R( ˆβΓ(λ)|X) = λ2tr(β(SX + λI)−2βT) + tr(Ω) n tr(SX(SX + λI)−2), where ΠX = Ip − S+ XSX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='1 Proof for envelope estimator when γ < 1 Let us consider the case where p/n → γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' From Theorem 1 of Bai and Yin (2008), σmin(SX) ≥ (1 − √γ)2/2 and σmax(SX) ≤ 2(1 + √γ)2 almost surely for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Therefore, in this case, SX is invertible and the bias term of R( ˆβΓ|X) is 0, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The variance term of R( ˆβΓ|X) is tr(Ω) n tr(S+ X) = p · tr(Ω) n � 1 sdFSX(s), where FSX(s) is the spectral measure of SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' By the Marchenko-Pastur theorem, which says that FSX → Fγ, and the Portmanteau theorem, � 2(1+√γ)2/ (1−√γ)2/2 1 sdFSX(s) → � 2(1+√γ)2/ (1−√γ)2/2 1 sdFγ(s) = � 1 sdFγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 23 The equality is because the support of Fγ is [(1 − √γ)2, (1 + √γ)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We can also remove the upper and lower limits of integration on the left-hand side by Theorem 1 of Bai and Yin (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Thus, combining above results, we arrive at R( ˆβΓ|X) → γ · tr(Ω) � 1 sdFγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The Stieltjes transformation of Fγ is given by m(z) = � 1 s − zdFγ(s) = (1 − γ − z) − � (1 − γ − z)2 − 4γz) 2γz , for any real z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' By taking the limit z → 0−, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2 Proof for envelope estimator when γ > 1 The variance term of R( ˆβΓ|X) is tr(Ω) n tr(S+ X) = tr(Ω) n tr((XXT/n)+) = tr(Ω) p tr((XXT/p)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Considering n/p → τ = 1/γ < 1, by the same arguments from the proof above, we conclude that tr(Ω) n tr(S+ X) → tr(Ω) 1 γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let β = [bT 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' bT r ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The bias term is vecT(β)[ΠX ⊗ Ir]vec(β) = r � i=1 bT i ΠXbi = r � i=1 lim z→0+ zbT i (SX + zI)−1bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' From Theorem 1 of Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' (2007), we have that zbT i (SX + zI)−1bi → z � 1 s + zFγ(s) = z∥bi∥2m(−z) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=', for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We further have that r � i=1 zbT i (SX + zI)−1bi → zc2m(−z) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' By the Arzela-Ascoli theorem and the Moore-Osgood theorem, we exchange limits and arrive at lim z→0+ r � i=1 zbT i (SX + zI)−1bi → c2 lim z→0+ zm(−z) = c2(1 − 1/γ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Combining the variance and the bias terms, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content='3 Proof for enhanced envelope estimator We use the similar techniques from the envelope estimator for both variance and bias terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The variance term of R( ˆβΓ(λ)) becomes tr(Ω) n tr(SX(SX + λI)−2) → γtr(Ω) � s (s + λ)2Fγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Let gn,λ(η) = λ · tr(β(SX + λ(1 + η)I)−1βT), η ∈ [−1/2, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The bias term of R( ˆβΓ(λ)) is λ2tr(β(SX + λI)−2βT) = − ∂ ∂ηgn(λ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' Because gn,λ(η) → λc2m(−λ(1 + η)) = λc2 � 1 s + λ(1 + η)dFγ(s), and derivative and limit are exchangeable, we have that λ2tr(β(SX + λI)−2βT) → λ2c2 � 1 (s + λ)2dFγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' We can conclude that, R( ˆβΓ(λ)) → � λ2c2 + s · γtr(Ω) (s + λ)2 Fγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' The right-hand side is minimized at λ∗ = γtr(Ω)/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' In such case, the right-hand side becomes γtr(Ω) · m(−λ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE3T4oBgHgl3EQfngpH/content/2301.04625v1.pdf'} diff --git a/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf b/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..42e70b452930285e4edd9140d0df2dd6311550ae --- /dev/null +++ b/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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+Preprint 6 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A Gaian Habitable Zone +Rudy Arthur,1★ Arwen Nicholson,2† +1University of Exeter, Department of Computer Science +2University of Exeter, Department of Physics and Astronomy +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +When searching for inhabited exoplanets, understanding the boundaries of the habitable zone around the parent star is key. If +life can strongly influence its global environment, then we would expect the boundaries of the habitable zone to be influenced +by the presence of life. Here using a simple abstract model of ‘tangled-ecology’ where life can influence a global parameter, +labelled as temperature, we investigate the boundaries of the habitable zone of our model system. As with other models of +life-climate interactions, the species act to regulate the temperature. However, the system can also experience ‘punctuations’, +where the system’s state jumps between different equilibria. Despite this, an ensemble of systems still tends to sustain or even +improve conditions for life on average, a feature we call Entropic Gaia. The mechanism behind this is sequential selection with +memory which is discussed in detail. With this modelling framework we investigate questions about how Gaia can affect and +ultimately extend the habitable zone to what we call the Gaian habitable zone. This generates concrete predictions for the size +of the habitable zone around stars, suggests directions for future work on the simulation of exoplanets and provides insight into +the Gaian bottleneck hypothesis and the habitability/inhabitance paradox. +Key words: Gaia – Habitable Zone – Biosignatures +1 INTRODUCTION +The Gaia hypothesis is that life influences the Earth’s feedback mech- +anisms to form a self-regulating system, and therefore life can help +maintain habitable conditions on its host planet Lovelock & Mar- +gulis (1974). Distinct from the biosphere Huggett (1999), Gaia is +the whole life-earth system, considered as a single entity. The impor- +tance of life’s interactions with the non-living environment are now +common sense, and the discipline of Earth System Science Lenton +& Watson (2013) studies the various feedback loops that constitute +‘Gaia’s body’ Volk (2012). Gaia theory itself takes a very broad +perspective, aiming to describe life at a planetary scale. Gaia theory +asks questions like: Is Gaia inevitable on a planet that hosts life, or +is it due to chance? What mechanisms can create a long lived Gaian +system? How will we detect other ‘Gaias’ beyond our solar system, +where direct planetary exploration is not an option? The astrophysical +point of view was crucial in the early development of Gaia, with the +search for life on Mars providing the initial inspiration for the Gaia +hypothesis Lovelock (1965). When looking at Earth from afar, Gaia +is what we see and the search for habitable or inhabited exoplanets is +the search for other Gaias. +Methods for exoplanet detection have developed considerably +since Gaia was first proposed. New telescopes, such as the James +Webb Space telescope and the Extremely Large Telescope (currently +under construction), and future missions, such as the Large Ultravi- +olet Optical Infrared Surveyor, mean that searching for signs of alien +★ E-mail: R.Arthur@exeter.ac.uk +† E-mail: A.E.Nicholson@exeter.ac.uk +life will be possible within the coming decades Snellen et al. (2021); +Quanz et al. (2021). While robotic missions to potentially habitable +exoplanets remain unfeasible, evidence for alien life will only be ob- +servable for exoplanets that have been dramatically shaped by their +biospheres. Exoplanets with newly emerging life, or those with the +remnants of a once-thriving biosphere that has since collapsed, will +be unlikely to produce a remotely observable signature. Answering +the key questions of Gaia theory not only informs how we think about +the history of life on Earth, but can form the theoretical foundation +for the study of life in the universe. +Catling et. al. Catling et al. (2018) proposed a framework for as- +sessing potential biosignatures using a probabilistic approach that +combines observations of the candidate planet and host star with +models of the possible abiotic and biotic planetary processes to de- +termine the probability of the planet being inhabited. With a great +diversity of exoplanets being found, any potential biosignature must +be considered within the context of its host planet Seager (2013); +Claudi. (2017); Kiang et al. (2018); Schwieterman et al. (2018); +Krissansen-Totton et al. (2022). Detailed abiotic models of exoplan- +ets are being developed for a wide range of detected planets, see e.g. +Amundsen et al. (2016); Boutle et al. (2017); Collins (2021); Fauchez +et al. (2021), and sophisticated models of biogeochemistry exist for +different points in Earth’s history, e.g. Kharecha et al. (2005); Daines +et al. (2017); Lenton et al. (2018b); Zakem et al. (2020). +Detailed and realistic modelling of life on other planets is im- +portant, however this paper will take a broader view that aims to +understand the generic mechanisms that lead to Gaia. We build on +recent work Ford Doolittle (2014); Lenton et al. (2018a); Arthur +& Nicholson (2022) on Gaian selection principles. We argued in +© 222 The Authors +arXiv:2301.02150v1 [astro-ph.EP] 5 Jan 2023 + +2 +Arthur & Nicholson +Arthur & Nicholson (2022) that some approaches to Gaian selec- +tion Ford Doolittle (2014); Lenton et al. (2018a) lead to anthropic +reasoning - we see Gaia because if we didn’t we wouldn’t exist. An- +thropic reasoning is controversial, with its opponents arguing that it +unfalsifiable with limited (if any) predictive power Smolin (2007). +The coming era of exoplanet astronomy gives new context and pur- +pose to these discussions. If our aim is for Gaia theory to inform our +search for life in the universe, then anthropic arguments are clearly +inadequate. +In Arthur & Nicholson (2022) we argue for ‘Entropic Gaia’ - that +the emergence of Gaia is a statistical tendency for planets that host +life. This means that life history on a single planet can be chaotic +and have periods of stability and collapse, however there is a trend +towards increasing biomass, stability, habitability and other Gaian +features. Any single planetary history for a life-bearing planet, such +as Earth, is likely to follow a (bumpy) trajectory towards Gaia. The +micro-mechanism leading to this behaviour was argued to be ‘Se- +quential Selection with Memory’ or an ‘entropic ratchet’. In brief, this +mechanism starts from the observation that coupled life-environment +systems move between regulation and disregulation. By definition, +disregulating systems quickly destroy themselves while regulating +systems persist, this is sequential selection Lenton et al. (2018a). In +models of ecology and Gaia (e.g. Becker & Sibani (2014); Hard- +ing (1999)) increasing diversity and complexity is associated with +increasing stability 1. More diverse ecosystems can generate more +novel species through mutation. Thus after every ecosystem collapse +(caused by internal or external factors) if a new ecosystem arises +it is likely to be more diverse, having started with a greater ‘pool’ +of species, and therefore also more stable. Sequential selection with +memory describes a sequence of distinct stable states that tends to +get ‘more Gaian’ over time. +This mechanism was originally proposed in the framework of the +Tangled Nature Model (TNM) Christensen et al. (2002). Originally +designed to study co-evolution, we demonstrated in Arthur & Nichol- +son (2017) that the TNM is closely related to the generalised Lotka- +Volterra model. The TNM is based on the idea that the growth rate +of a species is given by a fitness function that depends on the other +species present. Any model making this assumption will look like the +TNM close to equilibrium Arthur & Nicholson (2022). By studying +the model with agent based dynamics we can incorporate mutation, +giving us a very flexible, robust and general model of evolutionary +ecology. Since the TNM is quite general, conclusions drawn in this +framework are likely to have general applicability. +Artificial life modelling has been used extensively to study Gaia. +The original Daisy World Watson & Lovelock (1983) led to a large +number of variants Wood et al. (2008) and there are a variety of +other models such as the Guild Model Downing & Zvirinsky (1999), +Greenhouse World Worden (2010), Flask Model Williams & Lenton +(2007) and Exo-Gaia Nicholson et al. (2018) to name a few. We have +previously discussed Gaian models based on the TNM in Arthur & +Nicholson (2017, 2022). Here we propose a new variant on the TNM +that is more similar to other Gaian models with a very simple abiotic +(non-living) component. +While previous Gaian models have included mutation (such as +the Flask model and ExoGaia) the complexity of the biosphere in +these models has been limited and different species within the mod- +els only impact one another via the shared environment, e.g. via +1 See Landi et al. (2018) for a thorough discussion of the relationship be- +tween ecosystem complexity and stability, though most of these models don’t +consider coupling to the external environment +resource competition or via global parameters such as temperature. +When we look at life on Earth it is clear that different species can +have a large impact on each other beyond resource competition or +changing global parameters like temperature. For example, there are +complex interactions between worms, plants and soil that change the +structure, chemistry, water retention and other properties of soil for +the benefit of many species Le Bayon et al. (2021). These kinds of +symbiotic (and also antagonistic) interactions are usually missing in +Gaian models. We also observe that throughout Earth history there +have been dramatic and spontaneous changes in the diversity and +complexity of the biosphere, e.g. the Great Oxidation Event which +allowed for aerobic respiration to become an important energy source +for life Ligrone (2019). These types of events, crucial for the selec- +tion mechanism discussed above, are absent in other Gaian models. +In contrast, TNM species interact directly through antagonistic or +symbiotic inter-species couplings, the population varies consider- +ably due to spontaneously occurring ‘quakes’ and there is no rigid +upper bound on the population. Thus by combining elements of the +TNM with elements of earlier Gaian models we can explore how Ga- +ian regulation emerges within a system that allows for more complex +ecosystem dynamics. +With this model we hope to show that the arguments for Entropic +Gaia are robust by demonstrating how they work in a setting where +life needs to interact with and regulate an external environment. At the +same time we will explore how Gaia can inform the search for life in +the universe, in particular how Gaia predicts a larger ‘habitable-zone’. +In section 2 we describe the model and how we add temperature, +which is a combination of abiotic and biotic components. In section 3 +we study the model at constant background temperature to understand +how temperature is regulated and interacts with the spontaneous +‘quakes’ that occur in the TNM. Section 4 discusses the changes to +the habitable-zone in the presence of life and section 5 studies how +life adapts to deteriorating abiotic conditions. Finally we conclude in +section 6. +2 MODEL DESCRIPTION +2.1 The Tangled Nature Model +We start, as in Arthur & Nicholson (2022), with the generalised +Lotka-Volterra model +𝑑𝑁𝑖 +𝑑𝑡 = 𝑁𝑖 𝑓𝑖(�𝑛, 𝑁) +(1) +𝑁𝑖 is the population of species 𝑖, 𝑁 is the total population and 𝑛𝑖 = +𝑁𝑖 +𝑁 . 𝑓𝑖 is a fitness function that depends on the type and abundance +of the other species present through �𝑛 = (𝑛1, 𝑛2, . . . , 𝑛𝐷) and 𝑁. We +can expand 𝑓𝑖 to linear order around the equilibrium at 𝑁 = 0 +𝑑𝑁𝑖 +𝑑𝑡 = 𝑁𝑖 +� +𝑓𝑖(�0, 0) + +∑︁ +𝑗 +𝑑𝑓𝑖 +𝑑𝑛 𝑗 +(�0, 0)𝑛 𝑗 + 𝑑𝑓𝑖 +𝑑𝑁 (�0, 0)𝑁 . . . +� +(2) +The summations here and for the rest of this paper are over all extant +species. The three terms on the right hand side are the basic TNM +variables. +• 𝑟𝑖 ≡ 𝑓𝑖(�0, 0) is the growth rate of species 𝑖 in the absence of any +other species. We set this to zero, meaning that one species’ growth +depends entirely on the other species present. We could add some +species with non-zero growth rates to represent primary producers +but for simplicity and consistency with the rest of the TNM literature +every species has 𝑟𝑖 = 0. +MNRAS 000, 1–18 (222) + +A Gaian Habitable Zone +3 +• 𝐽𝑖 𝑗 ≡ 𝑑 𝑓𝑖 +𝑑𝑛𝑗 (�0, 0) is the inter-species coupling matrix where 𝐽𝑖 𝑗 +is the effect of species 𝑗 on species 𝑖. As usual Christensen et al. +(2002), we set the elements randomly from a symmetric distribution. +Here each element 𝐽𝑖 𝑗 is randomly chosen from a standard normal +product distribution times 𝑐 = 100. The exact functional form of the +distribution is not important, only that it has infinite support Arthur +et al. (2017). +• −𝜇 ≡ +𝑑 𝑓𝑖 +𝑑𝑁 (�0, 0) is the inverse carrying capacity, controlling +how much of the global ‘resource’ is consumed by each individual. +The growth equation now looks like +𝑑𝑁𝑖 +𝑑𝑡 = 𝑁𝑖 �� +� +∑︁ +𝑗 +𝐽𝑖 𝑗𝑛 𝑗 − 𝜇𝑁�� +� += 𝑁𝑖 𝑓 𝑇 𝑁 𝑀 +𝑖 +(3) +In Arthur & Nicholson (2022) we added higher order terms to +the fitness function and argued that these could be interpreted as +species-environment interactions, since their net effect was to modify +the 𝜇 term to create an “effective” carrying capacity. This kind of +‘endogenous’ environment (e.g. roughly analogous to atmospheric +composition or oceanic pH) is in contrast to most Gaian models +which represent the environment through one or more ‘exogenous’ +parameters, which the model agents aim to regulate. Daisyworld is the +paradigmatic example, where black and white daisies spontaneously +regulate a rising global temperature. We want to study this type of +regulation in the TNM framework and only deal with an abiotic +environment so, for simplicity, we do not include the higher order +terms. +While this is a common approach in Gaian modelling it is worth +some consideration. It was shown in Arthur & Nicholson (2022) and +Arthur & Nicholson (2017) that selection in the TNM tends to pro- +duce beneficial endogenous/biotic environments. If we included both +an abiotic and a biotic environment, TNM agents would be subject to +more selective pressure i.e. they would need to avoid degrading the +external parameters (temperature) and internal parameters (∼ pH). In +Arthur & Nicholson (2017) it was noted that environmental selection +isrelatively weak, because whennewspecies occurtheystart with low +populations and therefore minimal impact on the environment. This +must also be the case for an abiotic environment. Ultimately the rela- +tive weighting of each in the fitness function would determine which +environmental parameters are most ‘optimised’. Studying these ef- +fects is interesting but we leave it for future work, focusing here on +understanding the model with a purely exogenous environment. +2.2 Adding Temperature +To add temperature to the TNM we let the global temperature 𝑇 be +the sum of abiotic and biotic components: +𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒 +(4) +𝑇0 is the temperature in the absence of life and𝑇𝑙𝑖 𝑓 𝑒 is the effect of the +extant species in the model on the temperature. Every individual of +species𝑖 has an effect, 𝐻𝑖, on the global temperature. The values of 𝐻𝑖 +will be selected from a normal distribution with mean 0 and standard +deviation 𝜎𝐻 , so species are equally likely to have a warming or +cooling effect. The total effect of life on the temperature is +∑︁ +𝑖 +𝐻𝑖𝑁𝑖 +(5) +We describe how � +𝑖 𝐻𝑖𝑁𝑖 is related to 𝑇𝑙𝑖 𝑓 𝑒 in the next section. +We make the reproduction rate depend on the temperature by +modifying the fitness function to +𝑓 𝑇 𝑁 𝑀 +𝑖 +(𝑇) = +∑︁ +𝑗 +𝐽𝑖 𝑗 +1 + +�𝑇 −𝑇𝑃 +𝜏 +�2 𝑛 𝑗 − 𝜇𝑁 +(6) +𝑇𝑃 is the preferred temperature and 𝜏 is a tolerance parameter. The +functional form is chosen so that at temperatures, 𝑇, far from 𝑇𝑃 +the interaction strength is reduced, for example at 𝑇 = 𝑇𝑃 + 𝜏 the +inter-species interaction strength is halved. The functional form +1 +1+𝑥2 +is chosen for simplicity, any function that applies a smooth and +symmetric temperature ‘window’ would work. We have chosen 𝑇𝑃 +and 𝜏 to be constant for all species and interactions. We could, for +example, make the width different for every inter-species interaction: +𝜏 → 𝜏𝑖 𝑗 and similarly for 𝑇𝑃. In the interest of keeping this work +relatively brief and in line with other work such as the original +Daisyworld model Watson & Lovelock (1983), Flask model Williams +& Lenton (2007) and ExoGaia Nicholson et al. (2018), we use a +constant 𝑇𝑃. By keeping 𝑇𝑃 constant for all species we can focus on +and highlight life’s impact on its environment. If 𝑇𝑃 is kept constant, +then any improvement to a “planet’s” survival rate when including +life-environment interaction can only come from life improving its +environment rather than life simply adapting to it. As this is the part +of Gaia theory that is less well accepted Kirchner (2003) it makes +sense to explore scenarios where this effect isn’t potentially obscured +by species adaptation. +2.3 Running the Model +We solve the growth equation using agent based dynamics. This +means that we generate individual agents whose reproduction rate +is controlled by the fitness function 𝑓 𝑇 𝑁 𝑀 +𝑖 +(𝑇). Each agent is an +individual of some species𝑖 and each agent’s reproduction probability +is given by +𝑝𝑜 𝑓 𝑓 +𝑖 += +1 +1 + 𝑒− 𝑓 𝑇 𝑁 𝑀 +𝑖 +(𝑇 ) +(7) +The basic dynamics of the model are then (see also Arthur et al. +(2017)): +(i) Choose an individual and, with probability 𝑝𝑜 𝑓 𝑓 +𝑖 +, make a +copy of that individual. The copying step is meant to mimic asexual +reproduction. We take the 𝐿 = 20 bit binary representation of the +species-index 𝑖 and copy one bit at a time, with a probability 𝑝𝑚𝑢𝑡 = +0.01 to flip a bit during each copy operation. +(ii) Chose a random individual and kill it with probability 𝑝𝑘𝑖𝑙𝑙 = +0.1 +𝐿 is the genome length, where the value of 20 is standard Christensen +et al. (2002), meaning that the model can generate 2𝐿 ∼ 106 unique +species. A ‘generation’ is the time required to iterate over the basic +reproduction/death loop above 𝑁/𝑝𝑘𝑖𝑙𝑙 times, where this number +is recalculated at the end of each generation. This means in each +generation every individual has had a chance to be selected once +on average for a birth/death process. To update the temperature we +perform the following steps after every generation +• If required, update the abiotic temperature 𝑇0 (see Section 5). +• Update 𝑇𝑙𝑖 𝑓 𝑒 using +𝑇𝑙𝑖 𝑓 𝑒(𝑡) = 𝜆𝑇𝑙𝑖 𝑓 𝑒(𝑡 − 1) + (1 − 𝜆) +∑︁ +𝑖 +𝐻𝑖𝑁𝑖 +(8) +• Set 𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒 +MNRAS 000, 1–18 (222) + +4 +Arthur & Nicholson +Variable +Symbol +Value +Inverse carrying capacity +𝜇 +0.1 +Mutation rate +𝑝𝑚𝑢𝑡 +0.01 +Death rate +𝑝𝑘𝑖𝑙𝑙 +0.1 +Lag parameter +𝜆 +0.9 +Preferred temperature +𝑇𝑃 +100 +Temperature tolerance +𝜏 +2 +Temperature effect +𝜎𝐻 +0.05 +Table 1. A list of all the key parameters in the model and the values we +choose. The model has a large parameter space and the parameters are set +to convenient values used in previous work on the TNM. The qualitative +behaviour of the model is very robust to variations in these parameter values +Christensen et al. (2002); Arthur et al. (2017). +Here 𝑡 is the generation number, the timescale in this model and 𝜆 +is a lag-parameter that stops the temperature from changing instan- +taneously. This mimics the real behaviour of the Earth-system, e.g. +climate models have demonstrated a lag in the response of surface +temperatures over the ocean due to changes in atmospheric 𝐶𝑂2 +Boucher et al. (2012). The model is initialised with 500 individuals +of a randomly chosen species and all averages are taken over 1000 +model runs using different random seeds. +3 CONSTANT TEMPERATURE EXPERIMENTS +First we run the model with constant𝑇0. Figure 1 shows the behaviour +of the population and temperature in one ‘run’ of the model for +104 generations. The basic features of the standard TNM - quasi- +stable states punctuated by sharp transitions - persist Christensen +et al. (2002). The important features of ‘core’ and ‘cloud’ Becker +& Sibani (2014) are are retained as can be seen in Figure 2. The +core species are the only ones with significant population and these +are the primary drivers of the temperature. The cloud species are +mutants with small populations and random positive and negative +effects on the temperature. These two runs show that life can move +the temperature away from 𝑇𝑃 or towards it, the question is what +happens on average, in the long run. +Figures 3 (a) and (b) show the average population and average +temperature for 𝑇0 = 100 = 𝑇𝑃 and 𝑇0 = 105 = 𝑇𝑃 + 2.5𝜏 respec- +tively. For (a) 𝑇0 = 𝑇𝑃 and the temperature fluctuates close to the +abiotic temperature while the population increases logarithmically. +This behaviour, increasing population with constant temperature, in- +dicates that the TNM agents are optimising their mutual interactions, +� +𝑗 𝐽𝑖 𝑗 𝑁 𝑗, as in the standard model, while keeping the temperature +close to 𝑇𝑃. In (b) where 𝑇0 > 𝑇𝑃 we see that the population in- +creases while the temperature decreases. Thus the TNM agents are, +on average, simultaneously optimising their mutual interactions while +improving the temperature. +In Arthur & Nicholson (2022) we discussed Selection by Survival +(SBS) and Sequential Selection with memory (SSM). SBS is just dif- +ferential survival i.e. at late times we see systems with Gaian features +because those are the only ones that could survive that long. SBS is a +good null model, here it would predict that the average temperature +tends towards 𝑇𝑃 because runs that don’t maintain 𝑇𝑃 go extinct, +leaving a small number of surviving runs that happen to operate at +𝑇𝑃. SSM would predict that the punctuations during individual runs +drive the average temperature towards 𝑇𝑃. The numbers in the top +row of Figure 3 (a) and (b) show the proportion of runs which survive +up to that point in the experiment. In (b) for example, at 𝑇0 > 𝑇𝑃 +about 9% of the runs have gone completely extinct (𝑁 = 0) by 105 +generations compared to 3% when 𝑇0 = 𝑇𝑃. This is a relatively small +increase in extinction rate compared to the relatively large decrease +in the scaling factor 1/ +� +1 + +�𝑇0−𝑇𝑃 +𝜏 +�2� +≃ 0.14. +Figure 4 shows the model runs in more detail for 𝑇0 = 105 > 𝑇𝑃. +(a), (b) and (c) demonstrate that the runs can be split into two types: +low temperature, cooling core; and high temperature, heating core. +We will loosely call these ‘Gaian’ and ‘non-Gaian’ respectively. (d) +is the crucial plot. It shows the proportion of surviving runs over +time (dashed line) and the proportion of the surviving runs that have +𝑇 ≤ 𝑇𝑃. Here we see that while some runs do go extinct (SBS) +in the surviving runs the proportion of Gaian states increases. This +means that non-Gaian states transition to Gaian states, leading to +more of them over time. This is exactly as sequential selection with +memory predicts: (non-terminal) resets tend, on average, to improve +conditions for life. We will discuss the exact mechanism in detail +below. +This mechanism, Sequential Selection with Memory (SSM) was +discussed in Arthur & Nicholson (2022) and briefly in Secion 1. +Each model run consists of multiple quasi-equilibria interrupted by +quakes (Figure 1). These quakes completely reset the species which +make up the core. These core species are (by definition) the ones +with large populations which control the model dynamics, in this +case the total population and temperature. As has been discussed +in the TNM literature (especially Becker & Sibani (2014)), quakes +occur spontaneously due to the evolution of a ‘parasite’ that disrupts +the core. A parasite, 𝑎, is any species with significant reproduction +probability that isn’t a member of the core. To have a large probability +to reproduce, the sum of its interactions must be high enough that +its reproduction rate is higher than its death rate. Solving for fitness +gives: +∑︁ +𝑗 +𝐽𝑎 𝑗𝑛 𝑗 +1 + +�𝑇 −𝑇𝑃 +𝜏 +�2 ≥ 𝜇𝑁 + +� +1 − 1 +𝑝𝑘 +� +(9) +Lower total population makes it easier for a parasite to occur by +decreasing the 𝜇𝑁 term. Low total population can occur either due +to weak inter-species interactions in the core or unfavourable tem- +peratures. However because of the smaller number of reproduction +events at low 𝑁, fewer mutants are generated. On the other hand high +populations raise the barrier and increase the number of mutation +events. +Crossing the barrier requires finding a mutant 𝑎 with sufficiently +large, positive interactions with some or all species in the core. +Large values of 𝐽𝑎 𝑗 are rare (for our choice of distribution, expo- +nentially so) and the rate of generating new mutants is low. Con- +sidering each reproduction event as 𝐿 = 20 Bernoulli trials, the +expected number of mutations in a reproduction is given by a Bino- +mial distribution 𝐵(𝐿, 𝑝𝑚𝑢𝑡) with mean 𝐿𝑝𝑚𝑢𝑡 = 0.2 and variance +𝐿𝑝𝑚𝑢𝑡 (1 − 𝑝𝑚𝑢𝑡) ≃ 0.2. Thus the rate of exploration of the genetic +space is quite slow. Ultimately the barrier height is more important +than the increased rate of reproduction and is what explains the trend +of (slowly) increasing population and stability in the TNM. For much +more on this see Becker & Sibani (2014). +Here we have to analyse how the temperature interacts with this +mechanism. Assume we have a case where 𝑇0 > 𝑇𝑃 as in Figure +4. Temperatures far from 𝑇𝑃 make a quake more likely by reducing +the total population and hence the barrier height. When a quake +occurs a new core is selected on the basis of strong inter-species +interactions that allow it to quickly ‘use up’ the carrying capacity. +This new core has an equal chance to be warming or cooling, because +of the symmetry of 𝐻𝑖. If it is warming we stay in a non-Gaian state, +if not we move to a Gaian state. In a Gaian state the barrier can +MNRAS 000, 1–18 (222) + +A Gaian Habitable Zone +5 +Figure 1. The column (a) shows the population (top row) and temperature (bottom row) where the background temperature is 𝑇0 = 𝑇𝑃 = 100. Column (b) shows +the population and temperature where 𝑇0 = 105. The temperature in (a) is above 𝑇0 and 𝑇𝑃 while the temperature in (b) is below both 𝑇0 and 𝑇𝑃. +be significantly higher, leading to a much more stable, long lived +core. In a non-Gaian state the barrier is low, meaning the state will +be relatively short lived, being vulnerable to parasites and to large +population fluctuations which may result in total extinction. As shown +in Figure 4 (d) over time this leads to more and more model runs in +a Gaian state. +To summarise: both mechanisms, SBS and SSM operate. Ga- +ian states have temperatures close to 𝑇𝑃, and thus high populations +which, in this model, makes them more stable. Non-Gaian states are +far from 𝑇𝑃 and have low populations. This makes them vulnerable +to total extinction (SBS) and punctuation which can take a non-Gaian +to a Gaian state (SSM). In this model, for this particular temperature, +SSM is a more important mechanism than SBS, though the ratio can +vary with 𝑇0, as we will explore in the next section. +These ideas can help explain why the Earth today is in a habit- +able state. Since its conception the Gaia hypothesis has been defined +in numerous ways and ranging from a strong hypothesis that self- +regulating feedback loops are an expected property of a life-planet +coupled system, known as ‘probable Gaia’ Lenton & Wilkinson +(2003), to a weaker hypotheses that suggests that while the Earth +MNRAS 000, 1–18 (222) + +(a) +(b) +1400 +1000 +1200 +800- +1000 +800 +600 +N +N +600- +400 +400- +200 +200 +0 +0 +2000 +4000 +6000 +8000 +10000 +0 +2000 +4000 +6000 +8000 +10000 +t (generations) +t (generations) +108 +To +108 +Tp +Tp +106 +106 +104 +104 +Temperature +102 +102 +100 +100 +98 +98 - +96 +96 +94 +94 +0 +2000 +4000 +6000 +8000 +10000 +0 +2000 +4000 +6000 +8000 +10000 +t (generations) +t (generations)6 +Arthur & Nicholson +Figure 2. Model snapshot at 𝑡 = 9000 generations for the runs (a) and (b) from Figure 1. Each node represents a different species, with the size of the node an +indication of species’ population (upper and lower limits are applied to the point sizes for clarity). The colour of the nodes indicates the heating or cooling effect, +𝐻𝑖. The width of the arrows indicates the interaction strength 𝐽𝑖 𝑗𝑛𝑗. Only interactions with core species are shown. In (a) the red (bottom-right) core species +has a strong enough heating effect to overwhelm the cooling effect of the other core species, so this configuration has a net heating effect, as seen in Figure 1(a). +In (b) both core species have a (weak) cooling effect, reducing the temperature, as seen in Figure 1(b). +does have self-regulating feedback loops, these emerged merely by +chance and that Gaia is not an expected feature of a planet hosting +life, known as ‘lucky Gaia’ Watson (2004). As Figure 5 shows, in our +model the fraction of Gaian states is increasing over time. This sug- +gests that for early life starting out on a planet, a large amount of luck +might be needed to initially start off in a Gaian state, but for surviv- +ing runs over time the probability of being in a Gaian state increases. +This would suggest that when observing a biosphere ‘lucky Gaia’ +may be the case for young planets but ‘probable Gaia’ is operating +for older ones. +The experiments in Figure 5 have considered systems with only +internal perturbations, that is, those generated by the biosphere. How- +ever, real planets experience many abiotic perturbations, both rapid +and slower, such as changes in volcanic activity, changes in solar +luminosity or impacts by large objects Covey et al. (1994); Overpeck +& Cole (2006); Goldblatt & Zahnle (2011). Life is thought to have +emerged early on Earth during a time when debris left over from +the formation of the solar system was frequently colliding with the +Earth. Biospheres in a non-Gaian state will be more susceptible than +Gaian biospheres to perturbations and will have a higher risk of going +extinct. This is closely related to the ‘Gaian bottleneck’ hypothesis +Chopra & Lineweaver (2016) that proposes that early on in a planet’s +history, if life emerges it must quickly establish self-regulatory feed- +back loops to stabilise the climate of its planet in order to persist. +If the biosphere fails then life goes extinct, the planet’s abiotic pro- +cesses take over and the planet reverts to an inhospitable state. What +is novel here is the idea that apart from total extinction, a planet can +have a ‘near death experience’ where a mass extinction clears out a +large fraction of the extant species. These mass extinctions are cru- +cial for the exploration of the space of possible ecosystems Arthur +& Sibani (2017) and ultimately lead to the emergence of long-lived +stable states. Population diversity is known to significantly increase +the resilience of ecosystems to perturbations Peterson et al. (1998); +Luck et al. (2003), and additionally yeast Guan et al. (2012) and +bacteria Lambert & Kussell (2014) have been shown to develop in- +creased resilience to environmental stressors if exposed to them in +the past. It is possible that large perturbations that do not eliminate +all life are actually beneficial for evolving Gaia. Indeed, there may +be evidence of this in Earth history, as it is thought that a period of +global glaciation may have triggered the evolution of multi-cellular +life Hoffman et al. (1998); Hedges (2004); Vincent et al. (2004); +Boyle et al. (2007). +4 HABITABLE ZONE EXPERIMENTS +The habitable zone around a star is defined as the distance from a star +where liquid water could exist on the surface of a planet Kasting et al. +(1993). Models demonstrate that the habitable zone is impacted by +several factors, including the age and class of the host star Ramirez & +Kaltenegger (2016), planetary mass Kopparapu et al. (2014), plane- +tary atmospheric composition Pierrehumbert & Gaidos (2011), and +the surface water content of the planet Abe et al. (2011). Additionally +a planet being within the habitable zone doesn’t guarantee habitabil- +ity, as a planet may have more than one possible climate state for +the same stellar and orbital parameters, e.g. a temperate Earth versus +a frozen Earth Goldblatt & Zahnle (2011). For a more extreme ex- +ample, it is thought that Venus and Earth might represent alternate +end states for the same planetary system, with small perturbations +occurring early on in their history influencing their modern day states +Lenardic et al. (2016). +Existing exoplanet surveys and models have identified that rocky +MNRAS 000, 1–18 (222) + +(a) +(b) +0.04 +0.02 +0.00 +-0.02 +-0.04A Gaian Habitable Zone +7 +Figure 3. (a) shows the average (over all surviving model runs) of the population (top row) and temperature (bottom row) where the background temperature +𝑇0 = 100 = 𝑇𝑃. Column (b) shows the population and temperature where 𝑇0 = 105. The numbers next to the vertical dashed lines in the top row are the +proportion of runs which have survived for that number of generations. +planets can exist at a range of distances from their host star Domagal- +Goldman et al. (2016). Thus, it is a natural question to ask about the +stability and persistence of Gaia across a range of background tem- +peratures, some more conducive to life, some less. In this section we +run many experiments where we vary the background temperature +𝑇0 and look at averages over 1000 model histories. To mimic the +idea of a habitable zone with and without biotic influence we com- +pare two versions of the model: one where life cannot influence the +temperature, 𝜎𝐻 = 0, and one where life can influence it 𝜎𝐻 = 0.05. +In Figure 5 we show the fraction of runs which survive for 105 +generations in both scenarios. Perhaps surprisingly, the distributions +are roughly similar. As the background temperature changes, a similar +number of model runs survive for 105 generations whether life can +effect the environment or not. This shows, at least, that species- +environment interactions have little effect on the probability of total +extinction and therefore on the presence or absence of life. However, +as we saw in the previous section, the model runs can be split into +Gaian and non-Gaian states. Figure 5 also shows the proportion of +MNRAS 000, 1–18 (222) + +(a) +(b) +400 +400 +350- +350 +10.972 +i0.909 +300 - +300- +10.98 +10.989 +≥ 250 +≥ 250 +0.9 +i0.992 +10.981 +200 - +10.99 +150 - +150- +100 +100 +102 +103 +104 +105 +102 +103 +104 +105 +t (generations) +t (generations) +108 +108 +Tp +Tp +T +106 +106 +Temperature +104 - +Temperature +104 +102 +102 +100 +100 +98 +98 +102 +103 +104 +105 +102 +103 +104 +105 +t (generations) +t (generations)8 +Arthur & Nicholson +Figure 4. 𝑇0 = 105. (a) shows the temperature at 𝑡 = 105 generations versus population. Colour corresponds to the heating (red) or cooling (blue) effect of the +core. There are clearly two distinct clusters: one with (potentially) high population and low temperature and one with low population and high temperature. (b) +and (c) show histograms of the temperature and population respectively. (d) shows the proportion of surviving runs at each generation as well as the proportion +that have 𝑇 ≤ 𝑇𝑃. +runs that have 𝑁 > 200. The value of 200 is not itself significant, +what is important is the comparison between 𝜎𝐻 = 0 and 𝜎𝐻 = 0.05. +Far from 𝑇𝑃, only the Gaian states can have large populations, in the +other cases the total population is low and life is simply ‘clinging +on’. Importantly for exoplanet astronomy, a small pocket of life that +is clinging on to existence is unlikely to produce a detectable bio- +signature. +Figure 6 shows the population of the model runs as a function of +𝑇0. We see that when 𝜎𝐻 = 0 the total population at 𝑇𝑃 is larger. +At 𝜎𝐻 = 0 the TNM agents are only attempting to optimise inter- +species interactions, not interactions and temperature and thus can +find a better maxima. For example, strongly symbiotic cores may +have a detrimental effect on the temperature which is only relevant +in the 𝜎𝐻 = 0.05 case. However, the population falls rather rapidly +with 𝑇0 at 𝜎𝐻 = 0 compared to the 𝜎𝐻 = 0.05 case. We also see +(from the colour gradient) that at 𝜎𝐻 = 0.05, for 𝑇0 far from 𝑇𝑃 only +MNRAS 000, 1–18 (222) + +(a) +(b) +110.0 +110.0 +107.5 +107.5 +105.0 +105.0 +102.5 +102.5 +100.0 +100.0 +97.5 +97.5 +95.0 +95.0 +92.5 +92.5 +0 +200 +400 +600 +800 +1000 +1200 +1400 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +N +P(T) +(c) +(d) +0.0000 +1.0 +0.0005 +0.0010 +0.8 - +0.0015 +0.6 +0.0020 +0.0025 +0.4 - +0.0030 +0.2 +0.0035 +Fraction Surviving +0.0040 +0.0 +Fractionofsurvivors 200). +those runs which heat or cool as appropriate are capable of having +large populations. Figure 7 demonstrates that the runs split into two +clusters, as also shown in Figure 4, which can be labelled by their +temperature, in combination with Figure 6 this demonstrates that +large population Gaian states may be observed when 𝑇0 is far from +𝑇𝑃. +This simple Gaian model would therefore predicts that if life plays +only a minimal role in shaping its planet and we were looking at an +abiotic habitable zone, that there would be a narrow range of radii +around the host star where we might expect a detectable biospheres. +Outside this narrow range the chance of finding an inhabited planet +drops dramatically. If however life does play a strong role in shaping +and regulating its host planet then we would expect to observe a much +larger habitable zone. In the centre of this zone where conditions are +‘ideal’ large population states, and so therefore potentially detectable +biospheres, will be most probable but as we move towards the edges +of the habitable zone the probability of detectable biospheres will +be much higher than an abiotic habitable zone would predict. Our +MNRAS 000, 1–18 (222) + +OH= O +OH= 0.05 +TemperatureWindow +TemperatureWindow +Proportionnotextinct +Proportionnotextinct +1.2 +FractionofsurvivingrunswithN>2o0 +1.2 - +FractionofsurvivingrunswithN>2o0 +1.0 - +1.0- +0.8 +0.8 - +0.6 +0.6- +0.4 - +0.4 +0.2 +0.2 +0.0 - +0.0 - +80 +90 +100 +110 +120 +80 +90 +100 +110 +120 +To +To10 +Arthur & Nicholson +Figure 6. As Figure 5, 1000 experiments at different 𝑇0 values. The figure shows the population of each of the model runs. Some jitter in the x-direction is +applied to the points for clarity. The left hand shows the case where there is no species environment interaction 𝜎𝐻 = 0 and the right shows 𝜎𝐻 = 0.05 where +the colour of the points reflects a heating (red) or cooling (blue) core. +model suggests that there is a chance to detect biosignatures quite far +from the abiotic habitable zone, provided life can affect the global +temperature. Our model also predicts that looking at planets outside +the abiotic habitable zone will be more informative for testing ideas +of Gaia Theory, since within it we expect to see habitable planets +whether Gaia is operating or not. Our model also demonstrates that +finding a non-Gaian state within the biotic habitable zone is not +incompatible with Gaia theory. Where life can shape its planet there +remains the possibility for it to push its planet towards inhospitable +conditions. +5 INCREASING TEMPERATURE +Geological evidence on Earth suggests that life emerged on our planet +very soon after surface conditions allowed Nisbet & Sleep (2001) +implying that the probability for the emergence of life might be high +for planets with the correct prerequisites, however no alien life has yet +MNRAS 000, 1–18 (222) + +1400 +. +1400 +0 +. +. +.. +· +. +. +1200 +: +1200 +！ +: +: +: +: +1000 +1000 +0 +800 +800 +N +N +600 +. +0 +600 +400 +400 +200 +200 +9 +. +. +0 +0 +80 +90 +100 +110 +120 +80 +90 +100 +110 +120 +To +ToA Gaian Habitable Zone +11 +Figure 7. As Figure 5, showing the temperature in each of the model runs. The colour of the points reflects a heating (red) or cooling (blue) core. Only +𝜎𝐻 = 0.05 is shown, when 𝜎𝐻 = 0, 𝑇 = 𝑇0. +been detected. The Gaian bottleneck hypothesis suggests an answer to +this apparent contradiction and proposes that for newly emerged life +on a young planet, there is a small window of opportunity whereby +life can establish self-regulatory feedback loops to maintain habitable +conditions. If the biosphere succeeds, then planetary habitability can +be maintained for long time spans, however if the biosphere fails, +surface conditions on the planet will rapidly become inhospitable, +causing life to go extinct. This hypothesis is closely tied to ideas of an +inhabitance paradox Goldblatt (2016) - that the long term habitability +of a planet depends directly on whether or not it is inhabited. In this +section we investigate aspects of the inhabitance paradox in the TNM +setting. +The classic Daisyworld experiment studies temperature regulation +by life in the face of increasing solar luminosity. We can perform +a similar experiment by increasing 𝑇0 over the course of the model +runs. Figure 8 shows population and temperature for individual model +runs where the background temperature, 𝑇0, increases linearly from +𝑇𝑖𝑛𝑖𝑡 = 𝑇𝑃 = 100 up to 𝑇0 = 105 over the course of 104 genera- +tions. The key observation is that the actual temperature 𝑇 (bottom +row of Figure 8) increases more slowly than 𝑇0 - meaning that life +is regulating the temperature. The only way the TNM can regulate +without changing the composition of the core is by altering the pop- +ulations of the core species. In Figure 8 we can see the temperature +increase during an equilibrium is slowed by increasing or decreasing +the population, and thus life’s contribution to the total temperature. +Figure 9 shows the configuration of the model agents at a partic- +ular time in the history of the simulation where the core - the group +of species with significant reproduction probability - is stable and +life is adapting to the temperature change. There are two different +cases shown in (a) and (b). In case (a), between roughly 𝑡 = 4000 +and 𝑡 = 10000, the total population is increasing, which has the +effect of slowing the temperature increase. Figure 9 (a) shows that +the cloud (by definition species not in the core) has a roughly equal +number of heating and cooling species, and each of these species has +a small population, thus the cloud (i.e. the majority of species) does +not participate in temperature regulation. Of the 4 species making up +the core, 2 have a cooling effect, one is heating and one is approxi- +mately neutral. The upper left and lower right species happen to have +𝐻𝑖 = −0.047 and 𝐻𝑖 = 0.046 respectively, as well as roughly equal +populations, so their effects cancel out, resulting in a net cooling by +increasing the core population. +Note that in Figure 8 (a) during this period the temperature is +below 𝑇𝑃 = 100. As 𝑇0 increases it will push 𝑇 towards 𝑇𝑃, the +fitness of all species +𝑓𝑖 = +∑︁ +𝑗 +𝐽𝑖 𝑗𝑛 𝑗 +1 + +�𝑇 −𝑇𝑃 +𝜏 +�2 − 𝜇𝑁 +increases and therefore the population increases, which increases the +cooling effect to (partially) offset the abiotic temperature increase. +Figures 8 (b) and 9 (b) shows the opposite case. The core has a net +heating effect and the temperature is above 𝑇𝑃. Increasing 𝑇0 moves +the temperature further from 𝑇𝑃, reducing the fitness and also the +population, therefore reducing the heating effect of life. +This is a regulation mechanism known as ‘rein-control’ where the +temperature of the system can be thought of as being ‘pulled’ in +two different directions by different reins, in this case 𝑇0 and the +heating or cooling effect of life. As all species share the same 𝑇𝑃 +it is the overall heating or cooling impact of the TNM community +that is important for temperature regulation. Looking at the case of +a cooling community first, Figure 8 (a), after 𝑡 ≈ 4000 generations +has 𝑇 < 𝑇𝑃 < 𝑇0. In this case when 𝑇 < 𝑇𝑃, as 𝑇0 increases, this +moves 𝑇 closer to 𝑇𝑃 and boosts the growth rate and hence the size +of the cooling core, slowing the rate of heating. Once 𝑇 ≃ 𝑇𝑃 the +𝑇0 rein is pulling away from 𝑇𝑃, limiting further growth and so the +system stabilises. These feedback loops for an overall cooling TNM +community are shown in Figure 10. +MNRAS 000, 1–18 (222) + +120 +115 +110 - +105 - +← 100 - + S6 +90 - +85 - +80 - +80 +85 +90 +95 +100 +105 +110 +115 +120 +To12 +Arthur & Nicholson +Figure 8. Showing a single model run where background temperature 𝑇0 is increasing over time. The population is shown in the top row and temperature in the +bottom row. The two columns show the two different types of temperature regulation by the core. On the left, after 4000 generations the temperature is regulated +by increasing the population. On the right, between 4000 and 8000 generations, temperature is regulated by decreasing the population. +In Figure 8 (b) by 𝑡 ≈ 2000 the TNM community is overall heating +and 𝑇 > 𝑇0 > 𝑇𝑃. In this scenario any further growth of the com- +munity would increase 𝑇 which would decrease the growth rate. On +the other hand a reduction in population reduces its heating effect, +which partially offsets the increase in 𝑇0 and so the real temperature +𝑇 increases more slowly. Even though 𝑇0 > 𝑇𝑃 the heating TNM +community and 𝑇0 are still ‘pulling’ the temperature in opposite di- +rections as a reduction in the population will cool the environment +which will move 𝑇 closer to 𝑇𝑃. When 𝑇0 > 𝑇𝑃 a heating TNM +community can never achieve a 𝑇 close to 𝑇𝑃. +At 𝑡 ≈ 9000 we see that there is a quake and 𝑇 rapidly drops +below 𝑇𝑃 as the TNM community switches from an overall heating +one to overall cooling. This example demonstrates that a biosphere +in a non-Gaian state can become ‘unstuck’ and transition to a Ga- +ian state if life can cling on for long enough. Twice during Earth’s +history it is thought that the planet was covered in ice from poles to +MNRAS 000, 1–18 (222) + +(a) +(b) +1750 +1400 +1500 +1200 +1250- +1000 +1000 +800 +N +750 +600 +500 +400 +250 +200 +0- +. +2000 +4000 +6000 +8000 +10000 +12000 +2000 +4000 +6000 +8000 +10000 +12000 +t (generations) +t (generations) +108 +108 +IP +106 +106 +104 +104 +emperature +Temperature +102 +102 +100 +100 +98 - +98 - +96 +96 +94 - +94 - +0 +2000 +4000 +6000 +8000 +10000 +12000 +0 +2000 +4000 +6000 +8000 +10000 +12000 +t (generations) +t (generations)A Gaian Habitable Zone +13 +Figure 9. Model snapshot at 𝑡 = 7000 generations for the runs (a) and (b) from Figure 8. (a) The core has an overall cooling effect (b) the core has a heating +effect. +Figure 10. Feedback loops between community population, N, and environment temperature, T, for an overall cooling TNM community. A + symbol (also +indicated with a solid arrow) indicates an increase in the source leads to an increase in the sink, e.g. an increase in population leads to an increase in temperature. +A - symbol (also indicated with a dashed arrow) indicates an increase in the source leads to a decrease in the sink, e.g. an increase in the temperature leads to a +decrease in the total population. A feedback loop with an overall positive sign (determined by multiplying each sign in the loop) indicates a runaway feedback +loop, whereas a feedback look with an overall negative sign indicates a stable feedback loop. Therefore for a cooling TNM community, temperature regulation +occurs below 𝑇𝑃. +equator - known as a Snowball Earth Hoffman et al. (1998). These +Snowball Earth states persisted for millions of years and although +there is evidence that a diversity of life survived these states, a frozen +planet would present fewer niches for life than a thawed planet would +(indeed, on Earth the biodiversity is lowest at the poles Rutherford +et al. (1999)). Such a state could represent a non-Gaian biosphere +clinging on, and both Earth history and our experiments demonstrate +that observing a planet in a non-Gaian state doesn’t mean that it will +always remain so. +5.1 Averages +Again, we are interested in what happens in the long run on average. +Figure 11 shows that for our setup where 𝑇0 > 𝑇𝑃, on average the +temperature is regulated below 𝑇0. Only those communities that have +strong mutually symbiotic interactions and a cooling effect are likely +to survive. If the rate of heating is not too strong (top row) most of +the runs survive and the population grows logarithmically over time +while the proportion of runs at or below𝑇𝑃 falls at a much slower rate +than the increasing background temperature. Since most of the runs +survive we can’t have Selection by Survival, so Sequential Selection +with Memory must be responsible for this behaviour. The 𝑁 versus +MNRAS 000, 1–18 (222) + +N +N ++ +for T < Tp +for T > T.(a) +(b) +0.04 +0.02 +0.00 +-0.02 +-0.0414 +Arthur & Nicholson +Figure 11. Top row is the scenario where we heat from 𝑇0 = 100 = 𝑇𝑝 to 𝑇𝑓 𝑖𝑛 = 105 over the course of 104 generations. The first column is the average +population, the numbers in black are the proportion of runs which have survived, cyan italic shows the proportion of survivors which have 𝑇 ≤ 𝑇𝑃. The second +column shows the average temperature and 𝑇0. The final column shows 𝑁 versus 𝑇 for all the model runs after 104 generations when 𝑇0 = 𝑇𝑓 𝑖𝑛. The second +row is the same as the first but with 𝑇𝑓 𝑖𝑛 = 120. +𝑇 plot in the top row of Figure 11 shows that we still have a split +between model runs with a heating core and a cooling core, where +only those with a cooling core can have large 𝑁. +The bottom row of Figure 11 shows the case where the heating is +much more aggressive with 𝑇𝑓 𝑖𝑛 = 120. With a constant background +𝑇0 = 120 around 27% of model runs survive for 104 generations. +Figure 11 shows that once the temperature goes above ∼ 110 the +runs start to go extinct though a larger proportion, 68%, survive until +𝑡 = 104. Surviving runs are divided into two groups: runs where a +small population is ‘clinging on’ at high 𝑇 and runs where a large, +cooling population can be maintained. +We investigate this further in Figure 12, where we directly compare +runs with an constant background temperature 𝑇0 = 110 = 𝑇𝑃 + 5𝜏 +to runs where the temperature gradually increases to 𝑇0 = 110 over +104 generations. At 104 generations, when both systems experience +the same 𝑇0, the runs which have been heated gradually are doing +better i.e. more of them survive, they have higher populations and +lower temperatures. This simple observation has a few implications. +First it suggests that if life occurs earlier, as soon as conditions are +optimal for it, then it can survive longer and it can have a greater +influence on the long term habitability of its planet. Second it sug- +gests that more realistic models aiming to map out the habitable zone +around a star should consider if the planet has ever been hospitable +for life. In that case planets which would have inhospitable abiotic +parameters, like 𝑇0, at the time of observation may have been able to +maintain habitable temperatures. This phenomena - where life is key +in preserving habitability is known as the inhabitance paradox - that +long term habitability of a planet isn’t possible without life main- +taining habitability Goldblatt (2016). It also ties closely to the Gaian +Bottleneck hypothesis Chopra & Lineweaver (2016) - life emerging +during a window of opportunity can prevent the environment from +degrading, even as 𝑇0 changes. +Finally, Figure 13 studies the effect of the rate of heating by +comparing two scenarios where 𝑇0 is increased from 𝑇𝑃 = 100 +to 𝑇𝑓 𝑖𝑛 = 110 over 104 versus 105 generations. The ‘slow’ heating +scenario could be thought to mimic something like the gradually +increasing solar luminosity while the fast heating scenario is akin +to something like the rapid onset of global glaciation Overpeck & +Cole (2006). Figure 13 shows that, in general, slower heating leads +to more Gaian states. The population is higher and the final temper- +ature is lower. The average population in fact stops increasing in the +fast heating case, as abiotic conditions degrade faster than SSM can +operate, while the slow heating case shows a continuously increas- +ing population up to 105 generations. Thus, if SSM is to operate +the larger the separation between abiotic and biotic timescales (e.g. +geologic versus evolutionary) then the more likely we are to observe +a Gaia. +MNRAS 000, 1–18 (222) + +t= 104 +300 +107 +To +110.0 +106 +280 +Tp +(T) +107.5 +:0.47 +10.43 +105 +260 +105.0 +104 - +10.99 +10.98 +240 +10.50 +102.5 +M +220- +10.99 +102 +100.0 +101 +97.5 +180 +100 +95.0 +66 +92.5 +160 +98 +102 +103 +104 +2000 +4000 +6000 +8000 +10000 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +t (generations) +N +t (generations) +t= 104 +120 +280 - +120 +(T) +10.45 +260 +115 +115 +10.99 +10.48 +240 - +ature +110 +10.99 +220 - +105 +200 +:0.24 +105 - +10.68 +100 +180 - +160 - +100 +95 +102 +103 +104 +2000 +4000 +6000 +8000 +10000 +0 +200 +400 +600 +800 +1000 +1200 +1400 +t (generations) +t (generations) +NA Gaian Habitable Zone +15 +Figure 12. Comparing increasing 𝑇0 = 100 → 110 to constant 𝑇0 = 110. Top left is average population, the numbers show the number of surviving runs at each +time-step. Top right shows average temperature and 𝑇0. Bottom left shows all of the runs (see Figure 4) for the increasing temperature case, and bottom right +shows the runs for the constant temperature case. +6 CONCLUSIONS +Models such as the one described here help us to understand how +planetary regulation arises from ‘selfish’ individuals. Gaia is a prime +example of an emergent system - one where the whole has properties +its parts do not. However Gaia was first discussed some years before +emergence and complexity thinking were common. Lovelock and +others discussing Gaia at the macro level, for example talking about +her health with the notion of Geophysiology Lovelock (1989), have +been harshly criticised. There have been two primary criticisms: the +first argues that Gaia is simply a metaphor and not a scientific theory +Kirchner (1989) and the second argues that episodes from earth +history where life generates hostile conditions is strong evidence +against Gaia Ward (2009). We believe the notion of Entropic Gaia +Arthur & Nicholson (2022) and our discussion of selection principles +answers both of these criticisms. +First to address the charge that Gaia is ‘just a metaphor’ it is in- +structive to discuss some other emergent systems. A gas is ‘just’ a +collection of individual atoms. However emergent properties, like +MNRAS 000, 1–18 (222) + +300 +112 +Tinit = 100 +Tinit = 100 +Tinit = 110 +Tinit = 110 +:0.99 +110 +250 - +10.95 +10.99 +108- +200 +10.81 +106 +≤ 150 +10.94 +104 +100 +10.99 +102 +50 +100 +98 +102 +103 +104 +2000 +4000 +6000 +8000 +10000 +t (generations) +t (generations) +Tinit = 100, t = 104 +Tinit = 110, t = 104 +115.0 +112.5 +112.5 +110.0 +110.0 +107.5 +107.5 +105.0- +105.0 +102.5 +102.5 +100.0- +100.0 +97.5 +97.5 +95.0 - +95.0 +92.5 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +o +200 +400 +600 +800 +1000 +1200 +N +N16 +Arthur & Nicholson +Figure 13. Increasing 𝑇0 = 100 → 110 over 104 generations and 105 generations. Top left is average population, the numbers show the fraction of surviving +runs at each time-step. Top right shows average temperature and 𝑇0 (note the log scale makes the linearly increasing 𝑇0 look exponential). The numbers give the +average temperature at the end of the experiment. Bottom left shows all of the runs at the end of the fast heating experiment, and bottom right shows the runs in +the slow heating scenario. +pressure and temperature, are not features of individual atoms but +are still very much real. An organism is a ‘just’ a system of chemical +reactions. However biology is not just applied chemistry, it is legiti- +mate and useful to reason about cells. The economy is a phenomenon +that emerges out of the production and consumption patterns of mil- +lions of individuals. Depression, recession, asset bubbles and so on +are properties of the whole system that have real explanatory power. +Of course we can have incorrect theories about gases, cells or eco- +nomics, but these do not make it illegitimate to reason about whole +systems. When talking about life at planetary scale, we talk about +something called ‘Gaia’. This is a metaphor in the same sense as +an organism or an economy, a metaphor that admits rigorous micro +foundations and which can be very productive for understanding a +system or a collection of systems. In the context of bio-signature +detection, where we may have potentially very many ‘Gaias’ and +MNRAS 000, 1–18 (222) + +400 +112 +tmax =10000 +tmax = 10000 +tmax=100000 +tmax = 100000 +110 +350 +10.88 +108 +300 +10.98 +105.55 +M +10.98 +10.99 +104- +103/79 +250 - +10.99 +i0.95 +102 +200寸 +100 +150 +98 +102 +103 +104 +105 +102 +103 +104 +105 +t (generations) +t (generations) +To=110,t=104 +To=110, t= 105 +115.0 +115 +112.5 +110.0- +110 +107.5 +105.0 +105 +102.5 +100 +100.0 +97.5 +95 +95.0 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +o +500 +1000 +1500 +2000 +N +NA Gaian Habitable Zone +17 +very limited information about the processes going on inside them, +a holistic theory is crucial. +The second class of criticisms is directly addressed by our idea that +Gaia arises due to a selection principle operating on species networks. +To briefly re-iterate - sequential selection posits a type of punctuated +equilibrium Eldridge & Gould (1972), characterised by stable periods +interrupted by catastrophes. Models of co-evolution such as the TNM +and others (e.g. Kauffman & Weinberger (1989)) show exactly this +kind of behaviour. Entropic Gaia is the argument that these stable +periods get longer over time. In the TNM this is for the simple reason +that each punctuation is not a complete reset, the next stable period +emerges from the debris of the previous equilibrium. The species +networks that can establish themselves must have high population +growth rates so they saturate the carrying capacity, while also not +self limiting. High populations mean more diversity, which means +even more ‘debris’ during the next reset. In this view, periods of +disregulation are not evidence against Gaia, they are an integral part +of how she arises. +To show the use of such a theory in this work we have, within a +concrete and fairly general modelling framework, investigated some +pressing questions of astro-biology through Gaia theory. In section 3 +we studied the effect of life on the ability of a planet to sustain life in +suboptimal abiotic conditions. This leads us to propose the idea of the +Gaian habitable zone versus the standard abiotic habitable zone. +Our results predict that Gaia extends the habitable zone around a star +while making the abiotic habitable zone slightly less hospitable. This +has a straightforward and testable implication - search for life outside +the abiotic habitable zone as a signature of Gaia. +In section 5 we study the effect of a deteriorating abiotic envi- +ronment to address the idea of the Gaian bottleneck. Life’s chances +of long term survival, and the emergence of Gaia, are both more +likely if life can ‘catch’ a window of high habitability (in this model +where 𝑇0 = 𝑇𝑃). Life can then, on average, maintain better condi- +tions. Again this has implications in the search for life - planets which +were once inside but are currently outside the abiotic habitable zone +may host life. Again Gaia expands the boundary of habitability and +inhabitance. +Both Selection by Survival (SBS) and Sequential Selection with +Memory (SSM) play a role in determining the likelihood of a Gaian +planet. Nearer the centre of the abiotic habitable zone, SSM is the +main mechanism for generating Gaias and towards the edges SBS +becomes more important. Finding a non-Gaian planet at the center +of the abiotic habitable zone is not incompatible with Gaia theory. +If life can strongly influence its environment it can degrade it. The +results of this model suggest that if life can cling on, and abiotic +conditions do not degrade too much, then the planet can become +‘unstuck’ through the evolution of species which regulate the tem- +perature. To map out the Gaian habitable zone around a particular +star, or class of star, will require fusing detailed abiotic models with +models of biogeochemistry. Some steps in this direction were taken +in Nicholson et al. (2022), where the fine-details, such as lifespan +or maintenance energy requirements of the biosphere were shown +not to affect the general conclusion about life’s effect on potential +bio-signatures. If this is the case generally, and this framework can +be expanded to cover a range of biotic scenarios, then we may be able +to produce detailed predictions of the Gaian habitable zone without +needing to know the population-level details of any alien life. Iden- +tifying potential metabolic pathways and limiting abiotic factors on +microbial growth (e.g. resource limitation) would be sufficent for +robust biosignauture predictions. +In summary, we propose a statistical theory of planetary habitabil- +ity, with strong and testable implications on the search for alien life. +Our model, as well as Earth history, teaches us that a Gaian planet +can emerge from periods of disregulation and low habitability. Ulti- +mately, this suggests a wider range of habitable and inhabited planets +than abiotic models would predict. +ACKNOWLEDGEMENTS +This work was supported by a Leverhulme Trust research project +grant [RPG-2020-82]. +DATA AVAILABILITY +The code is available on request from the authors. +REFERENCES +Abe Y., Abe-Ouchi A., Sleep N. H., Zahnle K. J., 2011, Astrobiology, 11, 443 +Amundsen D. 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J., 2020, Nature communications, 11, 1 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–18 (222) + diff --git a/49A0T4oBgHgl3EQfNv86/content/tmp_files/load_file.txt b/49A0T4oBgHgl3EQfNv86/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..753ef903f381d518f0eb73ecd0a86392daef3c6e --- /dev/null +++ b/49A0T4oBgHgl3EQfNv86/content/tmp_files/load_file.txt @@ -0,0 +1,1007 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf,len=1006 +page_content='MNRAS 000, 1–18 (222) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='0 A Gaian Habitable Zone Rudy Arthur,1★ Arwen Nicholson,2† 1University of Exeter, Department of Computer Science 2University of Exeter, Department of Physics and Astronomy Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' in original form ZZZ ABSTRACT When searching for inhabited exoplanets, understanding the boundaries of the habitable zone around the parent star is key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If life can strongly influence its global environment, then we would expect the boundaries of the habitable zone to be influenced by the presence of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here using a simple abstract model of ‘tangled-ecology’ where life can influence a global parameter, labelled as temperature, we investigate the boundaries of the habitable zone of our model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As with other models of life-climate interactions, the species act to regulate the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' However, the system can also experience ‘punctuations’, where the system’s state jumps between different equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Despite this, an ensemble of systems still tends to sustain or even improve conditions for life on average, a feature we call Entropic Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The mechanism behind this is sequential selection with memory which is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' With this modelling framework we investigate questions about how Gaia can affect and ultimately extend the habitable zone to what we call the Gaian habitable zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This generates concrete predictions for the size of the habitable zone around stars, suggests directions for future work on the simulation of exoplanets and provides insight into the Gaian bottleneck hypothesis and the habitability/inhabitance paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Key words: Gaia – Habitable Zone – Biosignatures 1 INTRODUCTION The Gaia hypothesis is that life influences the Earth’s feedback mech- anisms to form a self-regulating system, and therefore life can help maintain habitable conditions on its host planet Lovelock & Mar- gulis (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Distinct from the biosphere Huggett (1999), Gaia is the whole life-earth system, considered as a single entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The impor- tance of life’s interactions with the non-living environment are now common sense, and the discipline of Earth System Science Lenton & Watson (2013) studies the various feedback loops that constitute ‘Gaia’s body’ Volk (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Gaia theory itself takes a very broad perspective, aiming to describe life at a planetary scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Gaia theory asks questions like: Is Gaia inevitable on a planet that hosts life, or is it due to chance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' What mechanisms can create a long lived Gaian system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' How will we detect other ‘Gaias’ beyond our solar system, where direct planetary exploration is not an option?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The astrophysical point of view was crucial in the early development of Gaia, with the search for life on Mars providing the initial inspiration for the Gaia hypothesis Lovelock (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' When looking at Earth from afar, Gaia is what we see and the search for habitable or inhabited exoplanets is the search for other Gaias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Methods for exoplanet detection have developed considerably since Gaia was first proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' New telescopes, such as the James Webb Space telescope and the Extremely Large Telescope (currently under construction), and future missions, such as the Large Ultravi- olet Optical Infrared Surveyor, mean that searching for signs of alien ★ E-mail: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='Arthur@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='uk † E-mail: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='Nicholson@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='uk life will be possible within the coming decades Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' While robotic missions to potentially habitable exoplanets remain unfeasible, evidence for alien life will only be ob- servable for exoplanets that have been dramatically shaped by their biospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Exoplanets with newly emerging life, or those with the remnants of a once-thriving biosphere that has since collapsed, will be unlikely to produce a remotely observable signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Answering the key questions of Gaia theory not only informs how we think about the history of life on Earth, but can form the theoretical foundation for the study of life in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Catling et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Catling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018) proposed a framework for as- sessing potential biosignatures using a probabilistic approach that combines observations of the candidate planet and host star with models of the possible abiotic and biotic planetary processes to de- termine the probability of the planet being inhabited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' With a great diversity of exoplanets being found, any potential biosignature must be considered within the context of its host planet Seager (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Claudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Kiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Krissansen-Totton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Detailed abiotic models of exoplan- ets are being developed for a wide range of detected planets, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Amundsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Boutle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Collins (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Fauchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2021), and sophisticated models of biogeochemistry exist for different points in Earth’s history, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Kharecha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Daines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Lenton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Zakem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Detailed and realistic modelling of life on other planets is im- portant, however this paper will take a broader view that aims to understand the generic mechanisms that lead to Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We build on recent work Ford Doolittle (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Lenton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Arthur & Nicholson (2022) on Gaian selection principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We argued in © 222 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='02150v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='EP] 5 Jan 2023 2 Arthur & Nicholson Arthur & Nicholson (2022) that some approaches to Gaian selec- tion Ford Doolittle (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Lenton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018a) lead to anthropic reasoning - we see Gaia because if we didn’t we wouldn’t exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' An- thropic reasoning is controversial, with its opponents arguing that it unfalsifiable with limited (if any) predictive power Smolin (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The coming era of exoplanet astronomy gives new context and pur- pose to these discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If our aim is for Gaia theory to inform our search for life in the universe, then anthropic arguments are clearly inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In Arthur & Nicholson (2022) we argue for ‘Entropic Gaia’ - that the emergence of Gaia is a statistical tendency for planets that host life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This means that life history on a single planet can be chaotic and have periods of stability and collapse, however there is a trend towards increasing biomass, stability, habitability and other Gaian features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Any single planetary history for a life-bearing planet, such as Earth, is likely to follow a (bumpy) trajectory towards Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The micro-mechanism leading to this behaviour was argued to be ‘Se- quential Selection with Memory’ or an ‘entropic ratchet’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In brief, this mechanism starts from the observation that coupled life-environment systems move between regulation and disregulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' By definition, disregulating systems quickly destroy themselves while regulating systems persist, this is sequential selection Lenton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In models of ecology and Gaia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Becker & Sibani (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Hard- ing (1999)) increasing diversity and complexity is associated with increasing stability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' More diverse ecosystems can generate more novel species through mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Thus after every ecosystem collapse (caused by internal or external factors) if a new ecosystem arises it is likely to be more diverse, having started with a greater ‘pool’ of species, and therefore also more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Sequential selection with memory describes a sequence of distinct stable states that tends to get ‘more Gaian’ over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This mechanism was originally proposed in the framework of the Tangled Nature Model (TNM) Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Originally designed to study co-evolution, we demonstrated in Arthur & Nichol- son (2017) that the TNM is closely related to the generalised Lotka- Volterra model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The TNM is based on the idea that the growth rate of a species is given by a fitness function that depends on the other species present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Any model making this assumption will look like the TNM close to equilibrium Arthur & Nicholson (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' By studying the model with agent based dynamics we can incorporate mutation, giving us a very flexible, robust and general model of evolutionary ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Since the TNM is quite general, conclusions drawn in this framework are likely to have general applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Artificial life modelling has been used extensively to study Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The original Daisy World Watson & Lovelock (1983) led to a large number of variants Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2008) and there are a variety of other models such as the Guild Model Downing & Zvirinsky (1999), Greenhouse World Worden (2010), Flask Model Williams & Lenton (2007) and Exo-Gaia Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018) to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We have previously discussed Gaian models based on the TNM in Arthur & Nicholson (2017, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here we propose a new variant on the TNM that is more similar to other Gaian models with a very simple abiotic (non-living) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' While previous Gaian models have included mutation (such as the Flask model and ExoGaia) the complexity of the biosphere in these models has been limited and different species within the mod- els only impact one another via the shared environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' via 1 See Landi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018) for a thorough discussion of the relationship be- tween ecosystem complexity and stability, though most of these models don’t consider coupling to the external environment resource competition or via global parameters such as temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' When we look at life on Earth it is clear that different species can have a large impact on each other beyond resource competition or changing global parameters like temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' For example, there are complex interactions between worms, plants and soil that change the structure, chemistry, water retention and other properties of soil for the benefit of many species Le Bayon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These kinds of symbiotic (and also antagonistic) interactions are usually missing in Gaian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We also observe that throughout Earth history there have been dramatic and spontaneous changes in the diversity and complexity of the biosphere, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' the Great Oxidation Event which allowed for aerobic respiration to become an important energy source for life Ligrone (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These types of events, crucial for the selec- tion mechanism discussed above, are absent in other Gaian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In contrast, TNM species interact directly through antagonistic or symbiotic inter-species couplings, the population varies consider- ably due to spontaneously occurring ‘quakes’ and there is no rigid upper bound on the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Thus by combining elements of the TNM with elements of earlier Gaian models we can explore how Ga- ian regulation emerges within a system that allows for more complex ecosystem dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' With this model we hope to show that the arguments for Entropic Gaia are robust by demonstrating how they work in a setting where life needs to interact with and regulate an external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' At the same time we will explore how Gaia can inform the search for life in the universe, in particular how Gaia predicts a larger ‘habitable-zone’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In section 2 we describe the model and how we add temperature, which is a combination of abiotic and biotic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In section 3 we study the model at constant background temperature to understand how temperature is regulated and interacts with the spontaneous ‘quakes’ that occur in the TNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Section 4 discusses the changes to the habitable-zone in the presence of life and section 5 studies how life adapts to deteriorating abiotic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Finally we conclude in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 2 MODEL DESCRIPTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='1 The Tangled Nature Model We start, as in Arthur & Nicholson (2022), with the generalised Lotka-Volterra model 𝑑𝑁𝑖 𝑑𝑡 = 𝑁𝑖 𝑓𝑖(�𝑛, 𝑁) (1) 𝑁𝑖 is the population of species 𝑖, 𝑁 is the total population and 𝑛𝑖 = 𝑁𝑖 𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 𝑓𝑖 is a fitness function that depends on the type and abundance of the other species present through �𝑛 = (𝑛1, 𝑛2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' , 𝑛𝐷) and 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We can expand 𝑓𝑖 to linear order around the equilibrium at 𝑁 = 0 𝑑𝑁𝑖 𝑑𝑡 = 𝑁𝑖 � 𝑓𝑖(�0, 0) + ∑︁ 𝑗 𝑑𝑓𝑖 𝑑𝑛 𝑗 (�0, 0)𝑛 𝑗 + 𝑑𝑓𝑖 𝑑𝑁 (�0, 0)𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' � (2) The summations here and for the rest of this paper are over all extant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The three terms on the right hand side are the basic TNM variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 𝑟𝑖 ≡ 𝑓𝑖(�0, 0) is the growth rate of species 𝑖 in the absence of any other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We set this to zero, meaning that one species’ growth depends entirely on the other species present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We could add some species with non-zero growth rates to represent primary producers but for simplicity and consistency with the rest of the TNM literature every species has 𝑟𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' MNRAS 000, 1–18 (222) A Gaian Habitable Zone 3 𝐽𝑖 𝑗 ≡ 𝑑 𝑓𝑖 𝑑𝑛𝑗 (�0, 0) is the inter-species coupling matrix where 𝐽𝑖 𝑗 is the effect of species 𝑗 on species 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As usual Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2002), we set the elements randomly from a symmetric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here each element 𝐽𝑖 𝑗 is randomly chosen from a standard normal product distribution times 𝑐 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The exact functional form of the distribution is not important, only that it has infinite support Arthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' −𝜇 ≡ 𝑑 𝑓𝑖 𝑑𝑁 (�0, 0) is the inverse carrying capacity, controlling how much of the global ‘resource’ is consumed by each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The growth equation now looks like 𝑑𝑁𝑖 𝑑𝑡 = 𝑁𝑖 �� � ∑︁ 𝑗 𝐽𝑖 𝑗𝑛 𝑗 − 𝜇𝑁�� � = 𝑁𝑖 𝑓 𝑇 𝑁 𝑀 𝑖 (3) In Arthur & Nicholson (2022) we added higher order terms to the fitness function and argued that these could be interpreted as species-environment interactions, since their net effect was to modify the 𝜇 term to create an “effective” carrying capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This kind of ‘endogenous’ environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' roughly analogous to atmospheric composition or oceanic pH) is in contrast to most Gaian models which represent the environment through one or more ‘exogenous’ parameters, which the model agents aim to regulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Daisyworld is the paradigmatic example, where black and white daisies spontaneously regulate a rising global temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We want to study this type of regulation in the TNM framework and only deal with an abiotic environment so, for simplicity, we do not include the higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' While this is a common approach in Gaian modelling it is worth some consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' It was shown in Arthur & Nicholson (2022) and Arthur & Nicholson (2017) that selection in the TNM tends to pro- duce beneficial endogenous/biotic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If we included both an abiotic and a biotic environment, TNM agents would be subject to more selective pressure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' they would need to avoid degrading the external parameters (temperature) and internal parameters (∼ pH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In Arthur & Nicholson (2017) it was noted that environmental selection isrelatively weak, because whennewspecies occurtheystart with low populations and therefore minimal impact on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This must also be the case for an abiotic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Ultimately the rela- tive weighting of each in the fitness function would determine which environmental parameters are most ‘optimised’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Studying these ef- fects is interesting but we leave it for future work, focusing here on understanding the model with a purely exogenous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='2 Adding Temperature To add temperature to the TNM we let the global temperature 𝑇 be the sum of abiotic and biotic components: 𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒 (4) 𝑇0 is the temperature in the absence of life and𝑇𝑙𝑖 𝑓 𝑒 is the effect of the extant species in the model on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Every individual of species𝑖 has an effect, 𝐻𝑖, on the global temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The values of 𝐻𝑖 will be selected from a normal distribution with mean 0 and standard deviation 𝜎𝐻 , so species are equally likely to have a warming or cooling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The total effect of life on the temperature is ∑︁ 𝑖 𝐻𝑖𝑁𝑖 (5) We describe how � 𝑖 𝐻𝑖𝑁𝑖 is related to 𝑇𝑙𝑖 𝑓 𝑒 in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We make the reproduction rate depend on the temperature by modifying the fitness function to 𝑓 𝑇 𝑁 𝑀 𝑖 (𝑇) = ∑︁ 𝑗 𝐽𝑖 𝑗 1 + �𝑇 −𝑇𝑃 𝜏 �2 𝑛 𝑗 − 𝜇𝑁 (6) 𝑇𝑃 is the preferred temperature and 𝜏 is a tolerance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The functional form is chosen so that at temperatures, 𝑇, far from 𝑇𝑃 the interaction strength is reduced, for example at 𝑇 = 𝑇𝑃 + 𝜏 the inter-species interaction strength is halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The functional form 1 1+𝑥2 is chosen for simplicity, any function that applies a smooth and symmetric temperature ‘window’ would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We have chosen 𝑇𝑃 and 𝜏 to be constant for all species and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We could, for example, make the width different for every inter-species interaction: 𝜏 → 𝜏𝑖 𝑗 and similarly for 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In the interest of keeping this work relatively brief and in line with other work such as the original Daisyworld model Watson & Lovelock (1983), Flask model Williams & Lenton (2007) and ExoGaia Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2018), we use a constant 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' By keeping 𝑇𝑃 constant for all species we can focus on and highlight life’s impact on its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If 𝑇𝑃 is kept constant, then any improvement to a “planet’s” survival rate when including life-environment interaction can only come from life improving its environment rather than life simply adapting to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As this is the part of Gaia theory that is less well accepted Kirchner (2003) it makes sense to explore scenarios where this effect isn’t potentially obscured by species adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='3 Running the Model We solve the growth equation using agent based dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This means that we generate individual agents whose reproduction rate is controlled by the fitness function 𝑓 𝑇 𝑁 𝑀 𝑖 (𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Each agent is an individual of some species𝑖 and each agent’s reproduction probability is given by 𝑝𝑜 𝑓 𝑓 𝑖 = 1 1 + 𝑒− 𝑓 𝑇 𝑁 𝑀 𝑖 (𝑇 ) (7) The basic dynamics of the model are then (see also Arthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017)): (i) Choose an individual and, with probability 𝑝𝑜 𝑓 𝑓 𝑖 , make a copy of that individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The copying step is meant to mimic asexual reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We take the 𝐿 = 20 bit binary representation of the species-index 𝑖 and copy one bit at a time, with a probability 𝑝𝑚𝑢𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='01 to flip a bit during each copy operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (ii) Chose a random individual and kill it with probability 𝑝𝑘𝑖𝑙𝑙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='1 𝐿 is the genome length, where the value of 20 is standard Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2002), meaning that the model can generate 2𝐿 ∼ 106 unique species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' A ‘generation’ is the time required to iterate over the basic reproduction/death loop above 𝑁/𝑝𝑘𝑖𝑙𝑙 times, where this number is recalculated at the end of each generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This means in each generation every individual has had a chance to be selected once on average for a birth/death process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' To update the temperature we perform the following steps after every generation If required, update the abiotic temperature 𝑇0 (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Update 𝑇𝑙𝑖 𝑓 𝑒 using 𝑇𝑙𝑖 𝑓 𝑒(𝑡) = 𝜆𝑇𝑙𝑖 𝑓 𝑒(𝑡 − 1) + (1 − 𝜆) ∑︁ 𝑖 𝐻𝑖𝑁𝑖 (8) Set 𝑇 = 𝑇0 + 𝑇𝑙𝑖 𝑓 𝑒 MNRAS 000, 1–18 (222) 4 Arthur & Nicholson Variable Symbol Value Inverse carrying capacity 𝜇 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='1 Mutation rate 𝑝𝑚𝑢𝑡 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='01 Death rate 𝑝𝑘𝑖𝑙𝑙 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='1 Lag parameter 𝜆 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='9 Preferred temperature 𝑇𝑃 100 Temperature tolerance 𝜏 2 Temperature effect 𝜎𝐻 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' A list of all the key parameters in the model and the values we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The model has a large parameter space and the parameters are set to convenient values used in previous work on the TNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The qualitative behaviour of the model is very robust to variations in these parameter values Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Arthur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here 𝑡 is the generation number, the timescale in this model and 𝜆 is a lag-parameter that stops the temperature from changing instan- taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This mimics the real behaviour of the Earth-system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' climate models have demonstrated a lag in the response of surface temperatures over the ocean due to changes in atmospheric 𝐶𝑂2 Boucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The model is initialised with 500 individuals of a randomly chosen species and all averages are taken over 1000 model runs using different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 3 CONSTANT TEMPERATURE EXPERIMENTS First we run the model with constant𝑇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Figure 1 shows the behaviour of the population and temperature in one ‘run’ of the model for 104 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The basic features of the standard TNM - quasi- stable states punctuated by sharp transitions - persist Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The important features of ‘core’ and ‘cloud’ Becker & Sibani (2014) are are retained as can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The core species are the only ones with significant population and these are the primary drivers of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The cloud species are mutants with small populations and random positive and negative effects on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These two runs show that life can move the temperature away from 𝑇𝑃 or towards it, the question is what happens on average, in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Figures 3 (a) and (b) show the average population and average temperature for 𝑇0 = 100 = 𝑇𝑃 and 𝑇0 = 105 = 𝑇𝑃 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='5𝜏 respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' For (a) 𝑇0 = 𝑇𝑃 and the temperature fluctuates close to the abiotic temperature while the population increases logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This behaviour, increasing population with constant temperature, in- dicates that the TNM agents are optimising their mutual interactions, � 𝑗 𝐽𝑖 𝑗 𝑁 𝑗, as in the standard model, while keeping the temperature close to 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In (b) where 𝑇0 > 𝑇𝑃 we see that the population in- creases while the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Thus the TNM agents are, on average, simultaneously optimising their mutual interactions while improving the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In Arthur & Nicholson (2022) we discussed Selection by Survival (SBS) and Sequential Selection with memory (SSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' SBS is just dif- ferential survival i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' at late times we see systems with Gaian features because those are the only ones that could survive that long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' SBS is a good null model, here it would predict that the average temperature tends towards 𝑇𝑃 because runs that don’t maintain 𝑇𝑃 go extinct, leaving a small number of surviving runs that happen to operate at 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' SSM would predict that the punctuations during individual runs drive the average temperature towards 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The numbers in the top row of Figure 3 (a) and (b) show the proportion of runs which survive up to that point in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In (b) for example, at 𝑇0 > 𝑇𝑃 about 9% of the runs have gone completely extinct (𝑁 = 0) by 105 generations compared to 3% when 𝑇0 = 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This is a relatively small increase in extinction rate compared to the relatively large decrease in the scaling factor 1/ � 1 + �𝑇0−𝑇𝑃 𝜏 �2� ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Figure 4 shows the model runs in more detail for 𝑇0 = 105 > 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (a), (b) and (c) demonstrate that the runs can be split into two types: low temperature, cooling core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' and high temperature, heating core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We will loosely call these ‘Gaian’ and ‘non-Gaian’ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (d) is the crucial plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' It shows the proportion of surviving runs over time (dashed line) and the proportion of the surviving runs that have 𝑇 ≤ 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here we see that while some runs do go extinct (SBS) in the surviving runs the proportion of Gaian states increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This means that non-Gaian states transition to Gaian states, leading to more of them over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This is exactly as sequential selection with memory predicts: (non-terminal) resets tend, on average, to improve conditions for life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We will discuss the exact mechanism in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This mechanism, Sequential Selection with Memory (SSM) was discussed in Arthur & Nicholson (2022) and briefly in Secion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Each model run consists of multiple quasi-equilibria interrupted by quakes (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These quakes completely reset the species which make up the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These core species are (by definition) the ones with large populations which control the model dynamics, in this case the total population and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As has been discussed in the TNM literature (especially Becker & Sibani (2014)), quakes occur spontaneously due to the evolution of a ‘parasite’ that disrupts the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' A parasite, 𝑎, is any species with significant reproduction probability that isn’t a member of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' To have a large probability to reproduce, the sum of its interactions must be high enough that its reproduction rate is higher than its death rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Solving for fitness gives: ∑︁ 𝑗 𝐽𝑎 𝑗𝑛 𝑗 1 + �𝑇 −𝑇𝑃 𝜏 �2 ≥ 𝜇𝑁 + � 1 − 1 𝑝𝑘 � (9) Lower total population makes it easier for a parasite to occur by decreasing the 𝜇𝑁 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Low total population can occur either due to weak inter-species interactions in the core or unfavourable tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' However because of the smaller number of reproduction events at low 𝑁, fewer mutants are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' On the other hand high populations raise the barrier and increase the number of mutation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Crossing the barrier requires finding a mutant 𝑎 with sufficiently large, positive interactions with some or all species in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Large values of 𝐽𝑎 𝑗 are rare (for our choice of distribution, expo- nentially so) and the rate of generating new mutants is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Con- sidering each reproduction event as 𝐿 = 20 Bernoulli trials, the expected number of mutations in a reproduction is given by a Bino- mial distribution 𝐵(𝐿, 𝑝𝑚𝑢𝑡) with mean 𝐿𝑝𝑚𝑢𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='2 and variance 𝐿𝑝𝑚𝑢𝑡 (1 − 𝑝𝑚𝑢𝑡) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Thus the rate of exploration of the genetic space is quite slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Ultimately the barrier height is more important than the increased rate of reproduction and is what explains the trend of (slowly) increasing population and stability in the TNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' For much more on this see Becker & Sibani (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Here we have to analyse how the temperature interacts with this mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Assume we have a case where 𝑇0 > 𝑇𝑃 as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Temperatures far from 𝑇𝑃 make a quake more likely by reducing the total population and hence the barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' When a quake occurs a new core is selected on the basis of strong inter-species interactions that allow it to quickly ‘use up’ the carrying capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This new core has an equal chance to be warming or cooling, because of the symmetry of 𝐻𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If it is warming we stay in a non-Gaian state, if not we move to a Gaian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In a Gaian state the barrier can MNRAS 000, 1–18 (222) A Gaian Habitable Zone 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The column (a) shows the population (top row) and temperature (bottom row) where the background temperature is 𝑇0 = 𝑇𝑃 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Column (b) shows the population and temperature where 𝑇0 = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The temperature in (a) is above 𝑇0 and 𝑇𝑃 while the temperature in (b) is below both 𝑇0 and 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' be significantly higher, leading to a much more stable, long lived core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In a non-Gaian state the barrier is low, meaning the state will be relatively short lived, being vulnerable to parasites and to large population fluctuations which may result in total extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As shown in Figure 4 (d) over time this leads to more and more model runs in a Gaian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' To summarise: both mechanisms, SBS and SSM operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Ga- ian states have temperatures close to 𝑇𝑃, and thus high populations which, in this model, makes them more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Non-Gaian states are far from 𝑇𝑃 and have low populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This makes them vulnerable to total extinction (SBS) and punctuation which can take a non-Gaian to a Gaian state (SSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In this model, for this particular temperature, SSM is a more important mechanism than SBS, though the ratio can vary with 𝑇0, as we will explore in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These ideas can help explain why the Earth today is in a habit- able state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Since its conception the Gaia hypothesis has been defined in numerous ways and ranging from a strong hypothesis that self- regulating feedback loops are an expected property of a life-planet coupled system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' known as ‘probable Gaia’ Lenton & Wilkinson (2003),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' to a weaker hypotheses that suggests that while the Earth MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 1–18 (222) (a) (b) 1400 1000 1200 800- 1000 800 600 N N 600- 400 400- 200 200 0 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t (generations) t (generations) 108 To 108 Tp Tp 106 106 104 104 Temperature 102 102 100 100 98 98 - 96 96 94 94 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t (generations) t (generations)6 Arthur & Nicholson Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Model snapshot at 𝑡 = 9000 generations for the runs (a) and (b) from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Each node represents a different species, with the size of the node an indication of species’ population (upper and lower limits are applied to the point sizes for clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The colour of the nodes indicates the heating or cooling effect, 𝐻𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The width of the arrows indicates the interaction strength 𝐽𝑖 𝑗𝑛𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Only interactions with core species are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In (a) the red (bottom-right) core species has a strong enough heating effect to overwhelm the cooling effect of the other core species, so this configuration has a net heating effect, as seen in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In (b) both core species have a (weak) cooling effect, reducing the temperature, as seen in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' does have self-regulating feedback loops, these emerged merely by chance and that Gaia is not an expected feature of a planet hosting life, known as ‘lucky Gaia’ Watson (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As Figure 5 shows, in our model the fraction of Gaian states is increasing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This sug- gests that for early life starting out on a planet, a large amount of luck might be needed to initially start off in a Gaian state, but for surviv- ing runs over time the probability of being in a Gaian state increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This would suggest that when observing a biosphere ‘lucky Gaia’ may be the case for young planets but ‘probable Gaia’ is operating for older ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The experiments in Figure 5 have considered systems with only internal perturbations, that is, those generated by the biosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' How- ever, real planets experience many abiotic perturbations, both rapid and slower, such as changes in volcanic activity, changes in solar luminosity or impacts by large objects Covey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Overpeck & Cole (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Goldblatt & Zahnle (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Life is thought to have emerged early on Earth during a time when debris left over from the formation of the solar system was frequently colliding with the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Biospheres in a non-Gaian state will be more susceptible than Gaian biospheres to perturbations and will have a higher risk of going extinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This is closely related to the ‘Gaian bottleneck’ hypothesis Chopra & Lineweaver (2016) that proposes that early on in a planet’s history, if life emerges it must quickly establish self-regulatory feed- back loops to stabilise the climate of its planet in order to persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' If the biosphere fails then life goes extinct, the planet’s abiotic pro- cesses take over and the planet reverts to an inhospitable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' What is novel here is the idea that apart from total extinction, a planet can have a ‘near death experience’ where a mass extinction clears out a large fraction of the extant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' These mass extinctions are cru- cial for the exploration of the space of possible ecosystems Arthur & Sibani (2017) and ultimately lead to the emergence of long-lived stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Population diversity is known to significantly increase the resilience of ecosystems to perturbations Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Luck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2003), and additionally yeast Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2012) and bacteria Lambert & Kussell (2014) have been shown to develop in- creased resilience to environmental stressors if exposed to them in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' It is possible that large perturbations that do not eliminate all life are actually beneficial for evolving Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Indeed, there may be evidence of this in Earth history, as it is thought that a period of global glaciation may have triggered the evolution of multi-cellular life Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Hedges (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Boyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 4 HABITABLE ZONE EXPERIMENTS The habitable zone around a star is defined as the distance from a star where liquid water could exist on the surface of a planet Kasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Models demonstrate that the habitable zone is impacted by several factors, including the age and class of the host star Ramirez & Kaltenegger (2016), planetary mass Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2014), plane- tary atmospheric composition Pierrehumbert & Gaidos (2011), and the surface water content of the planet Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Additionally a planet being within the habitable zone doesn’t guarantee habitabil- ity, as a planet may have more than one possible climate state for the same stellar and orbital parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' a temperate Earth versus a frozen Earth Goldblatt & Zahnle (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' For a more extreme ex- ample, it is thought that Venus and Earth might represent alternate end states for the same planetary system, with small perturbations occurring early on in their history influencing their modern day states Lenardic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Existing exoplanet surveys and models have identified that rocky MNRAS 000, 1–18 (222) (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='04A Gaian Habitable Zone 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (a) shows the average (over all surviving model runs) of the population (top row) and temperature (bottom row) where the background temperature 𝑇0 = 100 = 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Column (b) shows the population and temperature where 𝑇0 = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The numbers next to the vertical dashed lines in the top row are the proportion of runs which have survived for that number of generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' planets can exist at a range of distances from their host star Domagal- Goldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Thus, it is a natural question to ask about the stability and persistence of Gaia across a range of background tem- peratures, some more conducive to life, some less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In this section we run many experiments where we vary the background temperature 𝑇0 and look at averages over 1000 model histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' To mimic the idea of a habitable zone with and without biotic influence we com- pare two versions of the model: one where life cannot influence the temperature, 𝜎𝐻 = 0, and one where life can influence it 𝜎𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' In Figure 5 we show the fraction of runs which survive for 105 generations in both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Perhaps surprisingly, the distributions are roughly similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' As the background temperature changes, a similar number of model runs survive for 105 generations whether life can effect the environment or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' This shows, at least, that species- environment interactions have little effect on the probability of total extinction and therefore on the presence or absence of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' However, as we saw in the previous section, the model runs can be split into Gaian and non-Gaian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Figure 5 also shows the proportion of MNRAS 000, 1–18 (222) (a) (b) 400 400 350- 350 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='972 i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='909 300 - 300- 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='98 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='989 ≥ 250 ≥ 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='9 i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='992 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='981 200 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='99 150 - 150- 100 100 102 103 104 105 102 103 104 105 t (generations) t (generations) 108 108 Tp Tp T 106 106 Temperature 104 - Temperature 104 102 102 100 100 98 98 102 103 104 105 102 103 104 105 t (generations) t (generations)8 Arthur & Nicholson Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' 𝑇0 = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (a) shows the temperature at 𝑡 = 105 generations versus population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Colour corresponds to the heating (red) or cooling (blue) effect of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' There are clearly two distinct clusters: one with (potentially) high population and low temperature and one with low population and high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (b) and (c) show histograms of the temperature and population respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' (d) shows the proportion of surviving runs at each generation as well as the proportion that have 𝑇 ≤ 𝑇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' runs that have 𝑁 > 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' The value of 200 is not itself significant, what is important is the comparison between 𝜎𝐻 = 0 and 𝜎𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Far from 𝑇𝑃, only the Gaian states can have large populations, in the other cases the total population is low and life is simply ‘clinging on’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Importantly for exoplanet astronomy, a small pocket of life that is clinging on to existence is unlikely to produce a detectable bio- signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' Figure 6 shows the population of the model runs as a function of 𝑇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We see that when 𝜎𝐻 = 0 the total population at 𝑇𝑃 is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' At 𝜎𝐻 = 0 the TNM agents are only attempting to optimise inter- species interactions, not interactions and temperature and thus can find a better maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' For example, strongly symbiotic cores may have a detrimental effect on the temperature which is only relevant in the 𝜎𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' However, the population falls rather rapidly with 𝑇0 at 𝜎𝐻 = 0 compared to the 𝜎𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content=' We also see (from the colour gradient) that at 𝜎𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='05, for 𝑇0 far from 𝑇𝑃 only MNRAS 000, 1–18 (222) (a) (b) 110.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='0035 Fraction Surviving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49A0T4oBgHgl3EQfNv86/content/2301.02150v1.pdf'} +page_content='0 Fractionofsurvivors it should return .” +or “The function does not work. E.g. for the input +I get the following error message: ”, depending +on whether the failing test case from the QuixBugs dataset +returned an incorrect answer or threw an error. In the case of +5CoCoNut, solves overall only 2 instances less than best reported thus far +on the QuixBugs Python dataset [15], though details on patch ranking for +each program were missing from the later work. + +more complex inputs we made the following response: “The +function does not work. E.g., given the following call: The following should be the output: .”6 We +only provide one such hint and report results. This experiment +was run on the ChatGPT version from January 9, 2023. +III. RESULTS AND DISCUSSION +In this section, we present the results of the comparison +of ChatGPT, Codex, CoCoNut, and the standard APR ap- +proaches. We classify ChatGPT’s answers and report on short +discussions with the model. Furthermore, we describe what +we noticed while working with ChatGPT. +A. Automatic Bug Fixing Performance +Table I shows the achieved results of ChatGPT, Codex, +CoCoNut, and the dedicated APR approaches on the bench- +mark problems from QuixBugs. For the ChatGPT results, a +checkmark () indicates that a correct answer was given in +at least one of the four runs for a benchmark problem. A +cross () indicates that no correct answer was given in any +of the runs. In parentheses we additionally report the number +of runs that led to a successful solution. For the results from +the literature, a checkmark indicates that a correct bug fix is +reported. A cross means that no successful bug fix is reported. +We see that the results achieved by ChatGPT are similar +to Codex in performance and outperform the standard APR +approaches. Overall, we find bug fixes for 19 benchmark +problems with ChatGPT, 21 are reported for Codex, 19 for +CoCoNut, and only 7 for the standard approaches. +The large gap in performance between the language model +based approaches and the standard APR approaches can be +explained by the fact that the latter usually just use a small test +suite to define the problem, which can be easily overfitted. The +authors of [13] also report this problem. If only the test suite +is considered for evaluation, the standard approaches would +solve a total of 16 benchmark problems. However, as in real- +world applications only programs that work also on unseen +inputs are usable, we have only adopted the 7 generalizing +problems from [13] as correct. +If we take a closer look at the results for ChatGPT, we +see that benchmark problems are often only solved in one or +two runs. Only for the problems BUCKETSORT and FLATTEN +ChatGPT finds a bug fix in all four runs. So ChatGPT seems +to have a relatively high variance when fixing bugs. For an +end-user, however, this means that it can be helpful to execute +requests multiple times. +Furthermore, it is not surprising that ChatGPT solves about +the same number of problems as Codex, as ChatGPT and +Codex are from the same family of language models.7 How- +ever, we still see potential for improvement for ChatGPT, as +the given responses are often close to the correct solution +(for a detailed classification of ChatGPT’s responses see +Section III-B). +6The third case only appeared once. All queries are available online. +7https://beta.openai.com/docs/model-index-for-researchers (accessed Jan- +uary 18, 2023). +Nevertheless, we are very strict in our evaluation and +consider only patches as correct if the bug introduced by +QuixBugs is actually identified and corrected. E.g., for some +problems, ChatGPT suggests a complete re-implementation +which is then bug-free. However, these are probably no real +bug fixes, since the introduced bug is not localized. We assume +that ChatGPT simply reproduced what it has learned here. +Furthermore, we do not count a bug as fixed if additional +changes suggested by ChatGPT introduce new errors that +prevent the program from running properly. Moreover, by +sending just a single request in this evaluation, we are not +using the full potential of the dialogue system. Consequently, +we take a closer look at how ChatGPT behaves when we +interact more with the system and give it more information +about the bug in Section III-C. +B. A Classification of ChatGPT’s Answers +While working with ChatGPT, we noticed different types +of responses that ChatGPT gave to our requests, especially +when a bug could not be found. Therefore, we identified the +different types of answers from ChatGPT for the benchmark +problems from QuixBugs and analyzed their frequency. We +identified the following classes of ChatGPT answers: +• More information required: Asks for more information +on the program behavior to identify the bug. +• No bug found: Does not find a bug and states the program +is working correctly. +• Correct fix provided: Provides the correct fix for the +correct bug. +• Tries to fix something else: Does not find the intended +bug and tries to fix or advise on something else that is +not really a bug or adjusts for edge cases. +• Provides fix but introduces new bug: Provides the +correct fix for the target bug but introduces a new bug +somewhere else. +• Alternative implementation: Does not fix the bug but +gives a working alternative implementation. +Figure 2 shows the number of occurrences of identified +classes of ChatGPT answers given for the problems from +QuixBugs. +We see that for most of our requests, ChatGPT asks for more +information about the problem and the bug. With the second +most number of answers given, we observe ChatGPT claiming +that the given code snippet does not seem to have a bug. In +both cases it might be useful to fully utilize the possibilities +of the dialogue system ChatGPT offers, as further information +might lead to a correct bug fix. +Less often than the request for more information, we +observe that ChatGPT fixes the bug but at the same time +introduces new errors, or we see that ChatGPT not really +addresses the bug correctly but suggests a completely new +working re-implementation for the problem. +C. A Discussion with ChatGPT +In order to be able to compare ChatGPT with other systems +in a standardized form, we have so far studied how ChatGPT + +TABLE I: Results achieved by ChatGPT, Codex, CoCoNut, and the standard APR approaches on the problems from the +QuixBugs benchmark set. For ChatGPT, we also report the number of successful runs in brackets. +Benchmark problem +ChatGPT +Codex [15] +CoCoNut [14] +Standard APR [13] +bitcount + (0 / 4) + + + +breadth-first-search + (2 / 4) + + + +bucketsort + (4 / 4) + + + +depth-first-search + (0 / 4) + + + +detect-cycle + (0 / 4) + + + +find-first-in-sorted + (2 / 4) + + + +find-in-sorted + (3 / 4) + + + +flatten + (4 / 4) + + + +gcd + (0 / 4) + + + +get-factors + (1 / 4) + + + +hanoi + (0 / 4) + + + +is-valid-parenthesization + (2 / 4) + + + +kheapsort + (0 / 4) + + + +knapsack + (1 / 4) + + + +kth + (0 / 4) + + + +lcs-length + (0 / 4) + + + +levenshtein + (0 / 4) + + + +lis + (0 / 4) + + + +longest-common-subsequence + (0 / 4) + + + +max-sublist-sum + (0 / 4) + + + +mergesort + (1 / 4) + + + +minimum-spanning-tree + (0 / 4) + + + +next-palindrome + (1 / 4) + + + +next-permutation + (0 / 4) + + + +pascal + (1 / 4) + + + +possible-change + (1 / 4) + + + +powerset + (0 / 4) + + + +quicksort + (1 / 4) + + + +reverse-linked-list + (2 / 4) + + + +rpn-eval + (0 / 4) + + + +shortest-path-length + (1 / 4) + + + +shortest-path-lengths + (0 / 4) + + + +shortest-paths + (1 / 4) + + + +shunting-yard + (2 / 4) + + + +sieve + (0 / 4) + + + +sqrt + (1 / 4) + + + +subsequences + (1 / 4) + + + +to-base + (0 / 4) + + + +topological-ordering + (0 / 4) + + + +wrap + (0 / 4) + + + +ΣΣΣ (Solved) +19 +21 +19 +7 + +0 +10 +20 +30 +40 +50 +# of occurrences +other +alternative implementation +provides fix but introduces new bug +tries to fix something else +correct fix provided +no bug found +more information required +Fig. 2: Number of occurrences of identified classes of ChatGPT answers given for the problems from QuixBugs. +1 +Request +( 1 ) : +2 +Does +t h i s +program +have a bug ? How to +f i x +i t ? +3 +4 +def +gcd ( a , +b ) : +5 +i f +b == 0: +6 +r e t u r n +a +7 +e l s e : +8 +r e t u r n +gcd ( a % b , +b ) +9 +10 +Response +( 1 ) : +11 +This +i s +the +Euclidean +a l g o r i t h m +f o r +f i n d i n g +the +g r e a t e s t +common d i v i s o r +(GCD) +of two +12 +i n t e g e r s , +a and b . +I t +appears +to +be +implemented +c o r r e c t l y . +I f +you +are +e x p e r i e n c i n g +13 +i s s u e s +with +t h i s +program , +i t +i s +l i k e l y +t h a t +the +problem +l i e s +elsewhere +in +your +code . +14 +Without more +i n f o r m a t i o n +about +the +bug you +are +experiencing , +i t +i s +d i f f i c u l t +to +say +15 +how to +f i x +i t . +16 +17 +Request +( 2 ) : +18 +No , +the +f u n c t i o n +i s +not +c o r r e c t . E . g . , +f o r +the +i n p u t +a = 4 and b = 5 I +get +the +f o l l o w i n g +19 +e r r o r +message : +” RecursionError : maximum r e c u r s i o n +depth +exceeded +in +comparison ” . +20 +21 +Response +( 2 ) : +22 +The +i s s u e +i s +t h a t +the +f u n c t i o n +c a l l s +i t s e l f +r e c u r s i v e l y , +and +in some cases , +i t +never +23 +r e a c h e s +the +base +case +( b == 0) , +causing +i t +to +r e c u r s e +i n d e f i n i t e l y . +The base +case +of +24 +the +gcd +a l g o r i t h m +i s +when the +remainder +i s +zero , +but +in +t h i s +implementation , +i t ’ s when +25 +b +i s +zero . +For example , +when a=4 and b=5 , gcd (4 , +5) +i s +c a l l e d , +which +then +c a l l s +26 +gcd (4 % 5 , +5) +which +i s +gcd (4 , +5 ) . +This +causes +an +i n f i n i t e +loop . +27 +28 +You can +f i x +t h i s +by swapping +the +parameters +in +the +r e c u r s i v e +c a l l . +29 +30 +def +gcd ( a , +b ) : +31 +i f +a == 0: +32 +r e t u r n +b +33 +e l s e : +34 +r e t u r n +gcd ( b % a , +a ) +35 +36 +This way , +i f +a +i s +zero , +the +f u n c t i o n +r e t u r n s +b ( which +i s +the +gcd +of +a and b ) , +o t h e r w i s e +37 +i t +c a l l s +i t s e l f +with +the new values +of b % a and a . +Fig. 3: A discussion with ChatGPT about the GCD problem from QuixBugs. We inserted request and response markers in bold +font to make the chat history more readable. +behaves when making exactly one request (see Section III-A). +However, a potentially powerful advantage of ChatGPT is +that we can interact with the system in a dialogue to specify +a request in more detail. This is useful, as both, a textual +description as well as a definition given by a test suite, are +possibly ambiguous and could be misleading. + +1 +Request +( 1 ) : +2 +Does +t h i s +program +have a bug ? How to +f i x +i t ? +3 +4 +def +b i t c o u n t ( n ) : +5 +count = 0 +6 +while n : +7 +n ˆ= n − 1 +8 +count += 1 +9 +r e t u r n +count +10 +11 +Response +( 1 ) : +12 +I ’m unable +to +t e l l +i f +the +program +has a bug +without +more +i n f o r m a t i o n +on +the +expected +13 +behavior +and +the +i n p u t +t h a t +i s +causing +the +i s s u e . +[ . . . ] +14 +15 +Request +( 2 ) : +16 +The program +should +count +the +number +of 1− b i t s +in +the +bin ary +encoding +of n . +17 +18 +Response +( 2 ) : +19 +The +f u n c t i o n +b i t c o u n t ( n ) +appears +to +be +counting +the +number +of 1− b i t s +[ . . . ] +20 +21 +To confirm +t h i s , +you can +t e s t +the +f u n c t i o n +with some +t e s t +cases +to +check +i f +22 +i t +produces +the +expected +output : +23 +24 +t e s t +c a s e s += [0 , +1 , +2 , +3 , +4 , +5 , +6 , +7 , +8 , +9 , +10 , +11 , +12 , +13 , +14 , +15] +25 +exp ect ed output = [0 , +1 , +1 , +2 , +1 , +2 , +2 , +3 , +1 , +2 , +2 , +3 , +2 , +3 , +3 , +4] +26 +27 +Request +( 3 ) : +28 +The +f u n c t i o n +does +not work . +For +i n p u t +1 +i t +seems +to +run +i n t o +an +i n f i n i t e +loop . +29 +30 +Response +( 3 ) : +31 +You are +c o r r e c t +t h a t +the +f u n c t i o n +w i l l +run +i n t o +an +i n f i n i t e +loop when the +i n p u t +i s +1 . +32 +[ . . . ] +33 +34 +def +b i t c o u n t ( n ) : +35 +i f +n == 0: +36 +r e t u r n +0 +37 +count = 0 +38 +while n : +39 +n ˆ= n − 1 +40 +count += 1 +41 +r e t u r n +count +42 +43 +def +b i t c o u n t ( n ) : +44 +count = 0 +45 +while n > 0: +46 +n ˆ= n − 1 +47 +count += 1 +48 +r e t u r n +count +49 +50 +def +b i t c o u n t ( n ) : +51 +r e t u r n +bin ( n ) . count ( ’1 ’) +52 +53 +def +b i t c o u n t ( n ) : +54 +count = 0 +55 +while n : +56 +i f +n & 1: +57 +count += 1 +58 +n >>= 1 +59 +r e t u r n +count +Fig. 4: A discussion with ChatGPT about the BITCOUNT problem from QuixBugs. We inserted request and response markers +in bold font to make the chat history more readable. In addition, we shortened ChatGPT’s responses in some places to keep +the example as small as possible. When we left out large text passages, we marked this in the text. + +Consequently, we investigate for two benchmark problems +how ChatGPT behaves in a conversation and if it is possible +to find a working solution by discussing the problem with the +system. We choose GCD and BITCOUNT as benchmark prob- +lems because in our previous experiments the contained bug +could not be fixed correctly for both problems. Furthermore, +the problems consist of a relatively small number of code lines +which allows us to discuss these problems in detail. +Figure 3 shows an example discussion with ChatGPT about +the GCD problem (lines 1–8). In the first response (lines +10–15), ChatGPT does not present any solution. It asks for +more information about the bug (we observed this behavior +for many other problems, see Section III-B). Since the given +function causes recursion issues for many possible inputs, +we give ChatGPT an exact input example and the resulting +error message from Python (lines 17–19). By mentioning the +recursion issue, the final response goes in the right direction +and we get a correctly working patched version (lines 30–34). +In Figure 4 we see an example discussion with ChatGPT +about the BITCOUNT problem (lines 1–9). Again, ChatGPT +asks for more information about the problem and for an input +that causes an error (lines 11–13). As follow-up request, we +give ChatGPT a description of what the function should do +(based on a code comment from QuixBugs) and ignore the +request for an example input to see how ChatGPT reacts (lines +15 and 16). We can see in the following answer (lines 18–25) +that there is clearly a relation between ChatGPT’s first and +second answer because now we get an explanation of how +we can test the function with some test inputs. We respond +with a problem description for a test input and describe that +there is probably an issue with an infinite loop (lines 27 and +28). ChatGPT responds with four code snippets where the first +two (lines 34–48) do not solve the problem with the infinite +loop and the last two (lines 50–59) are complete but working +re-implementations which, however, not directly address the +contained bug. It seems that ChatGPT simply returns functions +here that somehow fit the content of the problem discussion, +even though the test cases mentioned by ChatGPT show +that the first two functions cannot work correctly. Also the +bug is not simply fixed by replacing n ˆ= n - 1 with +n &= n - 1 in the given function, but ChatGPT, as al- +ready mentioned, returns two complete re-implementations. +However, both observations are not particularly surprising for +a language model based approach. Nevertheless, the given +answers would be useful for a programmer as they help to +solve the problem. +D. Systematic Follow-up Requests for ChatGPT +Next, we conducted a study where we systematically discuss +with ChatGPT. For those programs for which the contained +bug was not correctly addressed by ChatGPT (see Table I), +we provide ChatGPT with a follow-up request giving a hint, +as specified in Section II-C. We report our results in Table II. +We use the same notation as before with the addition that a +checkmark with an asterisk (*) defines that a solution was +found without a follow-up request being necessary in this run. +TABLE II: Results achieved by ChatGPT with additional +information given in a follow-up request for the unsolved +benchmark problems (see Table I). +Benchmark problem +ChatGPT +bitcount + +depth-first-search +* +detect-cycle +* +gcd + +hanoi + +kheapsort + +kth + +lcs-length + +levenshtein + +lis + +longest-common-subsequence + +max-sublist-sum + +minimum-spanning-tree + +next-permutation + +powerset + +rpn-eval + +shortest-path-lengths + +sieve +* +to-base + +topological-ordering + +wrap + +ΣΣΣ (Solved) +9 (12) +For 9 benchmark problems we see that a more detailed +description of the bug is helpful for ChatGPT. For 3 bench- +mark problems no follow-up request was necessary in this run, +since the bug was correctly addressed in the response given +on our first request. Overall, adding a hint to ChatGPT vastly +improves its performance, with 31 out of 40 problems solved. +ChatGPT thus offers an exciting new way of approaching +automated program repair. +IV. THREATS TO VALIDITY +It is worth noting that ChatGPT is currently under active +development. During our study there was a major update to +it, which might have influenced our results. Although we +observed repairability rates before and after the update to be +similar. However, future releases might yield different results. +Furthermore, ChatGPT allows for conversation with its users. +Asking a different question than the ones presented in this +study could potentially have a different impact on results. +To mitigate this threat to validity, we conducted a pre-study, +varying the questions asked. We noted no significant influence +on the results. Moreover, the results might vary depending +on the programming language, size of the benchmarks, and + +the number of queries issued. To mitigate these threats, we +chose a standard benchmark set and targeted Python – the +most popular programming language.8 The classification of +the results was done manually and therefore represents the +subjective assessment of the authors. To enable a verification +of our results, we made our conversations with ChatGPT +available online. +V. CONCLUSIONS AND FUTURE WORK +To support programmers in finding and fixing software +bugs, several automated program repair (APR) methods have +been proposed. ChatGPT, a recently presented deep learning +(DL) based dialogue system, can also make suggestions for +improving erroneous source code. However, so far the quality +of these suggestions has been unclear. Therefore, we compared +in this work the automatic bug fixing performance of ChatGPT +with that of Codex and several dedicated APR approaches. +We find that ChatGPT has similar performance to Codex +and dedicated DL-based APR on a standard benchmark set. It +vastly outperforms standard APR methods (19 vs. 7 out of 40 +bugs fixed). Using ChatGPT’s dialogue option and giving the +system more information about the bug in a follow-up request +boosts the performance even further, giving an overall success +rate of 77.5%. This shows that human input can be of much +help to an automated APR system, with ChatGPT providing +means to do so. +Despite its great performance, the question arises whether +the mental cost required to verify ChatGPT answers outweighs +the advantages that ChatGPT brings. Perhaps incorporation +of automated approaches to provide ChatGPT with hints as +well as automated verification of its responses, e.g., through +automated testing, would yield ChatGPT to be a viable tool +that would help software developers in their daily tasks. +We hope our results and observations will be helpful for +future work with ChatGPT. +ACKNOWLEDGMENTS +This work was partially supported by UKRI EPSRC grant +no. EP/P023991/1. +REFERENCES +[1] W. E. Wong, X. Li, and P. A. Laplante, “Be more familiar with our +enemies and pave the way forward: A review of the roles bugs played +in software failures,” Journal of Systems and Software, vol. 133, pp. +68–94, 2017. +[2] C. Le Goues, T. Nguyen, S. Forrest, and W. Weimer, “GenProg: A +generic method for automatic software repair,” Ieee transactions on +software engineering, vol. 38, no. 1, pp. 54–72, 2011. +[3] L. Gazzola, D. Micucci, and L. 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Durieux and M. Monperrus, “Dynamoth: dynamic code synthesis +for automatic program repair,” in Proceedings of the 11th International +Workshop on Automation of Software Test, 2016, pp. 85–91. +[19] M. Martinez and M. Monperrus, “Astor: Exploring the design space +of generate-and-validate program repair beyond GenProg,” Journal of +Systems and Software, vol. 151, pp. 65–80, 2019. +[20] J. Xuan, M. Martinez, F. Demarco, M. Clement, S. L. Marcote, +T. Durieux, D. Le Berre, and M. Monperrus, “Nopol: Automatic repair +of conditional statement bugs in Java programs,” IEEE Transactions on +Software Engineering, vol. 43, no. 1, pp. 34–55, 2016. +[21] B. Cornu, T. Durieux, L. Seinturier, and M. Monperrus, “NPEfix: +Automatic runtime repair of null pointer exceptions in Java,” arXiv +preprint arXiv:1512.07423, 2015. + diff --git a/AdFAT4oBgHgl3EQfrh5X/content/tmp_files/load_file.txt b/AdFAT4oBgHgl3EQfrh5X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4801c41770e70d46e1da05b63ba045f2eb39664b --- /dev/null +++ b/AdFAT4oBgHgl3EQfrh5X/content/tmp_files/load_file.txt @@ -0,0 +1,699 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf,len=698 +page_content='An Analysis of the Automatic Bug Fixing Performance of ChatGPT Dominik Sobania Johannes Gutenberg University Mainz Email: dsobania@uni-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='de Carol Hanna University College London Email: carol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='hanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='21@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='uk Martin Briesch Johannes Gutenberg University Mainz Email: briesch@uni-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='de Justyna Petke University College London Email: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='petke@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='uk Abstract—To support software developers in finding and fixing software bugs, several automated program repair techniques have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Given a test suite, standard methods usually either synthesize a repair, or navigate a search space of software edits to find test-suite passing variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Recent program repair methods are based on deep learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' One of these novel methods, which is not primarily intended for automated program repair, but is still suitable for it, is ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The bug fixing performance of ChatGPT, however, is so far unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Therefore, in this paper we evaluate ChatGPT on the standard bug fixing benchmark set, QuixBugs, and compare the perfor- mance with the results of several other approaches reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We find that ChatGPT’s bug fixing performance is competitive to the common deep learning approaches CoCoNut and Codex and notably better than the results reported for the standard program repair approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In contrast to previous approaches, ChatGPT offers a dialogue system through which further information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', the expected output for a certain input or an observed error message, can be entered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' By providing such hints to ChatGPT, its success rate can be further increased, fixing 31 out of 40 bugs, outperforming state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Index Terms—Automated program repair, automatic bug fix- ing, ChatGPT, Codex, language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' INTRODUCTION Complex software usually contains undiscovered bugs in its source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The later these are found, the more far-reaching consequences these can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Uncorrected bugs in software can lead to failures of essential systems, which can result in high economic costs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In order to support programmers in finding and fixing software errors, automated program repair (APR) systems have been introduced that automatically suggest software patches to correct the detected errors [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For instance, Haralds- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' [4] suggest an approach based on genetic improve- ment (GI) [5] that tracks emerging bugs during a workday and searches for potential fixes for them overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The following morning the programmers get a list of suggestions which should help fix the detected bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Standard methods for automated program repair can be classified into two categories: the generate-and-validate ap- proaches mutate software guided by a search strategy, while semantics-driven (or synthesis-based) approaches use a con- straint solver to synthesize repairs [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The generate-and- validate ones have first seen industrial uptake [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' One of the key disadvantage of standard approaches to APR is their running cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The generate-and-validate ones usually rely on test suites to verify program correctness, while synthesis-based ones on calls to a constraint solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Both validation strategies are costly, making typical APR tools hours to run before a viable patch is presented to the developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Most recently, program repair tools based on deep learn- ing (DL) approaches have been introduced [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' These learn bug fixing patterns from existing databases and treat the automated program repair problem as a neural machine translation task, producing a ranking of, sometimes hundreds of, patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Unlike standard approaches, such generated patches are not usually evaluated against a test suite, or other automated verification strategy, so may not even compile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Nevertheless, DL-based program repair has shown competitive results to standard approaches [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In recent years, several large-scale language models based on the Transformer architecture [7] have been introduced, such as CodeBERT [8], PyMT5 [9], and Codex [10], which can also process and extend source code and achieve comparable results to standard approaches on various coding tasks [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A large- scale language model based on the Transformer architecture that has recently received great attention is ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='1 With ChatGPT not only text input can be extended, but it is even possible to have a conversation with the language model and the previous chat history is taken into account for answer generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In addition to very general or subject-specific topics, ChatGPT can also be used to discuss source code, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', to ask for a suggestion for a fix of incorrect code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, the quality of these suggestions is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Therefore, in this work we evaluate and analyse the au- tomatic bug fixing performance of ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Moreover, we provide a comparison with results reported in the literature obtained using state-of-the-art APR approaches and Codex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We chose the QuixBugs [12] benchmark set for our study, as it contains small, yet challenging programs for current APR 1https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='com/blog/chatgpt/ (accessed January 18, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='08653v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='SE] 20 Jan 2023 approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We consider all Python problems from QuixBugs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', 40 overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We first ask ChatGPT for bug fixes for the selected bench- marks and manually check whether the suggested solution is correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We repeat the query four times, to account for the heuristic nature of ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Next, we compare its perfor- mance with that of Codex and dedicated APR approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For the standard APR approaches, we take the results from a recent paper [13] that examines the performance of several methods on the QuixBugs benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For dedicated APR methods based on deep learning, we take results from CoCoNut [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='2 For the large-scale language model Codex, we take the results from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, we study and categorize ChatGPT’s answers to gain a deeper understanding of its behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Given that ChatGPT provides a unique opportunity for a conversation with the model, we provide a small hint to the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', a failing test input with an error it produces) to see if it improves ChatGPT’s fix rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We find that ChatGPT’s program repair performance is competitive to the results achieved with CoCoNut and Codex (19 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 19 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 21 instances solved, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Compared to the standard program repair approaches, ChatGPT achieves notably better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' With ChatGPT, we could fix bugs in 19 out of 40 problems while with the standard approaches only 7 can be fixed, even though we give ChatGPT only the incorrect code snippet without any additional information and without using the chat option in a conversational way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' If the chat function is actively used, we can fix even more instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' This shows the power of providing manual hints to a program repair system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' All our experimental data is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' CHATGPT FOR AUTOMATED PROGRAM REPAIR In this section we present our methodology for assessing ChatGPT’s program repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Benchamrk To evaluate the automatic bug fixing performance of Chat- GPT, we use the QuixBugs [12] benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Unlike many other benchmark suites for automated program repair, QuixBugs contains relatively small problems (small number of code lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' These are thus suitable for use in a dialogue sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For each of the 40 benchmark problems from QuixBugs, we take the erroneous Python code, remove all contained comments4, and ask ChatGPT if the code contains a bug and how it can be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For each benchmark problem, we make several independent requests to ChatGPT and manually check whether the given answer is correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We standardize our procedure by using the same format for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We ask: “Does this program have a bug?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' How to fix it?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' followed by an empty line and the buggy code without comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Figure 1 shows an example request to ChatGPT for the BITCOUNT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Lines 1-2 contain the question to ChatGPT where 2Although more recent approaches exist, we found this work is the most recent providing sufficient patch ranking detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 3https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='rlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='net/dsobania/chatgpt-apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 4This was necessary, as sometimes the comments contain the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 1 Does t h i s program have a bug ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' How to 2 f i x i t ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 3 4 def b i t c o u n t ( n ) : 5 count = 0 6 while n : 7 n ˆ= n − 1 8 count += 1 9 r e t u r n count Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 1: Request to ChatGPT for the BITCOUNT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' we ask how the bug can be fixed and starting from line 4 we present the erroneous code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For this example, we would expect from ChatGPT an answer that addresses the bug in line 7, where n ˆ= n - 1 should be replaced with n &= n - 1, either with a response containing the complete code snippet with the fixed bug (correctly addressed) or by giving an exact and correct description how to change the affected code lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Comparison Study We ran four independent requests to ChatGPT for each problem from the QuixBugs dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In order to compare the results of ChatGPT with the standard APR methods, we take the results from a comprehensive study from the literature [13] that reports the performance of ten different methods (Arja [16], Cardumen [17], Dynamoth [18], JGenProg [19], JKali [19], JMutRepair [19], Nopol [20], NPEfix [21], RSRe- pair [16], and Tibra [19]) on the problems from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For dedicated APR approaches based on deep learning we chose recent results reported by Lutellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='5 In Table I we report a fix only if the correct patch was ranked first by Lutellier et al.’s proposed approach, CoCoNut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For the large- scale language model Codex, we take the results from a recent paper [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We ran this experiment on ChatGPT versions from December 15, 2022 and January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Dialogue Study Given that ChatGPT provides a unique opportunity of a dialogue with the model, we also conduct a study where we provide ChatGPT with a hint, based on ChatGPT’s response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' If ChatGPT does not provide a correct answer to the first request (described in the previous paragraph), we tell ChatGPT in a standardized way that the function is not working correctly and additionally provide an input example that shows that the func- tion is not working properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' If ChatGPT incorrectly claimed the program was correct, we replied: “The function does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', for the input it should return .” or “The function does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' for the input I get the following error message: ”, depending on whether the failing test case from the QuixBugs dataset returned an incorrect answer or threw an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In the case of 5CoCoNut, solves overall only 2 instances less than best reported thus far on the QuixBugs Python dataset [15], though details on patch ranking for each program were missing from the later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' more complex inputs we made the following response: “The function does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', given the following call: The following should be the output: .”6 We only provide one such hint and report results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' This experiment was run on the ChatGPT version from January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' RESULTS AND DISCUSSION In this section, we present the results of the comparison of ChatGPT, Codex, CoCoNut, and the standard APR ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We classify ChatGPT’s answers and report on short discussions with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, we describe what we noticed while working with ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Automatic Bug Fixing Performance Table I shows the achieved results of ChatGPT, Codex, CoCoNut, and the dedicated APR approaches on the bench- mark problems from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For the ChatGPT results, a checkmark (\x13) indicates that a correct answer was given in at least one of the four runs for a benchmark problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A cross (\x17) indicates that no correct answer was given in any of the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In parentheses we additionally report the number of runs that led to a successful solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For the results from the literature, a checkmark indicates that a correct bug fix is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A cross means that no successful bug fix is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We see that the results achieved by ChatGPT are similar to Codex in performance and outperform the standard APR approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Overall, we find bug fixes for 19 benchmark problems with ChatGPT, 21 are reported for Codex, 19 for CoCoNut, and only 7 for the standard approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The large gap in performance between the language model based approaches and the standard APR approaches can be explained by the fact that the latter usually just use a small test suite to define the problem, which can be easily overfitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The authors of [13] also report this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' If only the test suite is considered for evaluation, the standard approaches would solve a total of 16 benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, as in real- world applications only programs that work also on unseen inputs are usable, we have only adopted the 7 generalizing problems from [13] as correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' If we take a closer look at the results for ChatGPT, we see that benchmark problems are often only solved in one or two runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Only for the problems BUCKETSORT and FLATTEN ChatGPT finds a bug fix in all four runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' So ChatGPT seems to have a relatively high variance when fixing bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For an end-user, however, this means that it can be helpful to execute requests multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, it is not surprising that ChatGPT solves about the same number of problems as Codex, as ChatGPT and Codex are from the same family of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='7 How- ever, we still see potential for improvement for ChatGPT, as the given responses are often close to the correct solution (for a detailed classification of ChatGPT’s responses see Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 6The third case only appeared once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' All queries are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 7https://beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='com/docs/model-index-for-researchers (accessed Jan- uary 18, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Nevertheless, we are very strict in our evaluation and consider only patches as correct if the bug introduced by QuixBugs is actually identified and corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', for some problems, ChatGPT suggests a complete re-implementation which is then bug-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, these are probably no real bug fixes, since the introduced bug is not localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We assume that ChatGPT simply reproduced what it has learned here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, we do not count a bug as fixed if additional changes suggested by ChatGPT introduce new errors that prevent the program from running properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Moreover, by sending just a single request in this evaluation, we are not using the full potential of the dialogue system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Consequently, we take a closer look at how ChatGPT behaves when we interact more with the system and give it more information about the bug in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A Classification of ChatGPT’s Answers While working with ChatGPT, we noticed different types of responses that ChatGPT gave to our requests, especially when a bug could not be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Therefore, we identified the different types of answers from ChatGPT for the benchmark problems from QuixBugs and analyzed their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We identified the following classes of ChatGPT answers: More information required: Asks for more information on the program behavior to identify the bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' No bug found: Does not find a bug and states the program is working correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Correct fix provided: Provides the correct fix for the correct bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Tries to fix something else: Does not find the intended bug and tries to fix or advise on something else that is not really a bug or adjusts for edge cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Provides fix but introduces new bug: Provides the correct fix for the target bug but introduces a new bug somewhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Alternative implementation: Does not fix the bug but gives a working alternative implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Figure 2 shows the number of occurrences of identified classes of ChatGPT answers given for the problems from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We see that for most of our requests, ChatGPT asks for more information about the problem and the bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' With the second most number of answers given, we observe ChatGPT claiming that the given code snippet does not seem to have a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In both cases it might be useful to fully utilize the possibilities of the dialogue system ChatGPT offers, as further information might lead to a correct bug fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Less often than the request for more information, we observe that ChatGPT fixes the bug but at the same time introduces new errors, or we see that ChatGPT not really addresses the bug correctly but suggests a completely new working re-implementation for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' A Discussion with ChatGPT In order to be able to compare ChatGPT with other systems in a standardized form, we have so far studied how ChatGPT TABLE I: Results achieved by ChatGPT, Codex, CoCoNut, and the standard APR approaches on the problems from the QuixBugs benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For ChatGPT, we also report the number of successful runs in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='Benchmark problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='ChatGPT ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 (0 / 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='wrap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 (0 / 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='\x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='ΣΣΣ (Solved) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='# of occurrences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='alternative implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='provides fix but introduces new bug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='tries to fix something else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='correct fix provided ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='no bug found ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='more information required ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 2: Number of occurrences of identified classes of ChatGPT answers given for the problems from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 1 Request ( 1 ) : 2 Does t h i s program have a bug ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' How to f i x i t ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 3 4 def gcd ( a , b ) : 5 i f b == 0: 6 r e t u r n a 7 e l s e : 8 r e t u r n gcd ( a % b , b ) 9 10 Response ( 1 ) : 11 This i s the Euclidean a l g o r i t h m f o r f i n d i n g the g r e a t e s t common d i v i s o r (GCD) of two 12 i n t e g e r s , a and b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' I t appears to be implemented c o r r e c t l y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' I f you are e x p e r i e n c i n g 13 i s s u e s with t h i s program , i t i s l i k e l y t h a t the problem l i e s elsewhere in your code .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 14 Without more i n f o r m a t i o n about the bug you are experiencing , i t i s d i f f i c u l t to say 15 how to f i x i t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 16 17 Request ( 2 ) : 18 No , the f u n c t i o n i s not c o r r e c t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' , f o r the i n p u t a = 4 and b = 5 I get the f o l l o w i n g 19 e r r o r message : ” RecursionError : maximum r e c u r s i o n depth exceeded in comparison ” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 20 21 Response ( 2 ) : 22 The i s s u e i s t h a t the f u n c t i o n c a l l s i t s e l f r e c u r s i v e l y , and in some cases , i t never 23 r e a c h e s the base case ( b == 0) , causing i t to r e c u r s e i n d e f i n i t e l y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' The base case of 24 the gcd a l g o r i t h m i s when the remainder i s zero , but in t h i s implementation , i t ’ s when 25 b i s zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For example , when a=4 and b=5 , gcd (4 , 5) i s c a l l e d , which then c a l l s 26 gcd (4 % 5 , 5) which i s gcd (4 , 5 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' This causes an i n f i n i t e loop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 27 28 You can f i x t h i s by swapping the parameters in the r e c u r s i v e c a l l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 29 30 def gcd ( a , b ) : 31 i f a == 0: 32 r e t u r n b 33 e l s e : 34 r e t u r n gcd ( b % a , a ) 35 36 This way , i f a i s zero , the f u n c t i o n r e t u r n s b ( which i s the gcd of a and b ) , o t h e r w i s e 37 i t c a l l s i t s e l f with the new values of b % a and a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 3: A discussion with ChatGPT about the GCD problem from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We inserted request and response markers in bold font to make the chat history more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' behaves when making exactly one request (see Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, a potentially powerful advantage of ChatGPT is that we can interact with the system in a dialogue to specify a request in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' This is useful, as both, a textual description as well as a definition given by a test suite, are possibly ambiguous and could be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 1 Request ( 1 ) : 2 Does t h i s program have a bug ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' How to f i x i t ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 3 4 def b i t c o u n t ( n ) : 5 count = 0 6 while n : 7 n ˆ= n − 1 8 count += 1 9 r e t u r n count 10 11 Response ( 1 ) : 12 I ’m unable to t e l l i f the program has a bug without more i n f o r m a t i o n on the expected 13 behavior and the i n p u t t h a t i s causing the i s s u e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ] 14 15 Request ( 2 ) : 16 The program should count the number of 1− b i t s in the bin ary encoding of n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 17 18 Response ( 2 ) : 19 The f u n c t i o n b i t c o u n t ( n ) appears to be counting the number of 1− b i t s [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ] 20 21 To confirm t h i s , you can t e s t the f u n c t i o n with some t e s t cases to check i f 22 i t produces the expected output : 23 24 t e s t c a s e s = [0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15] 25 exp ect ed output = [0 , 1 , 1 , 2 , 1 , 2 , 2 , 3 , 1 , 2 , 2 , 3 , 2 , 3 , 3 , 4] 26 27 Request ( 3 ) : 28 The f u n c t i o n does not work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For i n p u t 1 i t seems to run i n t o an i n f i n i t e loop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 29 30 Response ( 3 ) : 31 You are c o r r e c t t h a t the f u n c t i o n w i l l run i n t o an i n f i n i t e loop when the i n p u t i s 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 32 [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ] 33 34 def b i t c o u n t ( n ) : 35 i f n == 0: 36 r e t u r n 0 37 count = 0 38 while n : 39 n ˆ= n − 1 40 count += 1 41 r e t u r n count 42 43 def b i t c o u n t ( n ) : 44 count = 0 45 while n > 0: 46 n ˆ= n − 1 47 count += 1 48 r e t u r n count 49 50 def b i t c o u n t ( n ) : 51 r e t u r n bin ( n ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' count ( ’1 ’) 52 53 def b i t c o u n t ( n ) : 54 count = 0 55 while n : 56 i f n & 1: 57 count += 1 58 n >>= 1 59 r e t u r n count Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 4: A discussion with ChatGPT about the BITCOUNT problem from QuixBugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We inserted request and response markers in bold font to make the chat history more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In addition, we shortened ChatGPT’s responses in some places to keep the example as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' When we left out large text passages, we marked this in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Consequently, we investigate for two benchmark problems how ChatGPT behaves in a conversation and if it is possible to find a working solution by discussing the problem with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We choose GCD and BITCOUNT as benchmark prob- lems because in our previous experiments the contained bug could not be fixed correctly for both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, the problems consist of a relatively small number of code lines which allows us to discuss these problems in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Figure 3 shows an example discussion with ChatGPT about the GCD problem (lines 1–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In the first response (lines 10–15), ChatGPT does not present any solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' It asks for more information about the bug (we observed this behavior for many other problems, see Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Since the given function causes recursion issues for many possible inputs, we give ChatGPT an exact input example and the resulting error message from Python (lines 17–19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' By mentioning the recursion issue, the final response goes in the right direction and we get a correctly working patched version (lines 30–34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' In Figure 4 we see an example discussion with ChatGPT about the BITCOUNT problem (lines 1–9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Again, ChatGPT asks for more information about the problem and for an input that causes an error (lines 11–13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' As follow-up request, we give ChatGPT a description of what the function should do (based on a code comment from QuixBugs) and ignore the request for an example input to see how ChatGPT reacts (lines 15 and 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We can see in the following answer (lines 18–25) that there is clearly a relation between ChatGPT’s first and second answer because now we get an explanation of how we can test the function with some test inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We respond with a problem description for a test input and describe that there is probably an issue with an infinite loop (lines 27 and 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ChatGPT responds with four code snippets where the first two (lines 34–48) do not solve the problem with the infinite loop and the last two (lines 50–59) are complete but working re-implementations which, however, not directly address the contained bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' It seems that ChatGPT simply returns functions here that somehow fit the content of the problem discussion, even though the test cases mentioned by ChatGPT show that the first two functions cannot work correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Also the bug is not simply fixed by replacing n ˆ= n - 1 with n &= n - 1 in the given function, but ChatGPT, as al- ready mentioned, returns two complete re-implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, both observations are not particularly surprising for a language model based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Nevertheless, the given answers would be useful for a programmer as they help to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Systematic Follow-up Requests for ChatGPT Next, we conducted a study where we systematically discuss with ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For those programs for which the contained bug was not correctly addressed by ChatGPT (see Table I), we provide ChatGPT with a follow-up request giving a hint, as specified in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We report our results in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We use the same notation as before with the addition that a checkmark with an asterisk (\x13*) defines that a solution was found without a follow-up request being necessary in this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' TABLE II: Results achieved by ChatGPT with additional information given in a follow-up request for the unsolved benchmark problems (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Benchmark problem ChatGPT bitcount \x13 depth-first-search \x13* detect-cycle \x13* gcd \x13 hanoi \x13 kheapsort \x17 kth \x13 lcs-length \x17 levenshtein \x13 lis \x17 longest-common-subsequence \x17 max-sublist-sum \x13 minimum-spanning-tree \x13 next-permutation \x13 powerset \x13 rpn-eval \x17 shortest-path-lengths \x17 sieve \x13* to-base \x17 topological-ordering \x17 wrap \x17 ΣΣΣ (Solved) 9 (12) For 9 benchmark problems we see that a more detailed description of the bug is helpful for ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' For 3 bench- mark problems no follow-up request was necessary in this run, since the bug was correctly addressed in the response given on our first request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Overall, adding a hint to ChatGPT vastly improves its performance, with 31 out of 40 problems solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ChatGPT thus offers an exciting new way of approaching automated program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' THREATS TO VALIDITY It is worth noting that ChatGPT is currently under active development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' During our study there was a major update to it, which might have influenced our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Although we observed repairability rates before and after the update to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, future releases might yield different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Furthermore, ChatGPT allows for conversation with its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Asking a different question than the ones presented in this study could potentially have a different impact on results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' To mitigate this threat to validity, we conducted a pre-study, varying the questions asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We noted no significant influence on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Moreover, the results might vary depending on the programming language, size of the benchmarks, and the number of queries issued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' To mitigate these threats, we chose a standard benchmark set and targeted Python – the most popular programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='8 The classification of the results was done manually and therefore represents the subjective assessment of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' To enable a verification of our results, we made our conversations with ChatGPT available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK To support programmers in finding and fixing software bugs, several automated program repair (APR) methods have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ChatGPT, a recently presented deep learning (DL) based dialogue system, can also make suggestions for improving erroneous source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' However, so far the quality of these suggestions has been unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Therefore, we compared in this work the automatic bug fixing performance of ChatGPT with that of Codex and several dedicated APR approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We find that ChatGPT has similar performance to Codex and dedicated DL-based APR on a standard benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' It vastly outperforms standard APR methods (19 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' 7 out of 40 bugs fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Using ChatGPT’s dialogue option and giving the system more information about the bug in a follow-up request boosts the performance even further, giving an overall success rate of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' This shows that human input can be of much help to an automated APR system, with ChatGPT providing means to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Despite its great performance, the question arises whether the mental cost required to verify ChatGPT answers outweighs the advantages that ChatGPT brings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' Perhaps incorporation of automated approaches to provide ChatGPT with hints as well as automated verification of its responses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=', through automated testing, would yield ChatGPT to be a viable tool that would help software developers in their daily tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' We hope our results and observations will be helpful for future work with ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was partially supported by UKRI EPSRC grant no.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdFAT4oBgHgl3EQfrh5X/content/2301.08653v1.pdf'} diff --git a/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/2301.05323v1.pdf.txt b/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/2301.05323v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..033c08e7a7a291a0833beb0180b1b9ce593dfb7e --- /dev/null +++ b/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/2301.05323v1.pdf.txt @@ -0,0 +1,1535 @@ +Salient Object Detection for Images Taken by People With Vision Impairments +Jarek Reynolds*, Chandra Kanth Nagesh*, and Danna Gurari +* denotes equal contribution +University of Colorado Boulder +Abstract +Salient object detection is the task of producing a bi- +nary mask for an image that deciphers which pixels be- +long to the foreground object versus background. We in- +troduce a new salient object detection dataset using images +taken by people who are visually impaired who were seek- +ing to better understand their surroundings, which we call +VizWiz-SalientObject. Compared to seven existing datasets, +VizWiz-SalientObject is the largest (i.e., 32,000 human- +annotated images) and contains unique characteristics in- +cluding a higher prevalence of text in the salient objects +(i.e., in 68% of images) and salient objects that occupy a +larger ratio of the images (i.e., on average, ∼50% cover- +age). We benchmarked seven modern salient object detec- +tion methods on our dataset and found they struggle most +with images featuring salient objects that are large, have +less complex boundaries, and lack text as well as for lower +quality images. We invite the broader community to work on +our new dataset challenge by publicly sharing the dataset at +https://vizwiz.org/tasks-and-datasets/salient-object. +1. Introduction +Locating the most prominent foreground object in an im- +age is a core computer vision problem, often referred to +as salient object detection (as well as salient object seg- +mentation and foreground object detection/segmentation) +[8,12,32,40]. This work is motivated by the desire to have +salient object detection models work well for images taken +by people who are blind or with low vision1 (i.e., people +with vision impairments). Such a feature could offer sev- +eral benefits to this community. For example, it could con- +tribute to privacy-preservation for photographers who rely +on visual assistance technologies to learn about objects in +their daily lives, using mobile phone applications such as +Microsoft’s Seeing AI, Google Lookout, and TapTapSee.2 +1For people with low vision, solutions do not exist to correct their vi- +sion (e.g., by wearing glasses, surgery). +2Many companies record submitted data as evidence that potentially +could be needed for legal reasons. +Figure 1. +Example images demonstrating unique features of +our new VizWiz-SalientObject dataset when compared to other +datasets. The salient objects commonly contain text and occupy +a larger portion of the image (i.e., high coverage). +All content except the foreground content of interest could +be obfuscated, which is important since private information +is often inadvertently captured in the background of images +taken by these photographers [24]. Additionally, localiza- +tion of the foreground object would empower low vision +users to rapidly magnify content of interest and also enable +quick inspection of smaller details [21,39]. +Many salient object detection datasets have been created +to enable progress in algorithm development [7,8,22,42]. A +limitation of existing datasets is they are typically built us- +ing high-quality images collected from photo-sharing web- +sites on the Internet. As we will show in Section 3.2, such +images commonly lack many characteristics that can be ob- +served in real-world settings, especially for visual media +taken by visually impaired photographers who are trying +to learn about the content they photograph [24], often pho- +tographing distinct types of content such as objects showing +text [25], and cannot verify visual quality [13]. +To fill this gap, we introduce a new salient object de- +tection dataset based on images captured in an authentic +use case where visually impaired photographers shared their +images to solicit assistance in learning about the visual con- +tent. We created this dataset by crowdsourcing the collec- +1 +arXiv:2301.05323v1 [cs.CV] 12 Jan 2023 + +rCableSales +SUPPLY +ext Present +口 +WASYOURTRIP? +ANTTOHEARFR +OUR ON-LINB SUR +pse hcwyou want to take the +XILLL +uter +gli.ols.sgizmo.com/s3l +pne: +KUCLO LCINNO CICMO G +gli.olsiphone.sgizmo.co +Pad +LOLIUILKIUtion of salient object annotations for nearly 40,000 images +taken from the VizWiz-Captions dataset [25]. Examples +of resulting annotated images are shown in Figure 1. Af- +ter applying quality control filtration steps, our final dataset +consists of 32,000 annotated images. We call our dataset +VizWiz-SalientObject (or VizWiz-SO). We conduct a de- +tailed analysis revealing how this new dataset relates to ex- +isting datasets. When comparing our salient objects to the +visual evidence needed to answer questions the photogra- +phers asked about their images (i.e., taken from the VizWiz- +VQA-Grounding dataset [11]), we observe that over half +the time the necessary visual evidence is the salient ob- +ject. When comparing our dataset to seven existing datasets, +we observe VizWiz-SalientObject is the largest (i.e., 32,000 +human-annotated images) and is unique in its higher preva- +lence of text in the salient objects (i.e., in 68% of images) as +well as salient objects occupying a larger ratio of the images +(i.e., on average, ∼50%). +We also benchmark modern salient object detection al- +gorithms on our new dataset to uncover open challenges +for the research community. Experiments with seven al- +gorithms reveal that they struggle most for images with +salient objects that are large, have less complex bound- +aries, and lack text as well as for lower quality images. +To facilitate progress on these challenging problems, upon +publication, we will publicly-share the dataset and an +evaluation server with leaderboard at the following link: +https://vizwiz.org/tasks-and-datasets/salient-object. +In summary, our new dataset supports the development +of more generalized algorithms that not only address the in- +terests of people with vision impairments but also can ben- +efit related applications that encounter similar real world +challenges observed in our dataset. Relevant applications +include robotics, lifelogging, and privacy protection. +2. Related Work +Salient Object Detection Datasets. +Over the past cou- +ple of decades, many datasets were introduced to facili- +tate improving the design of algorithms that address salient +object detection problems. Several survey papers provide +comprehensive characterizations of the tens of datasets de- +signed for this task [7, 8, 22, 42]. A common observation +is that datasets were artificially constructed around high- +quality images which often feature salient objects in the +center of the images with a high contrast against the back- +ground. This is a mismatch from many real-world settings, +especially for visual media taken by visually impaired pho- +tographers who often photograph distinct types of content, +such as objects showing text [25], with the aim to learn +about that content. We introduce the first salient object de- +tection dataset based on images taken by visually impaired +people in an authentic use case where they were trying to +learn about their visual surroundings. Compared to seven +modern datasets, our dataset is larger, has a high prevalence +of salient objects containing textual information, and shows +objects that occupy larger portions of the images. +Salient Object Detection Algorithms. +Researchers have +designed novel algorithms to automatically perform salient +object detection for over 20 years, with the status quo since +2015 being that state-of-the-art methods employ neural net- +works trained on large-scale annotated datasets. +Several +survey papers provide comprehensive characterizations of +the hundreds of algorithms for this task [7,8,22,42]. While +convolutional neural network (CNN) based models became +the mainstream method [1, 33, 43] in 2015, transformer +based models [30, 44] have become the mainstream ap- +proach over the past few years. To assess how well mod- +ern methods perform on our new dataset, we benchmark +seven modern methods. We observe that existing methods +fall below human performance and struggle most for salient +objects that lack text and occupy a larger ratio of the image. +Visual Assistance Technologies. +Visually impaired peo- +ple can share their visual media (images and videos) with +various technologies [3, 4, 6, 14, 18, 27, 32, 40] in order to +receive assistance for daily tasks such as deciding what to +eat, wear, and buy [10,24]. The widespread impact of such +technologies for real users is exemplified by reports from +some of these companies that the technologies have 10s to +100s of thousands of users who have submitted millions of +assistance requests [5,9,14,17]. The most common reported +goal for using such technologies is to learn about a (salient) +object [9,10,23,28,47]. Given this common use case, salient +object detection models could help for privacy preservation. +Specifically, images (or video frames) could be edited be- +fore being shared with companies, by obfuscating the back- +ground, in order to reduce inadvertent disclosures of pri- +vate content that often appears in the background of images +taken by visually impaired photographers [24]. +3. VizWiz-SalientObject Dataset +We now introduce our new salient object detection +dataset, we call VizWiz-SalientObject (VizWiz-SO). +3.1. Dataset Creation +Image Source. +We focus on images taken by visually im- +paired people who shared them in an authentic use case +where they were soliciting visual assistance. Specifically, +we leverage the 39,181 labeled images from the VizWiz- +Captions dataset, each of which is paired with five crowd- +sourced captions [25]. Observing that images from these +photographers can have severe quality issues resulting in no +detectable salient object (e.g., extreme blur or inadequate +illumination), we did not use the images which were cap- +tioned as follows by at least four of the five crowdworkers: +2 + +“Quality issues are too severe to recognize visual content.” +We also did not use the small images (i.e., both the height +and width were less than 300 pixels) because of the chal- +lenges of collecting precise annotations for such images. +This left us with 37,120 images for our annotation task. +Task Design. +Our task interface for segmenting salient +objects begins with a comprehensive instruction set at the +top detailing both how to navigate the interface and how to +complete challenging annotation scenarios. Next, it shows +an image alongside two preliminary questions for verifying +there is a single, unambiguous foreground object. The first +question asks “Is the image showing a screenshot?” If the +answer is “yes”, we conclude the image lacks a salient ob- +ject. Next, we ask the more general, direct question of “Is +there a single unambiguous foreground object?” An anno- +tator is only prompted to segment the foreground object for +images deemed by these preliminary questions to show a +single, unambiguous foreground object. +To demarcate the boundary of the salient object, the in- +terface collects a series of points that are connected into +polygon(s). When segmenting the salient object, the an- +notator is required to remove any holes (e.g., donut) as well +as capture all object parts when occlusions break a salient +object into more than one polygon (e.g., hand obfuscates a +pencil into two parts). The annotator also has an option to +select a button indicating that the salient object occupies the +full image. We provide more details about the task interface +as well as a screenshot of it in the Supplementary Materials. +Annotation Collection. +We leveraged the benefits of +an around-the-clock distributed workforce by crowdsourc- +ing annotations via Amazon’s crowdsourcing marketplace, +Amazon Mechanical Turk (AMT). +Although AMT can support our large-scale annotation +needs, it brings concerns about annotation quality due to the +anonymous nature of the crowdsourced workforce. Con- +sequently, we implemented several measures to ensure the +collection of high-quality annotations, as summarized be- +low. First, we restricted who were potential candidates for +our task. +We only accepted workers who had at least a +98% acceptance rate while having completed at least 500 +Human Intelligence Tasks (HITs) on AMT. Moreover, to +encourage understanding of our initial and ongoing task in- +structions, we opted for crowdworkers only from the United +States since that provided us confidence that they have +English-language proficiency. In addition, we also required +crowdworkers to pass a qualification assessment covering +five challenging annotation scenarios documented in our in- +structions. The qualification images feature foreground ob- +jects consisting of complex boundaries, holes within the ob- +ject, and occlusions obfuscating portions of the foreground +object. Consequently, the task required crowdworkers to +demonstrate an understanding for how to generate multi- +ple polygons, annotate holes, handle occlusions, and draw +complex boundaries. +We employed 40 AMT crowdworkers who completed +our qualification task to complete annotations of all images. +For each of the 37,120 images, we collected two annotations +from the crowdworkers.3 During annotation collection, we +monitored ongoing quality by tracking each worker’s per- +formance with respect to their frequency of indicating the +presence of full-screen annotations or no prominent fore- +ground object as well as the level of detail they provided in +their segmentations (e.g., high prevalence of triangles). Cu- +mulatively, the crowdworkers took 1,290 annotation hours +over 11 days to complete annotating the 37,120 images. +Annotation Post-Processing. +We next analyzed the re- +dundant annotations per image to determine how to use each +annotated image in the final dataset. +First, we removed +3,662 images for which workers agreed there was no sin- +gle, unambiguous salient object, which occurred when both +annotators either answered “Yes” to “Is the image a screen- +shot?” or “No” to “Is there a single most prominent fore- +ground object?” Next, we manually inspected 7,443 images +for which workers disagreed on the answers to either of the +two preliminary questions and determined whether there is +indeed a single, unambiguous object. Finally, with all im- +ages deemed to have a single, unambiguous salient object, +we determined which annotation to assign as ground truth. +To assist in this process, we computed the intersection over +union (IoU) score between the two segmentations for all +images with two or more segmentations. With IoUs ≥ 0.90, +we deemed both annotations high quality and randomly se- +lected one as ground truth. For the remaining 2,951 images +with IoUs< 0.90, we manually reviewed the annotations to +decide whether one was correct or whether the image should +be discarded due to foreground object ambiguity. +3.2. Dataset Analysis +We now characterize the VizWiz-SalientObject (VizWiz- +SO) dataset and how it relates to existing datasets. +3.2.1 +Salient Objects vs Answer Groundings for VQA +We first explore how the target content the photographers +were asking about relates to an image’s salient object. To +do so, we compare the annotations of the visual evidence +needed to answer questions about the images, i.e., an- +swer groundings provided in the VizWiz-VQA-Grounding +dataset [11], to the annotations of the salient objects in our +dataset. We first identified all annotated images that were in +3For a subset of images, we collected four annotations to support fur- +ther analysis of human annotation performance, which we describe in the +Supplementary Materials. +3 + +Figure 2. The histogram summarizes for 6,540 images the fre- +quency of observing different levels of similarity between two +segmentations per image, which show the salient object and the +visual evidence needed to answer the photographer’s question re- +spectively. These findings reveal that visually impaired photogra- +phers often want to learn about the salient objects in their images. +common across the two datasets, yielding a total of 6,540 +images. For each image, we then measured the similarity +between the answer grounding and salient object segmenta- +tions using the IoU metric. We visualize our results using a +histogram where we categorize each image into one of ten +interval bins starting with IoU=[0.0, 0.1), incrementing in +intervals of 0.1, and ending with IoU=[0.9, 1.0). Results +are shown in Figure 2. +We observe that about half of the images have a high sim- +ilarity between the salient object and VQA answer ground- +ing; e.g., 46% had an IoU ≥ 0.9. This reveals that visually +impaired photographers often are trying to learn about the +salient object in their images when trying to get answers to +their visual questions. +We also observe that roughly one quarter of the images +have a very low similarity between the salient object and +VQA answer grounding; i.e., 25.7% of images had an IoU +< 0.1. We manually reviewed these 1,680 images with IoUs +less than 0.1 to understand the reasons for this finding. We +discovered that 95% (i.e., 1,599) of these images have a +salient object featuring a full-screen or large region while +the VQA answer grounding captures a small aspect of the +salient object. Examples include expiration dates on food +packages or the current page number of an open book. The +remaining 5% (i.e., 81) of these images featured a VQA an- +swer grounding unrelated to the salient object. +More generally, we observe that the IoU scores follow a +U-shaped distribution with only a small portion of images +having middling scores; e.g., 7.9% (i.e., 511) of images had +an IoU ≥ 0.3 and < 0.7. Among these images, we found the +salient object contained the VQA answer grounding region +100% of the time. There are two primary trends that led to +these less common IoU scores. The first trend is that larger +VQA answer grounding regions occur with smaller salient +objects. Examples include brands of cereal, types of soda, +and denominations of currency. The second trend was for +salient objects featuring holes. That is because the VizWiz- +VQA-Grounding dataset did not account for holes in their +annotation task. The absence of annotated holes in only one +of the two segmentations led to lower IoU scores. +Altogether, these findings highlight that a valuable step +for tackling many of this population’s VQA goals is to ini- +tially locate the salient object. That is because the answer +will likely only be grounded in the salient object or the +background rather than their intersection. +3.2.2 +VizWiz-SO vs Existing Datasets +We next compare our dataset to seven datasets: +• DUTS [41]: the most commonly used dataset to train +state-of-the-art algorithms (e.g., [1,30,33,38,43,44]) due +to its large size paired with diverse saliency challenges. +• DUT-OMRON [46]: consist of images showing multiple +salient objects, often with complex backgrounds. This is +a useful reference when considering extending our dataset +to when photographs taken by visually impaired photog- +raphers showing multiple salient objects. We share our +collected metadata indicating when this occurs to facili- +tate this line of future research. +• ECSSD [45]: consists of images featuring complex scenes +that present textures and structures expected to be com- +mon in real-world salient object detection scenarios. +• PASCAL-S [29]: derived from PASCAL VOC’s [16] val- +idation set, it is designed to facilitate salient object seg- +mentation generalization on realistic images. +• HRSOD [48]: explicitly designed for salient object de- +tection on high-resolution images; this is relevant for our +real-world application since images taken by people with +vision impairments often are relatively high resolution. +• UHRSD [44]: currently the largest ultra-high resolution +salient object detection dataset, which is relevant to our +work since images taken by people with vision impair- +ments can be ultra high resolution. +• DAVIS-S [48]: derived from DAVIS [36], a densely an- +notated video segmentation dataset. This is relevant for +our real-world application to analyze implications for +video frames since visually impaired photographers often +stream live video with their cameras when using visual +assistance technologies [4,18]. +Of note, images in six of these datasets originate from the +Internet on photo-sharing websites such as Flickr [29, 41, +44–46, 48], and so likely are high quality since they were +deemed of sufficient quality to upload to the Internet.4 +4The origins of the images for the final dataset is not reported [48]. +4 + +50% +46.0% +3.40% +Image +30% +25.7% +20% +9.0% +10% +4.4% +4.8% +2.4% +2.1% +1.7% +1.7% +2.2% +[0.8, 0.9] +[0.9. +[O. +[o. +0 +4. +0.6) +,1.0) +0.3) +IoU SimilarityDAVIS-S [48] +PASCAL-S [29] +HR [48] +ECSSD [45] +DUT-O [46] +UH [44] +DUTS [41] +Ours +Images +92 +850 +2,010 +1,000 +5,168 +5,920 +15,572 +32,000 +Text +13% +24% +15% +15% +11% +19% +13% +68% +MR +22% +31% +25% +9% +17% +35% +19% +1% +Holes +82% +50% +62% +29% +28% +75% +41% +4% +Table 1. Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to how many images +are included (“Images”), the percentage of images that have text present in the salient objects (“Text”), the percentage of images that have +salient objects consisting of more than one region (“MR”), and the percentage of images that have salient objects containing any holes +(“Holes”). As shown, our dataset is distinct in that it contains more images, more salient objects with text present, more salient objects +consisting of one region, and fewer salient objects containing holes. (HR=HRSOD; UH=UHRSD) +Figure 3. Summary statistics for ours and seven other datasets with respect to four measures. Each box reveals statistics about all salient +objects in a particular dataset with the central mark capturing the median value, box edges the 25th and 75th percentiles values, whiskers +the most extreme data points not considered outliers, and individually plotted points the outliers. Our dataset is unique in that salient objects +tend to have less complex boundaries, occupy larger portions of an image, and exhibit a greater diversity of sizes relative to the image. +For each salient object in every dataset, we characterize +it in six ways. Three measures focus on detecting the pres- +ence versus absence of particular properties for the salient +object. These are whether the salient object contains text 5, +consists of multiple regions 6, or contains any hole(s). The +remaining three measures characterize the salient region it- +self. First, we identify the position of an object within an +image by measuring its center of mass relative to the im- +age coordinates, resulting in x and y coordinate values in +the range between 0 to 1. Next, we characterize the ob- +ject’s boundary complexity by computing its isoperimetric +inequality, which is the ratio of the object’s area to the +length of its perimeter. +Values range from 0 to 1, with +larger values indicating simpler boundaries that are less +jagged/dented (e.g., a circle). Finally, to gauge the relative +size of a salient object in the image, we compute its cover- +age ratio, meaning the fraction of all image pixels that are +occupied by the salient object’s pixels. +We show summative statistics of our findings per dataset +in Table 1 and Figure 3. In particular, in Table 1, we re- +5We obfuscate all image content but the salient object and then check +whether Microsoft Azure’s OCR API returns text. +6Multiple regions means there are multiple separate polygons. This can +occur either because multiple salient objects were annotated or because of +occlusions that lead to more than one region for a single salient object. +port how many images are in each dataset paired with what +percentage of those images have salient objects with text, +multiple regions, and holes. In Figure 3, we visualize statis- +tics summarizing the values for each dataset’s salient ob- +jects with respect to center of mass, boundary complexity, +and coverage ratio using boxplots. +While our findings highlight that our VizWiz-SO dataset +has many distinct characteristics, one commonality it has +with most existing salient object detection datasets is that +the salient objects typically occupy centered positions +within an image. Specifically, in Figure 3, we observe this +trend for all datasets except HRSOD. We found this some- +what surprising since visually impaired photographers can- +not visually inspect their images to verify they are conform- +ing to the common photographer’s bias of centering con- +tents of interest they are trying to photograph. Yet, given +our findings from Section 3.2.1 that photographers often are +interested in learning about an image’s salient object, our +findings suggest these photographers have skills in center- +ing contents of interest in pictures they take. +A unique aspect of our VizWiz-SO dataset is that it fea- +tures more salient objects with textual data. Specifically, +68% of salient objects in VizWiz-SO contain text while the +dataset with the next highest prevalence of text, PASCAL- +S [29], only has it for 24% of the images (Table 1). A gap of +5 + +Center of mass Y-axis +Center of mass X-axis + Boundary Complexity +Coverage Ratio +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +DAVIS-S +PASCAL-S +HRSOD +ECSSD +DUT-OMRON +UHRSD +DUTS +Ours: VizWiz-SOthis magnitude (i.e., 44 percentage points) suggests that our +new dataset offers a considerable domain shift in the salient +object detection problem space. We suspect part of this shift +stems from the types of salient objects included, with many +more daily objects such as products (e.g., food packages) +included in our VizWiz-SO dataset. +Another unique aspect of VizWiz-SO is that far fewer +images feature salient objects that consist of multiple re- +gions; i.e., only 1% of images (Table 1). +We suspect +this distinction stems from our unique approach of adopt- +ing a rigorous annotation preprocessing step, where we re- +quire crowdworkers to verify images have one unambigu- +ous salient object before allowing them to annotate images +for use in our final dataset. Any remaining objects in our +dataset with multiple regions are therefore highly likely a +result of occlusions breaking a single salient object into +multiple polygons, which evidently is incredibly rare. +VizWiz-SO is also unique due to the rarity in which +salient objects contain holes; i.e., only observed for 4% +of images (Table 1). From visual inspection, we suspect +this finding reflects a domain shift in the types of content +found in the datasets. For example, examples from other +datasets of objects with holes include people riding bikes, +people dancing, and animals in intricate poses. +In con- +trast, in VizWiz-SO, objects with holes include retail pack- +aging made to hang from hooks, pairs of scissors, and coffee +mugs. We posit the lower prevalence of holes in VizWiz-SO +stems from the fact that images originate from an authentic +use case where photographers primarily photograph house- +hold and retail items, which naturally feature fewer holes. +A further distinction of our VizWiz-SO dataset is that the +salient objects tend to have less complex boundaries (Fig- +ure 3). We suspect this is again because of a domain shift in +the types of objects in our dataset, with many more human- +made items, such as food packaging boxes and cans, that by +design are typically more structured in shape. +A final distinction of salient objects in our VizWiz-SO +is how much of the image they occupy (Figure 3). First, +they tend to occupy a much larger amount of the image than +observed in other datasets. Specifically, they on average oc- +cupy roughly half of all image pixels, with a mean coverage +ratio of 0.5 and a median of 0.46. In contrast, the dataset +with the next highest coverage ratio statistics is PASCAL- +S [29], and over 75% of its images contain salient objects +that occupy less than half of the image pixels. We attribute +this distinction to the authentic use case of our dataset, +where visually impaired photographers attempting to learn +about the salient objects they are photographing seem to be +taking zoomed-in or close-to-camera images of the content +of interest. Another unique aspect of our salient objects, is +that they exhibit a larger range of sizes, as shown by the +gaps between the 25 and 75 percentile values of each box. +For example, PASCAL-S features the next largest interquar- +tile range with a 23% gap(i.e., 19% to 42%). In contrast, +the gap for VizWiz-SO is more than twice as large at 56% +(i.e., 22% to 78%). Consequently, a unique challenge of +our dataset for algorithms is that they no longer can assume +a strong bias regarding a salient object’s relative size. +4. Algorithm Benchmarking +We benchmark modern salient object detection algo- +rithms to show how they perform on our new dataset. We +conducted all experiments on a Nvidia A100 GPU. +4.1. Experimental Design +Dataset Splits. +We use the existing splits available for the +VizWiz-Captions dataset [25], which translates to approxi- +mately a 60/20/20 training, validation and test split for our +VizWiz-SO dataset. In particular, from the 32,000 anno- +tated images, the number of images in each split respec- +tively is 19,116, 6,105, and 6,779. +Evaluation Metrics. +We evaluate each model with re- +spect to five popular metrics for salient object detection +models: Mean Absolute Error (MAE), Structure Measure +(Sm), Mean F-Measure (Fm), Enhanced Alignment Mea- +sure (Em), and Intersection over Union (IoU). +Algorithms. +We benchmark the following seven methods +from the past three years to assess the difficulty of our new +dataset for modern salient object detection models: +• Boundary Aware Segmentation Network (BASNet) [38]: +an appealing model for real-time applications like our tar- +get use case because it can achieve 70fps during inference +time while achieving competitive performance (i.e., was +a top-performer in 2019). +• Fusion, Feedback and Focus Network (F3Net) [43]: state- +of-the-art performing model on five datasets in 2020. +• U2 Network (U2Net) [1]: an appealing model for real- +world applications like our target use case because it has +a very light footprint (4.7MB), and so is more suitable +for resource-constrained devices such as smartphones. It +achieved competitive performance in 2020. +• Visual Saliency Transformer (VST) [30]: achieved state- +of-the-art performance in 2021, and is based purely on a +transformer architecture. +• Pyramidal Feature Shrinking Network (PFSNet) [33]: +achieved state-of-the-art performance on five datasets in +2021; it consists of a decoder that aims at using aggre- +gated adjacent feature nodes hierarchically to avoid the +problem of leaping feature fusion. +• Pyramid Grafting Network (PGNet) [44]: introduced in +2022, it is a one-stage framework based on a transformer +6 + +HP +BASNet +F3Net +U2Net +VST +PFSNet +PGNet +DIS +VST-FT +VST-S +[38] +[43] +[1] +[30] +[33] +[44] +[37] +Attr. +Backbone +- +R-34 +R-50 +- +T2T-ViT +R-50 +R-18+SWIN +U2Net +VST +ViT +Training set +- +D +D +D +D +D +D+HR +DIS5K +D+VW +VW +Input size +- +2562 +3522 +3202 +2242 +3522 +2242, 10242 +10242 +2242 +2242 +Size (MB) +- +333 +98 +4.7 +171 +120 +280 +169 +171 +171 +VizWiz-SO +MAE ↓ +0.02 +0.28 +0.28 +0.26 +0.17 +0.32 +0.21 +0.36 +0.19 +0.21 +Sm ↑ +0.92 +0.59 +0.55 +0.61 +0.65 +0.48 +0.62 +0.46 +0.64 +0.63 +Fm ↑ +0.96 +0.77 +0.74 +0.80 +0.83 +0.70 +0.79 +0.61 +0.74 +0.72 +Em ↑ +0.97 +0.64 +0.65 +0.65 +0.76 +0.60 +0.74 +0.55 +0.77 +0.70 +IoU ↑ +0.94 +0.62 +0.53 +0.63 +0.73 +0.48 +0.67 +0.49 +0.70 +0.69 +Table 2. Analysis of existing algorithms that we benchmark on our VizWiz-SO dataset, including both off-the-shelf models (which are +cited) as well as those fine-tuned (-FT) and trained from scratch (-S). We first report differentiating attributes between the algorithm +architectures and then present the model performance with respect to five widely-used metrics. (HP=Human Performance; R=ResNet [26]; +ViT=Vision Transformer [15]; Swin=Shifted window transformer [31]; D=DUTS-TR [41]; VW=VizWiz-SO; HR=HRSOD [48]) +and CNN backbone that achieves state-of-the-art perfor- +mance on five benchmark datasets. [41,44,46,48]. +• Dichotomous Image Segmentation (DIS) [37]: also in- +troduced in 2022 as the state-of-the-art method for the +DIS5K [37] dataset; it is designed for detecting salient +object in high resolution images, which makes it relevant +for our use case where many images coming from people +with vision impairments are relatively high resolution. +We further characterize each model by identifying the +backbone architecture used in the architecture, datasets used +for training, image size used for training, and model foot- +print. These characteristics are reported in Table 2. +All models predict saliency maps that represent the +brightness of certain pixels within the same spatial reso- +lution as the input image; e.g., ∈ [0, 1] or alternatively +∈ [0, 255]. The predictions generated by salient object de- +tection models are converted into binary masks. +Humans. +We also evaluate human performance to estab- +lish an upper bound for what we should strive for from au- +tomated methods. Since, we get two human annotations per +image in our dataset, we calculate human performance by +comparing the two annotations in cases where the IoU is +greater than 0.90. +4.2. Performance for Off-The-Shelf Models +We first evaluate each of the algorithms as is in their orig- +inal design. Results are shown in Table 2. +We observe that VST [30] is the top-performing model. +Yet, it still falls short of human performance. For exam- +ple, the gap in performance is 0.15 in terms of MAE, 0.211 +in terms of IoU, 0.26 for Sm, and 0.2 for Em. +Conse- +quently, this dataset offers a new challenging benchmark +for the community. +A further observation is that the models perform poorly +on the VizWiz-SO dataset in comparison to their perfor- +mance on the original datasets for which they were bench- +marked. For example the MAE and Sm performance ob- +served by PGNet [44] on DUTS-TE is 0.028 and 0.912 re- +spectively versus 0.2123 and 0.6233 respectively for our +dataset. We hypothesize that part of the reason for this poor +performance is that models trained and evaluated on other +datasets are not able to learn how to generalize to the real- +world challenges that arise for images taken by visually im- +paired photographers. +4.3. Performance When Training on VizWiz-SO +We next explore whether training the top-performing al- +gorithm, VST [30], on our new dataset will lead to improved +performance. To do so we analyze two additional models: +(1) the pretrained VST [30] model fine-tuned on VizWiz- +SO (VST-FT) and (2) the pretrained VST [30] algorithm +trained from scratch on VizWiz-SO (VST-S). We use the +default hyperparameters reported in the VST [30] paper for +model training. Results are shown in Table 2. +We observe that both models, i.e., created by training +from scratch and fine-tuning on our VizWiz-SO dataset, +achieve worse results than the baseline of not training the al- +gorithm on our dataset. This suggests that the training data +used by algorithms is not the only culprit for what makes +our new dataset challenging. Rather, our findings suggest +that new algorithmic frameworks are also needed to achieve +strong generalization performance on our new dataset. +4.4. Fine-grained Analysis +We next conduct fine-grained analysis to better isolate +what makes our dataset challenging for modern algorithms. +To do so, we divide our VizWiz-SO test set according to the +following four factors, with the first three based on metadata +collected in Section 3.2 to characterize our dataset: +7 + +BASNet +F3Net +U2Net +VST +PFSNet +PGNet +DIS +VST-FT +VST-S +[38] +[43] +[1] +[30] +[33] +[44] +[37] +Text Presence +True +0.23 +0.22 +0.22 +0.13 +0.25 +0.16 +0.32 +0.16 +0.17 +False +0.35 +0.38 +0.32 +0.24 +0.42 +0.29 +0.40 +0.24 +0.26 +Coverage +Small +0.06 +0.16 +0.07 +0.11 +0.16 +0.12 +0.10 +0.09 +0.11 +Medium +0.15 +0.20 +0.15 +0.09 +0.24 +0.15 +0.25 +0.09 +0.10 +Large +0.60 +0.47 +0.54 +0.30 +0.54 +0.35 +0.70 +0.38 +0.39 +Complexity +High +0.15 +0.21 +0.15 +0.12 +0.24 +0.16 +0.21 +0.11 +0.12 +Low +0.38 +0.34 +0.35 +0.21 +0.38 +0.25 +0.48 +0.26 +0.27 +Image Quality +Good +0.22 +0.23 +0.21 +0.14 +0.26 +0.17 +0.30 +0.16 +0.17 +Poor +0.44 +0.43 +0.41 +0.27 +0.47 +0.34 +0.50 +0.30 +0.31 +Table 3. Fine-grained analysis of existing algorithms with respect to presence of text on the salient object (“Text Presence”), relative size +of the salient object in the image (“Coverage”), relative complexity of the salient object’s boundary (“Complexity”), and image quality +(“Image quality”). As shown the algorithms perform worse when there is salient objects lack text, occupy a large portion of the image, +have less complex boundarys as well as when the image quality is poor. +• Text Presence: +two groups based on whether text is +present in the salient object. +• Coverage Ratio (Coverage): three groups based on the +33rd and 66th quartile values in our dataset. All images +with coverage ratio less than 0.32 has small coverage, be- +tween 0.32 and 0.62 has medium coverage, and greater +than 0.62 has large coverage. +• Boundary Complexity (Complexity): two groups by split- +ting them around the mean score for boundary complex- +ity (i.e., 0.66) with high boundary complexity when the +score is less than the mean and low boundary complexity +otherwise. +• Image Quality: leveraging metadata from prior work [25], +which indicates how many of the five crowdworkers in- +dicated an image as insufficient quality to recognize the +content, we split the images into groups with good qual- +ity being when none of the crowdworkers indicate insuf- +ficient quality and poor otherwise. +Due to space constraints, we only report results in the main +paper with respect to the Mean Absolute Error [35]. Results +for all benchmarked models are shown in Table 3. +In terms of text presence, we see that the models perform +better when there is text present as opposed to when there +is none. For example, the performance drops by 0.11 for +the best performing model, VST. We suspect visual patterns +that arise with text may serve as a valuable cue to models in +locating salient objects. +Next, we see that as the coverage ratio of the salient ob- +jects increase, the models tend to perform worse. For in- +stance, the best performing model, VST, has a performance +dropoff of 0.19 when predicting images with small cover- +age ratios as opposed to large coverage ratios. We see an +even greater performance dropoff from other models such +as 0.60 for DIS. We suspect this performance gap arises in +part from the fact that existing datasets largely lack such +large salient objects, which both could have affected what +algorithms were designed to handle as well what they could +learn from the data they observed. +Further observed trends are that performance drops for +salient objects with lower boundary complexity and for +poorer quality images. These are two additional factors that +reflect domain shifts between our dataset and prior datasets +that could have affected the design of algorithms as well +what they could learn from the data training data. +5. Conclusions +We introduce the VizWiz-SalientObject dataset to en- +courage the community to design more generalized salient +object detection models that can handle a larger range of +challenges motivated by our authentic use case that also can +occur in many real-world applications. We offer our exper- +imental findings from benchmarking modern salient object +detection algorithms as a valuable starting point for iden- +tifying valuable future research directions. To summarize, +new models are needed to better handle salient objects that +are large, have less complex boundaries, and lack text as +well as work well in the presence of lower quality images. +We now close with a discussion of some ethical impli- +cations of our work. While we are motivated to better as- +sist a population that is traditionally marginalized in society, +we acknowledge our work can lead to potentially adverse +social effects. Our concern is primarily centered on bad- +actor behaviors intended to exploit the privacy, autonomy, +and livelihoods of a population demographic inherently sus- +ceptible to such behavior. 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In Computer Vision and Pattern Recogni- +tion (CVPR), 2013 IEEE Conference on, pages 3166–3173. +IEEE, 2013. 4, 5, 7, 13 +[47] Xiaoyu Zeng, Yanan Wang, Tai-Yin Chiu, Nilavra Bhat- +tacharya, and Danna Gurari. +Vision skills needed to an- +swer visual questions. Proceedings of the ACM on Human- +Computer Interaction, 4(CSCW2):1–31, 2020. 2 +[48] Yi Zeng, Pingping Zhang, Jianming Zhang, Zhe Lin, and +Huchuan Lu. Towards high-resolution salient object detec- +tion. In Proceedings of the IEEE/CVF International Confer- +ence on Computer Vision, pages 7234–7243, 2019. 4, 5, 7, +13 +Appendix +This document supplements the main paper with additional +information concerning: +1. Dataset Creation (supplements Section 3.1) +• Annotation Task Interface +• Worker Qualification Task +• Analysis of Workers’ Annotation Differences +2. Dataset Analysis: VizWiz-SO vs Existing Datasets +(supplements Section 3.2.2) +3. Experimental Design (supplements Section 4.1) +A. Dataset Creation +A.1. Annotation Task Interface +The task interface displays five images within a tabbed +container on the left and preliminary questions with task +instructions on the right. A screenshot of the task interface +(without instructions) is shown in Figure 4. +To account for occlusions and holes while keeping the +task simple for annotators, we permitted annotators to gen- +erate multiple polygons. For occlusions, annotators could +use as many polygons as necessary for demarcating fore- +ground objects partitioned into multiple polygons. +For +holes, we apply an even-odd fill rule to images featuring +foreground objects with holes. With an even-odd fill rule, +every area inside an even number of enclosed areas be- +comes hollow, and every region inside an odd number of +enclosed areas becomes filled [34]. By treating the image’s +four corners as the first enclosed area, the outermost bound- +ary of the foreground object becomes the second enclosed +area. Moreover, holes within foreground objects represent +the third layer of enclosed areas and become filled, allowing +annotators to demarcate foreground objects featuring holes. +In practice, annotators first trace the outermost boundary of +the foreground object and close the path by clicking the first +point a second time. We then instructed annotators to trace +any holes within the foreground object, and so those holes +end up in odd-numbered layers. +A.2. Worker Qualification Task +We administered a qualification task for workers to sup- +port our collection of high-quality ground truth annotations. +The qualification task required annotating five images, each +of which features a distinct challenging annotation scenario. +All five images are shown in Figure 5. The first two images +show a table and a bench, offering examples with complex +boundaries and holes. The next two images feature a per- +son holding a coffee mug, to support educating a crowd- +worker about our expectations for annotating objects with +10 + +Figure 4. A screenshot of our annotation task interface. +Figure 5. The five images used for the worker qualification task. +Each was selected to demonstrate a challenging annotation sce- +nario such as complex boundaries, holes, and occlusions. +complex geometries that have many curves and occlusions +that require annotating multiple polygons. The final image +is a spatula. This task verified a crowdworker’s ability to +correctly identify and annotate multiple holes that can arise +within the salient object. +After crowdworkers annotated each qualification image, +the backend code of our website checked if their annotation +was sufficiently similar to the GT annotation (i.e., IoU sim- +ilarity of at least 0.90). Crowdworkers could only proceed +to the following image after they obtained an IoU ≥ 0.90 +on the current image. Crowdworkers obtaining an IoU ≥ +0.90 on all five qualification assessment images on a per- +image basis gave us substantial confidence that they would +be able to successfully handle complex and challenging out- +lier cases within the original VizWiz Dataset.7 +A.3. Analysis of Workers’ Annotation Differences +We collected a larger number of redundant annotations +per image for a random subset of images to better explore +when and why annotation differences are observed from dif- +ferent workers. Specifically, for this analysis, we collected +four annotations as opposed to two for a subset of 1,237 im- +ages. Examples of the redundant annotations collected per +image are shown in Figure 6. +The first example (i.e., row 1 of Figure 6) highlights that +annotation differences can stem from challenging annota- +tion scenarios where objects contain holes (e.g., in mug han- +dle) or are occluded (e.g., by the straw). For instance, the +hole was not annotated in the third annotation. Addition- +ally, only the fourth annotation captured the occlusion that +arises from the straw. +The second example (i.e., row 2 of Figure 6) highlights +that annotation differences can stem from ambiguity regard- +7Some crowdworkers did not pass the qualification assessment due to +time constraints. In these cases, crowdworkers would contact us with the +images they annotated. If we were confident in their annotation abilities, +we manually added these crowdworkers to the qualified worker pool. +11 + +Image 1 +Image 2 +Image 3 +Image 4 +Image 5 +Work may be rejected for not following instructions +Step 1: Is the image showing a screenshot? +O Yes +O No +WASYOURTRIP? +Step 2: Is there a single prominent foreground object? +OYes +ONO +ANTTOHEARFRO +OUR ON-LINE SURVEYTONACHANCETO WIN SU +Step 3: Demarcate the prominent foreground object +osehcwyou wantto takethesunvey +typethatweb address inyourbrowse +puter +one: +gli.ols.sgizmo.com/s3/ +Prev Image +Next Image +Pad +gli.olsiphone.sgizmo.com/s3 +gli.olsipad.sgizmo.com/s3/ +Select Full +Undo Last +Clear All +Image +Point +PolygonsFigure 6. Example annotations from our random subset where we +collected four annotations as opposed to two. We find worker dif- +ferences primarily occur in challenging annotation scenarios such +as holes, occlusions, complex boundaries, and object saliency. +ing what is the salient object. As shown, the first two an- +notations flag the image as lacking a foreground object, +the third annotation identifies the child holding the cup as +the salient object, and the fourth annotation identified the +child’s cup as the salient object. +The third example (i.e., in row 3 of Figure 6) highlights +that annotation differences also can arise for objects that +simultaneously have complex boundaries and holes. In an- +notation one, the worker did not fully annotate the salient +object, cutting out part of the object from the annotation. +Only the third and fourth annotations accurately annotate +all holes that are present in the salient object’s boundary +while also having tight boundaries in the annotation. +In summary, we found occlusions, holes, and saliency +ambiguity to be the primary factors contributing to annota- +tion differences. In the case of occlusions, worker differ- +ences can arise when deciding whether to include objects +that are a composite part of the salient object. In the case +of holes, annotation differences can arise regarding which +holes to annotate. Last, we found that it can be ambiguous +as to which object is the most salient. +Figure 7. Example ground truth annotations from the HRSOD +dataset which exemplify that salient objects are not usually not +centered in the image. This is a common trend in the dataset. +B. Dataset Analysis +B.1. VizWiz-SO vs Existing Datasets +We present finer-grained details about typical image res- +olutions for the different salient object detection datasets +to expand upon discussions in the main paper about how +VizWiz-SO relates to other datasets. Specifically, we report +the median image width (Med. W), median image height +(Med. H), and whether the dataset supports high resolu- +tion images (High Res.) as defined by whether the median +image height and width are greater than 1080 and 1920 re- +spectively. Results are reported in Table 4. We observe that +our new dataset, overall, provides higher resolution images +than most datasets. +We also expand on a surprising finding reported in our +main paper that the HRSOD dataset is the only one for +which salient objects do not occupy the typical center po- +sitions. To do so, we visualize the ground truth masks of +some non-centered objects in Figure 7. In row one, we see +that objects are horizontally distributed to left and right po- +sitions of the images. Similarly, we observe in row two that +the salient objects are vertically distributed to the top and +bottom positions of the images. +C. Algorithmic Benchmarking +C.1. Experimental Design +We compute the five metrics used in the benchmarking +section using the following definitions: +Mean Absolute Error [35] represents the average abso- +lute difference between the predicted saliency map and its +12 + +Annotation 1 +Annotation 2 +Annotation 3 +Annotation 4 +WRKDAVIS-S [48] +PASCAL-S [29] +HR [48] +ECSSD [45] +DUT-O [46] +UH [44] +DUTS [41] +Ours +Med. W +1080 +375 +2704 +300 +300 +3612 +300 +1296 +Med. H +1920 +500 +3264 +400 +400 +5000 +400 +968 +High Res. + + + + + + + + +Table 4. Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to metrics showcasing +the image resolution. This includes median image width (“Med. W”), median image height (“Med. H”), and flag indicating if high +resolution (“High Res.”). (HR=HRSOD; UH=UHRSD) +ground truth per pixel. It can be given as: +MAE = +1 +H ∗ W +H +� +r=1 +W +� +c=1 +|pred(r, c) − gt(r, c)| +(1) +where pred represents the predicted saliency map, gt repre- +sents the ground truth, (H, W) represents the height and +width of the image, and (r, c) represents the pixel co- +ordinates for the given image. +Structure Measure [19] is used to measure the similarity +between the predicted saliency map and the ground truth. +Since, we convert both the predictions and ground truths +into the [0, 1] range, we apply the formula directly to the +predictions and maps. It can defined as follows: +Sm = (1 − α)Sr + αSo +(2) +where, Sr is defined as the region aware similarity score, So +is defined as the object aware similarity score, and α repre- +sents the weight that is used to sum up the values. We set +α = 0.5, therefore making sure that we see equal contribu- +tion from both region and object aware scores. +F-Measure [2] represents the precision and recall ratio +for the given prediction. It can be represented as: +Fm = (1 + β2) ∗ Precision ∗ Recall +β2 ∗ Precision + Recall +(3) +Here precision = +T P +T P +F P and recall = +T P +T P +F N on the +entire prediction image by pixels. We set β2 = 0.3 and re- +port the average of all F-measures as Fm similar to previous +works. +Enhanced Alignment Measure [20] is used as the met- +ric to measure the effectiveness of the saliency prediction +against the ground truth. It captures the pixel-level match- +ing information and image-level statistics into one single +metric by the means of an enhanced alignment matrix φ. It +is defined as follows: +Em = +1 +H ∗ W +H +� +r=1 +W +� +c=1 +φF M(r, c) +(4) +where, φF M represents the enhanced alignment matrix for +the foreground map, (H, W) represents the height and +width of the image, and (r, c) represents the pixel co- +ordinates for the given image. +Intersection over Union also known as Jaccard Index is +used to determine the similarity between sample sets. In +this case it captures the overlap between the ground truth +and prediction map of the salient object. We convert the +predictions in binary map and compute the Jaccard Index +over two classes. It can be defined as follows: +IoU = J(A, B) = |A ∩ B| +|A ∪ B| +(5) +where, A and B are images of same size, consisting of inte- +ger class values {0, 1}. +We further show how the models performed on VizWiz- +SO with qualitative examples shown in Figure 8. These ex- +amples feature a variety of challenges we observed for the +models, such as a large salient object, less complex bound- +aries, lack of text on the salient object, and lower qual- +ity images. For example, we observe how the models fail +to perform adequately in identifying larger salient objects +(rows 4 and 5). We also observe the models perform bet- +ter when salient objects contain text (rows 1 and 2) versus +lack text (rows 5 and 6). Further, we see models perform +worse for images that are lower quality (rows 3, 4, and 5). +Our fine-grained analysis in the main paper suggests each +of these factors offer unique challenges for modern salient +object detection models. +13 + +Figure 8. Examples of difficult images present in VizWiz-SO, with characteristics such as high coverage ratio, presence of text, less +complex boundaries, and lower image quality. We show how the seven models perform on these cases as compared to the human annotation +(GT=Ground Truth). We see that models such as PFSNet [33], DIS [37], and F3Net [43] do not always give us the correct salient objects +or sometime no predictions at all. We also notice that VST [30] usually predicts salient objects with better accuracy compared to other +models, but also suffer from not detecting the correct salient object. +14 + +Image +GT +BASNet +F3Net +U2Net +VST +PFSNet +PGNet +DIS \ No newline at end of file diff --git a/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/load_file.txt b/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..24fa4840a85284287b203a38e6a5564483b60824 --- /dev/null +++ b/B9E4T4oBgHgl3EQf5Q72/content/tmp_files/load_file.txt @@ -0,0 +1,866 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf,len=865 +page_content='Salient Object Detection for Images Taken by People With Vision Impairments Jarek Reynolds*, Chandra Kanth Nagesh*, and Danna Gurari denotes equal contribution University of Colorado Boulder Abstract Salient object detection is the task of producing a bi- nary mask for an image that deciphers which pixels be- long to the foreground object versus background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We in- troduce a new salient object detection dataset using images taken by people who are visually impaired who were seek- ing to better understand their surroundings, which we call VizWiz-SalientObject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Compared to seven existing datasets, VizWiz-SalientObject is the largest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 32,000 human- annotated images) and contains unique characteristics in- cluding a higher prevalence of text in the salient objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', in 68% of images) and salient objects that occupy a larger ratio of the images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', on average, ∼50% cover- age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We benchmarked seven modern salient object detec- tion methods on our dataset and found they struggle most with images featuring salient objects that are large, have less complex boundaries, and lack text as well as for lower quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We invite the broader community to work on our new dataset challenge by publicly sharing the dataset at https://vizwiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='org/tasks-and-datasets/salient-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Introduction Locating the most prominent foreground object in an im- age is a core computer vision problem, often referred to as salient object detection (as well as salient object seg- mentation and foreground object detection/segmentation) [8,12,32,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This work is motivated by the desire to have salient object detection models work well for images taken by people who are blind or with low vision1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', people with vision impairments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Such a feature could offer sev- eral benefits to this community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example, it could con- tribute to privacy-preservation for photographers who rely on visual assistance technologies to learn about objects in their daily lives, using mobile phone applications such as Microsoft’s Seeing AI, Google Lookout, and TapTapSee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2 1For people with low vision, solutions do not exist to correct their vi- sion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', by wearing glasses, surgery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 2Many companies record submitted data as evidence that potentially could be needed for legal reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Example images demonstrating unique features of our new VizWiz-SalientObject dataset when compared to other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The salient objects commonly contain text and occupy a larger portion of the image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', high coverage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' All content except the foreground content of interest could be obfuscated, which is important since private information is often inadvertently captured in the background of images taken by these photographers [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Additionally, localiza- tion of the foreground object would empower low vision users to rapidly magnify content of interest and also enable quick inspection of smaller details [21,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Many salient object detection datasets have been created to enable progress in algorithm development [7,8,22,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A limitation of existing datasets is they are typically built us- ing high-quality images collected from photo-sharing web- sites on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' As we will show in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2, such images commonly lack many characteristics that can be ob- served in real-world settings, especially for visual media taken by visually impaired photographers who are trying to learn about the content they photograph [24], often pho- tographing distinct types of content such as objects showing text [25], and cannot verify visual quality [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To fill this gap, we introduce a new salient object de- tection dataset based on images captured in an authentic use case where visually impaired photographers shared their images to solicit assistance in learning about the visual con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We created this dataset by crowdsourcing the collec- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='05323v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='CV] 12 Jan 2023 rCableSales SUPPLY ext Present 口 WASYOURTRIP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' ANTTOHEARFR OUR ON-LINB SUR pse hcwyou want to take the XILLL uter gli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='ols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='sgizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='com/s3l pne: KUCLO LCINNO CICMO G gli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='olsiphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='sgizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='co Pad LOLIUILKIUtion of salient object annotations for nearly 40,000 images taken from the VizWiz-Captions dataset [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Examples of resulting annotated images are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Af- ter applying quality control filtration steps, our final dataset consists of 32,000 annotated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We call our dataset VizWiz-SalientObject (or VizWiz-SO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We conduct a de- tailed analysis revealing how this new dataset relates to ex- isting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' When comparing our salient objects to the visual evidence needed to answer questions the photogra- phers asked about their images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', taken from the VizWiz- VQA-Grounding dataset [11]), we observe that over half the time the necessary visual evidence is the salient ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' When comparing our dataset to seven existing datasets, we observe VizWiz-SalientObject is the largest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 32,000 human-annotated images) and is unique in its higher preva- lence of text in the salient objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', in 68% of images) as well as salient objects occupying a larger ratio of the images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', on average, ∼50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also benchmark modern salient object detection al- gorithms on our new dataset to uncover open challenges for the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Experiments with seven al- gorithms reveal that they struggle most for images with salient objects that are large, have less complex bound- aries, and lack text as well as for lower quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To facilitate progress on these challenging problems, upon publication, we will publicly-share the dataset and an evaluation server with leaderboard at the following link: https://vizwiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='org/tasks-and-datasets/salient-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In summary, our new dataset supports the development of more generalized algorithms that not only address the in- terests of people with vision impairments but also can ben- efit related applications that encounter similar real world challenges observed in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Relevant applications include robotics, lifelogging, and privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Related Work Salient Object Detection Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Over the past cou- ple of decades, many datasets were introduced to facili- tate improving the design of algorithms that address salient object detection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Several survey papers provide comprehensive characterizations of the tens of datasets de- signed for this task [7, 8, 22, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A common observation is that datasets were artificially constructed around high- quality images which often feature salient objects in the center of the images with a high contrast against the back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This is a mismatch from many real-world settings, especially for visual media taken by visually impaired pho- tographers who often photograph distinct types of content, such as objects showing text [25], with the aim to learn about that content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We introduce the first salient object de- tection dataset based on images taken by visually impaired people in an authentic use case where they were trying to learn about their visual surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Compared to seven modern datasets, our dataset is larger, has a high prevalence of salient objects containing textual information, and shows objects that occupy larger portions of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Salient Object Detection Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Researchers have designed novel algorithms to automatically perform salient object detection for over 20 years, with the status quo since 2015 being that state-of-the-art methods employ neural net- works trained on large-scale annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Several survey papers provide comprehensive characterizations of the hundreds of algorithms for this task [7,8,22,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' While convolutional neural network (CNN) based models became the mainstream method [1, 33, 43] in 2015, transformer based models [30, 44] have become the mainstream ap- proach over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To assess how well mod- ern methods perform on our new dataset, we benchmark seven modern methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We observe that existing methods fall below human performance and struggle most for salient objects that lack text and occupy a larger ratio of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Visual Assistance Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Visually impaired peo- ple can share their visual media (images and videos) with various technologies [3, 4, 6, 14, 18, 27, 32, 40] in order to receive assistance for daily tasks such as deciding what to eat, wear, and buy [10,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The widespread impact of such technologies for real users is exemplified by reports from some of these companies that the technologies have 10s to 100s of thousands of users who have submitted millions of assistance requests [5,9,14,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The most common reported goal for using such technologies is to learn about a (salient) object [9,10,23,28,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Given this common use case, salient object detection models could help for privacy preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, images (or video frames) could be edited be- fore being shared with companies, by obfuscating the back- ground, in order to reduce inadvertent disclosures of pri- vate content that often appears in the background of images taken by visually impaired photographers [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' VizWiz-SalientObject Dataset We now introduce our new salient object detection dataset, we call VizWiz-SalientObject (VizWiz-SO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Creation Image Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We focus on images taken by visually im- paired people who shared them in an authentic use case where they were soliciting visual assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, we leverage the 39,181 labeled images from the VizWiz- Captions dataset, each of which is paired with five crowd- sourced captions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Observing that images from these photographers can have severe quality issues resulting in no detectable salient object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', extreme blur or inadequate illumination), we did not use the images which were cap- tioned as follows by at least four of the five crowdworkers: 2 “Quality issues are too severe to recognize visual content.” We also did not use the small images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', both the height and width were less than 300 pixels) because of the chal- lenges of collecting precise annotations for such images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This left us with 37,120 images for our annotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Task Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Our task interface for segmenting salient objects begins with a comprehensive instruction set at the top detailing both how to navigate the interface and how to complete challenging annotation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Next, it shows an image alongside two preliminary questions for verifying there is a single, unambiguous foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The first question asks “Is the image showing a screenshot?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' If the answer is “yes”, we conclude the image lacks a salient ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Next, we ask the more general, direct question of “Is there a single unambiguous foreground object?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' An anno- tator is only prompted to segment the foreground object for images deemed by these preliminary questions to show a single, unambiguous foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To demarcate the boundary of the salient object, the in- terface collects a series of points that are connected into polygon(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' When segmenting the salient object, the an- notator is required to remove any holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', donut) as well as capture all object parts when occlusions break a salient object into more than one polygon (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', hand obfuscates a pencil into two parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The annotator also has an option to select a button indicating that the salient object occupies the full image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We provide more details about the task interface as well as a screenshot of it in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Annotation Collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We leveraged the benefits of an around-the-clock distributed workforce by crowdsourc- ing annotations via Amazon’s crowdsourcing marketplace, Amazon Mechanical Turk (AMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Although AMT can support our large-scale annotation needs, it brings concerns about annotation quality due to the anonymous nature of the crowdsourced workforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Con- sequently, we implemented several measures to ensure the collection of high-quality annotations, as summarized be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' First, we restricted who were potential candidates for our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We only accepted workers who had at least a 98% acceptance rate while having completed at least 500 Human Intelligence Tasks (HITs) on AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Moreover, to encourage understanding of our initial and ongoing task in- structions, we opted for crowdworkers only from the United States since that provided us confidence that they have English-language proficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In addition, we also required crowdworkers to pass a qualification assessment covering five challenging annotation scenarios documented in our in- structions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The qualification images feature foreground ob- jects consisting of complex boundaries, holes within the ob- ject, and occlusions obfuscating portions of the foreground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Consequently, the task required crowdworkers to demonstrate an understanding for how to generate multi- ple polygons, annotate holes, handle occlusions, and draw complex boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We employed 40 AMT crowdworkers who completed our qualification task to complete annotations of all images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For each of the 37,120 images, we collected two annotations from the crowdworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3 During annotation collection, we monitored ongoing quality by tracking each worker’s per- formance with respect to their frequency of indicating the presence of full-screen annotations or no prominent fore- ground object as well as the level of detail they provided in their segmentations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', high prevalence of triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Cu- mulatively, the crowdworkers took 1,290 annotation hours over 11 days to complete annotating the 37,120 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Annotation Post-Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We next analyzed the re- dundant annotations per image to determine how to use each annotated image in the final dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' First, we removed 3,662 images for which workers agreed there was no sin- gle, unambiguous salient object, which occurred when both annotators either answered “Yes” to “Is the image a screen- shot?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' or “No” to “Is there a single most prominent fore- ground object?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Next, we manually inspected 7,443 images for which workers disagreed on the answers to either of the two preliminary questions and determined whether there is indeed a single, unambiguous object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Finally, with all im- ages deemed to have a single, unambiguous salient object, we determined which annotation to assign as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To assist in this process, we computed the intersection over union (IoU) score between the two segmentations for all images with two or more segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' With IoUs ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90, we deemed both annotations high quality and randomly se- lected one as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For the remaining 2,951 images with IoUs< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90, we manually reviewed the annotations to decide whether one was correct or whether the image should be discarded due to foreground object ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Analysis We now characterize the VizWiz-SalientObject (VizWiz- SO) dataset and how it relates to existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1 Salient Objects vs Answer Groundings for VQA We first explore how the target content the photographers were asking about relates to an image’s salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To do so, we compare the annotations of the visual evidence needed to answer questions about the images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', an- swer groundings provided in the VizWiz-VQA-Grounding dataset [11], to the annotations of the salient objects in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We first identified all annotated images that were in 3For a subset of images, we collected four annotations to support fur- ther analysis of human annotation performance, which we describe in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The histogram summarizes for 6,540 images the fre- quency of observing different levels of similarity between two segmentations per image, which show the salient object and the visual evidence needed to answer the photographer’s question re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' These findings reveal that visually impaired photogra- phers often want to learn about the salient objects in their images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' common across the two datasets, yielding a total of 6,540 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For each image, we then measured the similarity between the answer grounding and salient object segmenta- tions using the IoU metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We visualize our results using a histogram where we categorize each image into one of ten interval bins starting with IoU=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1), incrementing in intervals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1, and ending with IoU=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Results are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We observe that about half of the images have a high sim- ilarity between the salient object and VQA answer ground- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 46% had an IoU ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This reveals that visually impaired photographers often are trying to learn about the salient object in their images when trying to get answers to their visual questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also observe that roughly one quarter of the images have a very low similarity between the salient object and VQA answer grounding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7% of images had an IoU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We manually reviewed these 1,680 images with IoUs less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1 to understand the reasons for this finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We discovered that 95% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 1,599) of these images have a salient object featuring a full-screen or large region while the VQA answer grounding captures a small aspect of the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Examples include expiration dates on food packages or the current page number of an open book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The remaining 5% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 81) of these images featured a VQA an- swer grounding unrelated to the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' More generally, we observe that the IoU scores follow a U-shaped distribution with only a small portion of images having middling scores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='9% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 511) of images had an IoU ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3 and < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Among these images, we found the salient object contained the VQA answer grounding region 100% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' There are two primary trends that led to these less common IoU scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The first trend is that larger VQA answer grounding regions occur with smaller salient objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Examples include brands of cereal, types of soda, and denominations of currency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The second trend was for salient objects featuring holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' That is because the VizWiz- VQA-Grounding dataset did not account for holes in their annotation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The absence of annotated holes in only one of the two segmentations led to lower IoU scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Altogether, these findings highlight that a valuable step for tackling many of this population’s VQA goals is to ini- tially locate the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' That is because the answer will likely only be grounded in the salient object or the background rather than their intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2 VizWiz-SO vs Existing Datasets We next compare our dataset to seven datasets: DUTS [41]: the most commonly used dataset to train state-of-the-art algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', [1,30,33,38,43,44]) due to its large size paired with diverse saliency challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' DUT-OMRON [46]: consist of images showing multiple salient objects, often with complex backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This is a useful reference when considering extending our dataset to when photographs taken by visually impaired photog- raphers showing multiple salient objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We share our collected metadata indicating when this occurs to facili- tate this line of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' ECSSD [45]: consists of images featuring complex scenes that present textures and structures expected to be com- mon in real-world salient object detection scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' PASCAL-S [29]: derived from PASCAL VOC’s [16] val- idation set, it is designed to facilitate salient object seg- mentation generalization on realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' HRSOD [48]: explicitly designed for salient object de- tection on high-resolution images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' this is relevant for our real-world application since images taken by people with vision impairments often are relatively high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' UHRSD [44]: currently the largest ultra-high resolution salient object detection dataset, which is relevant to our work since images taken by people with vision impair- ments can be ultra high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' DAVIS-S [48]: derived from DAVIS [36], a densely an- notated video segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This is relevant for our real-world application to analyze implications for video frames since visually impaired photographers often stream live video with their cameras when using visual assistance technologies [4,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Of note, images in six of these datasets originate from the Internet on photo-sharing websites such as Flickr [29, 41, 44–46, 48], and so likely are high quality since they were deemed of sufficient quality to upload to the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='4 4The origins of the images for the final dataset is not reported [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4 50% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='40% Image 30% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7% 20% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0% 10% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='4% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='8% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='4% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='9] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' [O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' [o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='6) ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3) IoU SimilarityDAVIS-S [48] PASCAL-S [29] HR [48] ECSSD [45] DUT-O [46] UH [44] DUTS [41] Ours Images 92 850 2,010 1,000 5,168 5,920 15,572 32,000 Text 13% 24% 15% 15% 11% 19% 13% 68% MR 22% 31% 25% 9% 17% 35% 19% 1% Holes 82% 50% 62% 29% 28% 75% 41% 4% Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to how many images are included (“Images”), the percentage of images that have text present in the salient objects (“Text”), the percentage of images that have salient objects consisting of more than one region (“MR”), and the percentage of images that have salient objects containing any holes (“Holes”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' As shown, our dataset is distinct in that it contains more images, more salient objects with text present, more salient objects consisting of one region, and fewer salient objects containing holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' (HR=HRSOD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' UH=UHRSD) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Summary statistics for ours and seven other datasets with respect to four measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Each box reveals statistics about all salient objects in a particular dataset with the central mark capturing the median value, box edges the 25th and 75th percentiles values, whiskers the most extreme data points not considered outliers, and individually plotted points the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Our dataset is unique in that salient objects tend to have less complex boundaries, occupy larger portions of an image, and exhibit a greater diversity of sizes relative to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For each salient object in every dataset, we characterize it in six ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Three measures focus on detecting the pres- ence versus absence of particular properties for the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' These are whether the salient object contains text 5, consists of multiple regions 6, or contains any hole(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The remaining three measures characterize the salient region it- self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' First, we identify the position of an object within an image by measuring its center of mass relative to the im- age coordinates, resulting in x and y coordinate values in the range between 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Next, we characterize the ob- ject’s boundary complexity by computing its isoperimetric inequality, which is the ratio of the object’s area to the length of its perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Values range from 0 to 1, with larger values indicating simpler boundaries that are less jagged/dented (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Finally, to gauge the relative size of a salient object in the image, we compute its cover- age ratio, meaning the fraction of all image pixels that are occupied by the salient object’s pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We show summative statistics of our findings per dataset in Table 1 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In particular, in Table 1, we re- 5We obfuscate all image content but the salient object and then check whether Microsoft Azure’s OCR API returns text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 6Multiple regions means there are multiple separate polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This can occur either because multiple salient objects were annotated or because of occlusions that lead to more than one region for a single salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' port how many images are in each dataset paired with what percentage of those images have salient objects with text, multiple regions, and holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In Figure 3, we visualize statis- tics summarizing the values for each dataset’s salient ob- jects with respect to center of mass, boundary complexity, and coverage ratio using boxplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' While our findings highlight that our VizWiz-SO dataset has many distinct characteristics, one commonality it has with most existing salient object detection datasets is that the salient objects typically occupy centered positions within an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, in Figure 3, we observe this trend for all datasets except HRSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We found this some- what surprising since visually impaired photographers can- not visually inspect their images to verify they are conform- ing to the common photographer’s bias of centering con- tents of interest they are trying to photograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Yet, given our findings from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1 that photographers often are interested in learning about an image’s salient object, our findings suggest these photographers have skills in center- ing contents of interest in pictures they take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A unique aspect of our VizWiz-SO dataset is that it fea- tures more salient objects with textual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, 68% of salient objects in VizWiz-SO contain text while the dataset with the next highest prevalence of text, PASCAL- S [29], only has it for 24% of the images (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A gap of 5 Center of mass Y-axis Center of mass X-axis Boundary Complexity Coverage Ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='0 DAVIS-S PASCAL-S HRSOD ECSSD DUT-OMRON UHRSD DUTS Ours: VizWiz-SOthis magnitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 44 percentage points) suggests that our new dataset offers a considerable domain shift in the salient object detection problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We suspect part of this shift stems from the types of salient objects included, with many more daily objects such as products (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', food packages) included in our VizWiz-SO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Another unique aspect of VizWiz-SO is that far fewer images feature salient objects that consist of multiple re- gions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', only 1% of images (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We suspect this distinction stems from our unique approach of adopt- ing a rigorous annotation preprocessing step, where we re- quire crowdworkers to verify images have one unambigu- ous salient object before allowing them to annotate images for use in our final dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Any remaining objects in our dataset with multiple regions are therefore highly likely a result of occlusions breaking a single salient object into multiple polygons, which evidently is incredibly rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' VizWiz-SO is also unique due to the rarity in which salient objects contain holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', only observed for 4% of images (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' From visual inspection, we suspect this finding reflects a domain shift in the types of content found in the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example, examples from other datasets of objects with holes include people riding bikes, people dancing, and animals in intricate poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In con- trast, in VizWiz-SO, objects with holes include retail pack- aging made to hang from hooks, pairs of scissors, and coffee mugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We posit the lower prevalence of holes in VizWiz-SO stems from the fact that images originate from an authentic use case where photographers primarily photograph house- hold and retail items, which naturally feature fewer holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A further distinction of our VizWiz-SO dataset is that the salient objects tend to have less complex boundaries (Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We suspect this is again because of a domain shift in the types of objects in our dataset, with many more human- made items, such as food packaging boxes and cans, that by design are typically more structured in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A final distinction of salient objects in our VizWiz-SO is how much of the image they occupy (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' First, they tend to occupy a much larger amount of the image than observed in other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, they on average oc- cupy roughly half of all image pixels, with a mean coverage ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='5 and a median of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In contrast, the dataset with the next highest coverage ratio statistics is PASCAL- S [29], and over 75% of its images contain salient objects that occupy less than half of the image pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We attribute this distinction to the authentic use case of our dataset, where visually impaired photographers attempting to learn about the salient objects they are photographing seem to be taking zoomed-in or close-to-camera images of the content of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Another unique aspect of our salient objects, is that they exhibit a larger range of sizes, as shown by the gaps between the 25 and 75 percentile values of each box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example, PASCAL-S features the next largest interquar- tile range with a 23% gap(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 19% to 42%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In contrast, the gap for VizWiz-SO is more than twice as large at 56% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 22% to 78%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Consequently, a unique challenge of our dataset for algorithms is that they no longer can assume a strong bias regarding a salient object’s relative size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Algorithm Benchmarking We benchmark modern salient object detection algo- rithms to show how they perform on our new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We conducted all experiments on a Nvidia A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Experimental Design Dataset Splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We use the existing splits available for the VizWiz-Captions dataset [25], which translates to approxi- mately a 60/20/20 training, validation and test split for our VizWiz-SO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In particular, from the 32,000 anno- tated images, the number of images in each split respec- tively is 19,116, 6,105, and 6,779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We evaluate each model with re- spect to five popular metrics for salient object detection models: Mean Absolute Error (MAE), Structure Measure (Sm), Mean F-Measure (Fm), Enhanced Alignment Mea- sure (Em), and Intersection over Union (IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We benchmark the following seven methods from the past three years to assess the difficulty of our new dataset for modern salient object detection models: Boundary Aware Segmentation Network (BASNet) [38]: an appealing model for real-time applications like our tar- get use case because it can achieve 70fps during inference time while achieving competitive performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', was a top-performer in 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Fusion, Feedback and Focus Network (F3Net) [43]: state- of-the-art performing model on five datasets in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' U2 Network (U2Net) [1]: an appealing model for real- world applications like our target use case because it has a very light footprint (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7MB), and so is more suitable for resource-constrained devices such as smartphones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It achieved competitive performance in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Visual Saliency Transformer (VST) [30]: achieved state- of-the-art performance in 2021, and is based purely on a transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Pyramidal Feature Shrinking Network (PFSNet) [33]: achieved state-of-the-art performance on five datasets in 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' it consists of a decoder that aims at using aggre- gated adjacent feature nodes hierarchically to avoid the problem of leaping feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Pyramid Grafting Network (PGNet) [44]: introduced in 2022, it is a one-stage framework based on a transformer 6 HP BASNet F3Net U2Net VST PFSNet PGNet DIS VST-FT VST-S [38] [43] [1] [30] [33] [44] [37] Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Backbone R-34 R-50 T2T-ViT R-50 R-18+SWIN U2Net VST ViT Training set D D D D D D+HR DIS5K D+VW VW Input size 2562 3522 3202 2242 3522 2242, 10242 10242 2242 2242 Size (MB) 333 98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7 171 120 280 169 171 171 VizWiz-SO MAE ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='21 Sm ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='65 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='72 Em ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='70 IoU ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='69 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Analysis of existing algorithms that we benchmark on our VizWiz-SO dataset, including both off-the-shelf models (which are cited) as well as those fine-tuned (-FT) and trained from scratch (-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We first report differentiating attributes between the algorithm architectures and then present the model performance with respect to five widely-used metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' (HP=Human Performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' R=ResNet [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' ViT=Vision Transformer [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Swin=Shifted window transformer [31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' D=DUTS-TR [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' VW=VizWiz-SO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' HR=HRSOD [48]) and CNN backbone that achieves state-of-the-art perfor- mance on five benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' [41,44,46,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dichotomous Image Segmentation (DIS) [37]: also in- troduced in 2022 as the state-of-the-art method for the DIS5K [37] dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' it is designed for detecting salient object in high resolution images, which makes it relevant for our use case where many images coming from people with vision impairments are relatively high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We further characterize each model by identifying the backbone architecture used in the architecture, datasets used for training, image size used for training, and model foot- print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' These characteristics are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' All models predict saliency maps that represent the brightness of certain pixels within the same spatial reso- lution as the input image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', ∈ [0, 1] or alternatively ∈ [0, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The predictions generated by salient object de- tection models are converted into binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also evaluate human performance to estab- lish an upper bound for what we should strive for from au- tomated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Since, we get two human annotations per image in our dataset, we calculate human performance by comparing the two annotations in cases where the IoU is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Performance for Off-The-Shelf Models We first evaluate each of the algorithms as is in their orig- inal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We observe that VST [30] is the top-performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Yet, it still falls short of human performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For exam- ple, the gap in performance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='15 in terms of MAE, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='211 in terms of IoU, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='26 for Sm, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2 for Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Conse- quently, this dataset offers a new challenging benchmark for the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A further observation is that the models perform poorly on the VizWiz-SO dataset in comparison to their perfor- mance on the original datasets for which they were bench- marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example the MAE and Sm performance ob- served by PGNet [44] on DUTS-TE is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='028 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='912 re- spectively versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2123 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='6233 respectively for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We hypothesize that part of the reason for this poor performance is that models trained and evaluated on other datasets are not able to learn how to generalize to the real- world challenges that arise for images taken by visually im- paired photographers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Performance When Training on VizWiz-SO We next explore whether training the top-performing al- gorithm, VST [30], on our new dataset will lead to improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To do so we analyze two additional models: (1) the pretrained VST [30] model fine-tuned on VizWiz- SO (VST-FT) and (2) the pretrained VST [30] algorithm trained from scratch on VizWiz-SO (VST-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We use the default hyperparameters reported in the VST [30] paper for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We observe that both models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', created by training from scratch and fine-tuning on our VizWiz-SO dataset, achieve worse results than the baseline of not training the al- gorithm on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This suggests that the training data used by algorithms is not the only culprit for what makes our new dataset challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Rather, our findings suggest that new algorithmic frameworks are also needed to achieve strong generalization performance on our new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Fine-grained Analysis We next conduct fine-grained analysis to better isolate what makes our dataset challenging for modern algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To do so, we divide our VizWiz-SO test set according to the following four factors, with the first three based on metadata collected in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2 to characterize our dataset: 7 BASNet F3Net U2Net VST PFSNet PGNet DIS VST-FT VST-S [38] [43] [1] [30] [33] [44] [37] Text Presence True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='16 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='31 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Fine-grained analysis of existing algorithms with respect to presence of text on the salient object (“Text Presence”), relative size of the salient object in the image (“Coverage”), relative complexity of the salient object’s boundary (“Complexity”), and image quality (“Image quality”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' As shown the algorithms perform worse when there is salient objects lack text, occupy a large portion of the image, have less complex boundarys as well as when the image quality is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Text Presence: two groups based on whether text is present in the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Coverage Ratio (Coverage): three groups based on the 33rd and 66th quartile values in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' All images with coverage ratio less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='32 has small coverage, be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='32 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='62 has medium coverage, and greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='62 has large coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Boundary Complexity (Complexity): two groups by split- ting them around the mean score for boundary complex- ity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='66) with high boundary complexity when the score is less than the mean and low boundary complexity otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Image Quality: leveraging metadata from prior work [25], which indicates how many of the five crowdworkers in- dicated an image as insufficient quality to recognize the content, we split the images into groups with good qual- ity being when none of the crowdworkers indicate insuf- ficient quality and poor otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Due to space constraints, we only report results in the main paper with respect to the Mean Absolute Error [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Results for all benchmarked models are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In terms of text presence, we see that the models perform better when there is text present as opposed to when there is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example, the performance drops by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='11 for the best performing model, VST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We suspect visual patterns that arise with text may serve as a valuable cue to models in locating salient objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Next, we see that as the coverage ratio of the salient ob- jects increase, the models tend to perform worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For in- stance, the best performing model, VST, has a performance dropoff of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='19 when predicting images with small cover- age ratios as opposed to large coverage ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We see an even greater performance dropoff from other models such as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='60 for DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We suspect this performance gap arises in part from the fact that existing datasets largely lack such large salient objects, which both could have affected what algorithms were designed to handle as well what they could learn from the data they observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Further observed trends are that performance drops for salient objects with lower boundary complexity and for poorer quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' These are two additional factors that reflect domain shifts between our dataset and prior datasets that could have affected the design of algorithms as well what they could learn from the data training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Conclusions We introduce the VizWiz-SalientObject dataset to en- courage the community to design more generalized salient object detection models that can handle a larger range of challenges motivated by our authentic use case that also can occur in many real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We offer our exper- imental findings from benchmarking modern salient object detection algorithms as a valuable starting point for iden- tifying valuable future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To summarize, new models are needed to better handle salient objects that are large, have less complex boundaries, and lack text as well as work well in the presence of lower quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We now close with a discussion of some ethical impli- cations of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' While we are motivated to better as- sist a population that is traditionally marginalized in society, we acknowledge our work can lead to potentially adverse social effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Our concern is primarily centered on bad- actor behaviors intended to exploit the privacy, autonomy, and livelihoods of a population demographic inherently sus- ceptible to such behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Bad actors could use our work to deceive visually impaired individuals in harmful ways, such as through fraud, scams, and other deceptive practices, by for example intercepting their visual media and replacing automatically detected salient objects with misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 8 Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This project was supported in part by a National Science Foundation SaTC award (#2148080) and Amazon Mechanical Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We thank Leah Findlater and Yang Wang for contributing to this research idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' References [1] U2-net: Going deeper with nested u-structure for salient ob- ject detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Pattern Recognition, 106:107404, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} 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Tai-Yin Chiu, Nilavra Bhat- tacharya, and Danna Gurari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Vision skills needed to an- swer visual questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Proceedings of the ACM on Human- Computer Interaction, 4(CSCW2):1–31, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 2 [48] Yi Zeng, Pingping Zhang, Jianming Zhang, Zhe Lin, and Huchuan Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Towards high-resolution salient object detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Confer- ence on Computer Vision, pages 7234–7243, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 4, 5, 7, 13 Appendix This document supplements the main paper with additional information concerning: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Creation (supplements Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1) Annotation Task Interface Worker Qualification Task Analysis of Workers’ Annotation Differences 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Analysis: VizWiz-SO vs Existing Datasets (supplements Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Experimental Design (supplements Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Creation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Annotation Task Interface The task interface displays five images within a tabbed container on the left and preliminary questions with task instructions on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A screenshot of the task interface (without instructions) is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To account for occlusions and holes while keeping the task simple for annotators, we permitted annotators to gen- erate multiple polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For occlusions, annotators could use as many polygons as necessary for demarcating fore- ground objects partitioned into multiple polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For holes, we apply an even-odd fill rule to images featuring foreground objects with holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' With an even-odd fill rule, every area inside an even number of enclosed areas be- comes hollow, and every region inside an odd number of enclosed areas becomes filled [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' By treating the image’s four corners as the first enclosed area, the outermost bound- ary of the foreground object becomes the second enclosed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Moreover, holes within foreground objects represent the third layer of enclosed areas and become filled, allowing annotators to demarcate foreground objects featuring holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In practice, annotators first trace the outermost boundary of the foreground object and close the path by clicking the first point a second time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We then instructed annotators to trace any holes within the foreground object, and so those holes end up in odd-numbered layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Worker Qualification Task We administered a qualification task for workers to sup- port our collection of high-quality ground truth annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The qualification task required annotating five images, each of which features a distinct challenging annotation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' All five images are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The first two images show a table and a bench, offering examples with complex boundaries and holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The next two images feature a per- son holding a coffee mug, to support educating a crowd- worker about our expectations for annotating objects with 10 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' A screenshot of our annotation task interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The five images used for the worker qualification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Each was selected to demonstrate a challenging annotation sce- nario such as complex boundaries, holes, and occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' complex geometries that have many curves and occlusions that require annotating multiple polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The final image is a spatula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This task verified a crowdworker’s ability to correctly identify and annotate multiple holes that can arise within the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' After crowdworkers annotated each qualification image, the backend code of our website checked if their annotation was sufficiently similar to the GT annotation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', IoU sim- ilarity of at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Crowdworkers could only proceed to the following image after they obtained an IoU ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90 on the current image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Crowdworkers obtaining an IoU ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='90 on all five qualification assessment images on a per- image basis gave us substantial confidence that they would be able to successfully handle complex and challenging out- lier cases within the original VizWiz Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Analysis of Workers’ Annotation Differences We collected a larger number of redundant annotations per image for a random subset of images to better explore when and why annotation differences are observed from dif- ferent workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, for this analysis, we collected four annotations as opposed to two for a subset of 1,237 im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Examples of the redundant annotations collected per image are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The first example (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', row 1 of Figure 6) highlights that annotation differences can stem from challenging annota- tion scenarios where objects contain holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', in mug han- dle) or are occluded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', by the straw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For instance, the hole was not annotated in the third annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Addition- ally, only the fourth annotation captured the occlusion that arises from the straw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The second example (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', row 2 of Figure 6) highlights that annotation differences can stem from ambiguity regard- 7Some crowdworkers did not pass the qualification assessment due to time constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In these cases, crowdworkers would contact us with the images they annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' If we were confident in their annotation abilities, we manually added these crowdworkers to the qualified worker pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 11 Image 1 Image 2 Image 3 Image 4 Image 5 Work may be rejected for not following instructions Step 1: Is the image showing a screenshot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' O Yes O No WASYOURTRIP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Step 2: Is there a single prominent foreground object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' OYes ONO ANTTOHEARFRO OUR ON-LINE SURVEYTONACHANCETO WIN SU Step 3: Demarcate the prominent foreground object osehcwyou wantto takethesunvey typethatweb address inyourbrowse puter one: gli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='ols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='sgizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='com/s3/ Prev Image Next Image Pad gli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='olsiphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='sgizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='com/s3 gli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='olsipad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='sgizmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='com/s3/ Select Full Undo Last Clear All Image Point PolygonsFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Example annotations from our random subset where we collected four annotations as opposed to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We find worker dif- ferences primarily occur in challenging annotation scenarios such as holes, occlusions, complex boundaries, and object saliency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' ing what is the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' As shown, the first two an- notations flag the image as lacking a foreground object, the third annotation identifies the child holding the cup as the salient object, and the fourth annotation identified the child’s cup as the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' The third example (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=', in row 3 of Figure 6) highlights that annotation differences also can arise for objects that simultaneously have complex boundaries and holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In an- notation one, the worker did not fully annotate the salient object, cutting out part of the object from the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Only the third and fourth annotations accurately annotate all holes that are present in the salient object’s boundary while also having tight boundaries in the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In summary, we found occlusions, holes, and saliency ambiguity to be the primary factors contributing to annota- tion differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In the case of occlusions, worker differ- ences can arise when deciding whether to include objects that are a composite part of the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In the case of holes, annotation differences can arise regarding which holes to annotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Last, we found that it can be ambiguous as to which object is the most salient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Example ground truth annotations from the HRSOD dataset which exemplify that salient objects are not usually not centered in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This is a common trend in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Dataset Analysis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' VizWiz-SO vs Existing Datasets We present finer-grained details about typical image res- olutions for the different salient object detection datasets to expand upon discussions in the main paper about how VizWiz-SO relates to other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Specifically, we report the median image width (Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' W), median image height (Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' H), and whether the dataset supports high resolu- tion images (High Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=') as defined by whether the median image height and width are greater than 1080 and 1920 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Results are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We observe that our new dataset, overall, provides higher resolution images than most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also expand on a surprising finding reported in our main paper that the HRSOD dataset is the only one for which salient objects do not occupy the typical center po- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' To do so, we visualize the ground truth masks of some non-centered objects in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In row one, we see that objects are horizontally distributed to left and right po- sitions of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Similarly, we observe in row two that the salient objects are vertically distributed to the top and bottom positions of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Algorithmic Benchmarking C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Experimental Design We compute the five metrics used in the benchmarking section using the following definitions: Mean Absolute Error [35] represents the average abso- lute difference between the predicted saliency map and its 12 Annotation 1 Annotation 2 Annotation 3 Annotation 4 WRKDAVIS-S [48] PASCAL-S [29] HR [48] ECSSD [45] DUT-O [46] UH [44] DUTS [41] Ours Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' W 1080 375 2704 300 300 3612 300 1296 Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' H 1920 500 3264 400 400 5000 400 968 High Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' \x13 \x17 \x13 \x17 \x17 \x13 \x17 \x13 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Characterization of our VizWiz-SO dataset and seven existing salient object detection datasets with respect to metrics showcasing the image resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' This includes median image width (“Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' W”), median image height (“Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' H”), and flag indicating if high resolution (“High Res.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' (HR=HRSOD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' UH=UHRSD) ground truth per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It can be given as: MAE = 1 H ∗ W H � r=1 W � c=1 |pred(r, c) − gt(r, c)| (1) where pred represents the predicted saliency map, gt repre- sents the ground truth, (H, W) represents the height and width of the image, and (r, c) represents the pixel co- ordinates for the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Structure Measure [19] is used to measure the similarity between the predicted saliency map and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Since, we convert both the predictions and ground truths into the [0, 1] range, we apply the formula directly to the predictions and maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It can defined as follows: Sm = (1 − α)Sr + αSo (2) where, Sr is defined as the region aware similarity score, So is defined as the object aware similarity score, and α repre- sents the weight that is used to sum up the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='5, therefore making sure that we see equal contribu- tion from both region and object aware scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' F-Measure [2] represents the precision and recall ratio for the given prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It can be represented as: Fm = (1 + β2) ∗ Precision ∗ Recall β2 ∗ Precision + Recall (3) Here precision = T P T P +F P and recall = T P T P +F N on the entire prediction image by pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We set β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content='3 and re- port the average of all F-measures as Fm similar to previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Enhanced Alignment Measure [20] is used as the met- ric to measure the effectiveness of the saliency prediction against the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It captures the pixel-level match- ing information and image-level statistics into one single metric by the means of an enhanced alignment matrix φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It is defined as follows: Em = 1 H ∗ W H � r=1 W � c=1 φF M(r, c) (4) where, φF M represents the enhanced alignment matrix for the foreground map, (H, W) represents the height and width of the image, and (r, c) represents the pixel co- ordinates for the given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Intersection over Union also known as Jaccard Index is used to determine the similarity between sample sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' In this case it captures the overlap between the ground truth and prediction map of the salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We convert the predictions in binary map and compute the Jaccard Index over two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' It can be defined as follows: IoU = J(A, B) = |A ∩ B| |A ∪ B| (5) where, A and B are images of same size, consisting of inte- ger class values {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We further show how the models performed on VizWiz- SO with qualitative examples shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' These ex- amples feature a variety of challenges we observed for the models, such as a large salient object, less complex bound- aries, lack of text on the salient object, and lower qual- ity images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' For example, we observe how the models fail to perform adequately in identifying larger salient objects (rows 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also observe the models perform bet- ter when salient objects contain text (rows 1 and 2) versus lack text (rows 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Further, we see models perform worse for images that are lower quality (rows 3, 4, and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Our fine-grained analysis in the main paper suggests each of these factors offer unique challenges for modern salient object detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 13 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' Examples of difficult images present in VizWiz-SO, with characteristics such as high coverage ratio, presence of text, less complex boundaries, and lower image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We show how the seven models perform on these cases as compared to the human annotation (GT=Ground Truth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We see that models such as PFSNet [33], DIS [37], and F3Net [43] do not always give us the correct salient objects or sometime no predictions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' We also notice that VST [30] usually predicts salient objects with better accuracy compared to other models, but also suffer from not detecting the correct salient object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} +page_content=' 14 Image GT BASNet F3Net U2Net VST PFSNet PGNet DIS' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E4T4oBgHgl3EQf5Q72/content/2301.05323v1.pdf'} diff --git a/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/2301.13464v1.pdf.txt b/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/2301.13464v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..af255ac087627a9f41d73a8dda04c108312a3131 --- /dev/null +++ b/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/2301.13464v1.pdf.txt @@ -0,0 +1,3033 @@ +TRAINING WITH MIXED-PRECISION FLOATING-POINT ASSIGNMENTS +Wonyeol Lee 1 Rahul Sharma 2 Alex Aiken 1 +ABSTRACT +When training deep neural networks, keeping all tensors in high precision (e.g., 32-bit or even 16-bit floats) +is often wasteful. However, keeping all tensors in low precision (e.g., 8-bit floats) can lead to unacceptable +accuracy loss. Hence, it is important to use a precision assignment—a mapping from all tensors (arising in +training) to precision levels (high or low)—that keeps most of the tensors in low precision and leads to sufficiently +accurate models. We provide a technique that explores this memory-accuracy tradeoff by generating precision +assignments that (i) use less memory and (ii) lead to more accurate models at the same time, compared to the +precision assignments considered by prior work in low-precision floating-point training. Our method typically +provides > 2× memory reduction over a baseline precision assignment while preserving training accuracy, and +gives further reductions by trading off accuracy. Compared to other baselines which sometimes cause training +to diverge, our method provides similar or better memory reduction while avoiding divergence. +1 +INTRODUCTION +In deep neural network training, floating-point formats are +usually used to represent tensors and it is worthwhile to use +the smallest bitwidth format that gives acceptable results. +For example, it is common to replace tensors using 32-bit +floats with tensors that use 16-bit floats (Micikevicius et al., +2018; Kalamkar et al., 2019). The benefits are easy to under- +stand: computations using lower-precision floats not only +use less memory but are also faster (due to improved vec- +tor parallelism, locality, and reduced data movement). The +downside is that there is generally some loss of training accu- +racy, and in the worst case training may not even converge. +For such low-precision floating-point training, the most +common approaches use two floating-point formats—one +for lower-precision floats (e.g., 8-bit floats) and the other +for higher-precision floats (e.g., 16-bit floats)—and assign +one of the two formats to each tensor (including weights, +activations, and their gradients). The precision assignments +studied in previous work fall into one of two assignment +schemes (which both have several variants): the uniform +assignment uses low precision for almost all tensors +(often excepting those in the first and/or last few layers) +(Micikevicius et al., 2018), while the operator-based +assignment limits low precision to the input tensors of +certain operators (e.g., convolutions) (Sun et al., 2019). +Prior work has shown that both precision assignment +schemes (with well-chosen low-bitwidth floating-point +formats) can match the accuracy of 32-bit-float training +1Stanford University, USA +2Microsoft Research, India. +Correspondence to: Wonyeol Lee . +Preprint. Under review. +(Micikevicius et al., 2018; Kalamkar et al., 2019; Wang +et al., 2018; Sun et al., 2019; Chmiel et al., 2021; Drumond +et al., 2018; Cambier et al., 2020; Fox et al., 2021). +There is an important limitation in all prior approaches +to low-precision floating-point training: they use very +few precision assignments (most often just one) for a +given set of models, but there are some other models and +inputs where the chosen precision assignment (i) results +in noticeably worse accuracy than 32-bit-float training, +(ii) causes training to even diverge, or (iii) admits a more +efficient assignment that achieves similar training accuracy +(see Figures 1, 3, and 4). +In this paper, we present a new, automated method for choos- +ing precision assignments that removes the limitations de- +scribed above. To do so, we formally introduce the memory- +accuracy tradeoff problem: given a dataset, a model, and +two floating-point precision levels (i.e., bitwidths; high and +low), find a mixed precision assignment (a mapping from +all tensors arising in training to high/low precision) for the +model that maximizes training accuracy subject to a given +upper bound on the model aggregate (i.e., the total number +of bits of all tensors appearing in training). The model aggre- +gate is a proxy for the memory and time required for training, +as it is roughly proportional to memory footprint and also +well-correlated with training time (since training is often +dominated by data movement) (Micikevicius et al., 2018). +We prove that the memory-accuracy tradeoff problem +is theoretically difficult (namely NP-hard) partly due +to the exponential number of possible mixed precision +assignments (which we often refer to simply as precision +assignments for brevity). The large number of possible +assignments makes the problem difficult in practice as +arXiv:2301.13464v1 [cs.LG] 31 Jan 2023 + +Training with Mixed-Precision Floating-Point Assignments +(a) SqueezeNet +(b) ShuffleNet-v2 +(c) MobileNet-v2 +Figure 1: Training trajectory of various models on CIFAR-100. Colors denote precision assignments: all-32-bit πfp32 (red), +uniform πunif (yellow), and operator-based πop (blue) (see §3.1); the latter two use the 8-bit (and 16-bit) floats in (Sun et al., +2019) as low (and high) precision numbers. Markers denote the “width multiplier” of a model, which controls the capacity +of the model (see §5.3): 1.0 (•), 0.5 (■), 0.25 (▲), and 0.1 ( +). Some lines of πunif are missing as they converge to small +values or diverge. Observe that neither πunif nor πop works best for all models: in some models, πop has a similar accuracy +to πfp32; but in other (and all) models, the accuracy drop of πop (and πunif) from πfp32 are noticeably large (i.e., >1%). +well: there is no known analytical method for predicting +the training accuracy of a given precision assignment, and +for any practical model there are far too many precision +assignments to simply test them all. +We propose a simple (heuristic) approach to the tradeoff +problem that prioritizes tensors for low-precision formats +based on the tensor’s size (with an additional step described +below). More specifically, our algorithm takes as input +a single parameter giving a desired upper bound on the +model aggregate. Starting with the largest tensor in the +model, tensors are assigned low precision in size order +(from largest to smallest) until the model aggregate falls +below the given upper bound; all remaining tensors are +assigned high precision. Our main result is that this method +discovers mixed precision assignments that use less memory +while achieving higher training accuracy than previous +approaches. While we cannot show that our method finds +Pareto-optimal memory-accuracy tradeoffs, we do show that +our results are closer to Pareto-optimal than prior methods. +Some precision assignments initially generated by our +algorithm cause training to diverge due to an excessive +number of overflows. To address this issue, we propose an +overflow handling technique that promotes tensors causing +too many overflows from low precision to high precision +during training. +In our experiments, these promotions +consume only a small amount of additional memory (< 3% +of the maximum model aggregate) and prevent training +from diverging. The overflow handling technique is not +specific to our algorithm and can be applied to other +precision assignment methods as well. +We evaluate a PyTorch implementation of our method using +experiments on standard image classification tasks. We first +demonstrate that the precision assignments computed by our +method alleviate the limitations of existing methods: they +indeed explore the tradeoff between memory and accuracy +and exhibit a better tradeoff than the uniform and operator- +based assignments. We then show the two main components +of our method (i.e., precision demotion of larger tensors +and precision promotion of overflowing tensors) are both +important to produce competitive precision assignments. +We also provide some guidance on how users may apply +our method to navigate the memory-accuracy tradeoff. +To summarize, this work makes four main contributions: +• We formally introduce the memory-accuracy tradeoff +problem to explore better mixed precision assignments +for low-precision floating-point training and prove the +NP-hardness of the problem. +• We present a novel precision assignment technique, as a +heuristic solution to the tradeoff problem, that proposes +assignments based on a single parameter denoting a +desired upper bound on the model aggregate. +• We present a novel technique that can handle an exces- +sive number of overflows arising in training while using +a small amount of additional memory. The technique can +be applied to any (not just our) precision assignments. +• We demonstrate that the mixed precision assignments +found by our method do explore the tradeoff between +memory and training accuracy, and outperform existing +precision assignment methods. +We remark that this work focuses on low-precision +floating-point training, not fixed-point training (which uses +fixed-point formats), since we want to target upcoming +hardware (e.g., (Andersch et al., 2022)) with native support +for low-precision floats (e.g., 8-bit floats) and their oper- +ations. Also, this work focuses on low-precision training +(which trains a model from scratch), not inference (which +assumes a pre-trained model). More discussion is in §2. +Our precision assignment method typically provides > 2× +memory reduction over the operator-based assignment +while maintaining similar training accuracy and gives +further reductions by trading off accuracy. Our method +also provides similar memory reduction to the uniform +assignment, while avoiding the divergence of training often +caused by a uniform assignment. + +70 +accuracy (%) +60 +50 +test +40 +0 +50 +100 +150 +200 +epoch75 +70 +test accuracy (%) +65 +60 +55 +50 +45 +0 +50 +100 +150 +200 +epoch75 +test accuracy (%) +70 +65 +60 +55 +50 +45 +0 +50 +100 +150 +200 +epochTraining with Mixed-Precision Floating-Point Assignments +The paper is organized as follows. +After discussing +related work (§2), we define the memory-accuracy tradeoff +problem and study its hardness (§3). We then describe our +algorithm for the problem (§4) and our evaluation (§5). We +conclude with limitations and future work (§6). +2 +RELATED WORK +Low-precision floating-point training has been exten- +sively studied since the work of (Micikevicius et al., 2018). +One active research direction is to select appropriate floating- +point formats (or their variants) for low- and high-precision +numbers in training. Various floating-point formats have +been proposed, including FP16 (Micikevicius et al., 2018), +BF16 (Kalamkar et al., 2019), FP8 (Wang et al., 2018), +HFP8 (Sun et al., 2019), and FP6 (Chmiel et al., 2021), +along with some variants such as HBFP (Drumond et al., +2018), S2FP8 (Cambier et al., 2020), and BM (Fox et al., +2021). Recently, the problem of automatically selecting +such floating-point formats has been considered: e.g., (Yang +et al., 2022). Another research direction is to develop algo- +rithmic techniques that improve training accuracy under low +precision: e.g., (Sa et al., 2018; Yang et al., 2019a; Zamirai +et al., 2020; Björck et al., 2021). Our work is orthogonal +and complementary to all these prior works: they consider +various floating-point formats or training algorithms but use +a fixed precision assignment, which is either the uniform +or operator-based assignment (or their variants); our work +explores various precision assignments once floating-point +formats and training algorithms are fixed (e.g., based on +the prior works). The tradeoff between memory and accu- +racy in training is also considered in (Yang et al., 2022), +but the work differs from ours: they vary floating-point for- +mats when a precision assignment is fixed, while we vary +precision assignments when floating-point formats are fixed. +Low-precision fixed-point training uses fixed-point for- +mats as a low-precision representation instead of a floating- +point format. Some works use fixed-point formats for for- +ward tensors and floating-point formats for backward ten- +sors: e.g., (Courbariaux et al., 2015; Jacob et al., 2018; Choi +et al., 2018; Yang et al., 2019b; Sun et al., 2020). Other +works use only fixed-point formats for all tensors: e.g., +(Gupta et al., 2015; Zhou et al., 2016; Wu et al., 2018; Das +et al., 2018; Banner et al., 2018; Sakr & Shanbhag, 2019; +Zhang et al., 2020; Rajagopal et al., 2020). Among all these +works, some consider various mixed precision assignments +with different bitwidth (fixed-point) formats (e.g., (Sakr & +Shanbhag, 2019; Zhang et al., 2020)); but they are not ap- +plicable to our context (i.e., floating-point training) since +they rely on some properties of fixed-point formats that do +not hold for floating-point formats (e.g., all numbers in a +given format are equally distributed). The approach taken in +(Rajagopal et al., 2020) is orthogonal and complementary +to ours: they use only the uniform precision assignment, but +change the underlying low-precision formats during train- +ing; we consider various mixed precision assignments, but +fix the underlying low-precision formats during training. +Low-precision inference, often called neural network quan- +tization (in a narrow sense), aims at reducing the latency +or memory of neural network inference (instead of train- +ing) by using low-precision numbers (Nagel et al., 2021). +Existing approaches typically assume a pre-trained model +and try to find low-precision formats for each part of +the inference computation, either by retraining the model +(called quantization-aware training) or without any retrain- +ing (called post-training quantization); see, e.g., (Gholami +et al., 2022; Qin et al., 2022) for surveys. Some works on +inference consider various mixed precision assignments, but +they are not applicable to our context: they focus on making +inference more efficient and usually assume a pre-trained +model; we focus on making training more efficient and aim +at learning a model from scratch. +Floating-point tuning is another related topic, which con- +siders the following problem: given a program, assign ap- +propriate formats (among given candidates) to the program’s +floating-point variables such that the program’s output has +an error smaller than a given threshold for all given inputs, +while also maximizing performance (Rubio-González et al., +2013; 2016; Chiang et al., 2017; Guo & Rubio-González, +2018; Menon et al., 2018). This problem is different from +the problem we consider: the former is concerned with the +floating-point error after a single run of a program, while +we are concerned with the training accuracy after a large +number of runs of a program (i.e., a gradient computation) +where each run affects the next run; further, the former con- +siders general-purpose programs, while we consider deep +learning programs and exploit their unique features. +3 +PROBLEM +In this section, we first provide background on low- +precision floating-point training (§3.1), based on which +the memory-accuracy tradeoff problem is introduced (§3.2). +We then prove the NP-hardness of the problem (§3.3). Our +approach in §3–4 is more formal than most related works +for two reasons: (i) we show the problem is NP-hard, which +has not been considered in prior work; and (ii) to clearly +describe the precision assignments to be considered. +3.1 Background: Low-Precision Floating-Point Training +Let T be the set of real-valued tensors and let [n] denote the +set {1, · · · , n}. For a supervised learning task, we usually +consider a model network M = (f1, · · · , fn) parameter- +ized by θ = (θ1, · · · , θn) ∈ Tn, and a loss network L = +(fn+1, · · · , fm), where fi : T2 → T is a primitive operator +on tensors (e.g., convolution, batch normalization, maxpool, +and softmax). Given an input-output pair (x, y) ∈ T2, the + +Training with Mixed-Precision Floating-Point Assignments +𝑓! +𝑑𝑓!,! +𝑑𝑓!,# +𝑓$%! +𝑑𝑓$%!,! +⋯ +𝑓$ +𝑑𝑓$,# +𝑑𝑓$,! +𝑓& +𝑑𝑓&,! +⋯ +⋯ +⋯ +# +𝑑𝑣& +%𝑣# +&𝜃! +# +𝑑𝑣# +# +𝑑𝜃! +%𝑣$%! +%𝑣$%# +# +𝑑𝑣$%! +# +𝑑𝑣$%# +&𝜃$ +# +𝑑𝜃$ +%𝑣&%! +# +𝑑𝑣&%! +𝑦 +: forward computation +: backward computation +%𝑣! +%𝑣$ +%𝑣& +# +𝑑𝑣! +# +𝑑𝑣$ +Figure 2: A diagram showing the tensors and operators +used in a gradient computation; see Eq. (1) for details. For +brevity, rounding functions rndπ(·) are omitted. +model M computes a predicted output y′ of x by iteratively +applying fi(·, θi) to x (i ∈ [n]), and L computes a loss +from y′ by iteratively applying fi′(·, y) to y′ (i′ ∈ [m]\[n]). +A standard way to train M is to minimize the loss value +using the gradient descent algorithm: iteratively update θ +by following the gradient of the loss with respect to θ. +Floating-point training. In practice, we perform a gradient +computation usually with tensors represented in floating- +point formats. Let π : TS → FP be a precision assignment +giving the floating-point format of each tensor, where TS = +∆ +{vi, dv i, θj, dθj | i ∈ [m + 1], j ∈ [n]} is the set of tensors +arising in a gradient computation (explained below), and +FP = +∆ {fp(e, m, b) | e, m ∈ N, b ∈ Z} is the set of floating- +point formats. Here fp(e, m, b) denotes a floating-point +format that consists of a 1-bit sign, an e-bit exponent, and +an m-bit mantissa, and has an (additional) exponent bias +of b ∈ Z. A common choice of π is πfp32(t) = +∆ fp32 for +all t ∈ TS, where fp32 = +∆ fp(8, 23, 0) is the standard 32-bit +floating-point format. +Given a precision assignment π, a gradient computation is +typically performed by the backpropagation algorithm: with +ˆv1 = rndπ(v1)(x) and ˆdv m+1 = rndπ(dv m+1)(1), compute +ˆvi+1 = rndπ(vi+1)(fi(ˆvi, ˆui)), +ˆθj = rndπ(θj)(θj), +ˆdv i = rndπ(dv i)(df i,1( ˆdv i+1, ˆvi, ˆui)), +ˆ +dθj = rndπ(dθj)(df j,2( ˆdv j+1, ˆvj, ˆθj)), +(1) +for i ∈ [m] and j ∈ [n]; see Figure 2 for a diagram. Here +rnd : FP × T → T is a function rounding a given input +to a given floating-point format, df i,1, df i,2 : T3 → T are +the backward operators of fi with respect to its first and +second arguments, respectively, and ˆui = ˆθi if i ∈ [n] and +y otherwise. We call vi and θj the forward tensors, and +dv i and dθj the backward tensors. We put a hat over each +tensor to emphasize that its value is the output of a rounding +function to a possibly low-precision format; remark that +such a rounding function is not used within fi, df i,1, and +df i,2, since they typically use large bitwidth floats (e.g., +fp32) and no low-precision floats internally (Kalamkar +et al., 2019; Cambier et al., 2020). After the computation, +ˆ +dθj stores the gradient of the loss value with respect to θj. +The overall picture of floating-point training is now de- +scribed as follows. In each iteration of the gradient descent +algorithm, we compute ˆ +dθj via Eq. (1) using a given preci- +sion assignment π, training data (x, y), and current weights +θ. We then update each θj by θj ← rndfp32(θj − η · ˆ +dθj) +given a learning rate η > 0, and proceed to the next iteration +until the training ends. Here we use fp32 to represent θj +by following the convention in low-precision floating-point +training: a “master copy” of weights (i.e., θj) is stored sep- +arately from the weight values (i.e., ˆθj) used in a gradient +computation, and is usually represented by fp32 (Micike- +vicius et al., 2018; Kalamkar et al., 2019; Cambier et al., +2020). The memory overhead of this master copy is very +small compared to the memory required to store other ten- +sors (e.g., activation tensors vi) (Micikevicius et al., 2018). +Low-precision floating-point training. In low-precision +training, we use a precision assignment π where some +tensors have a smaller bitwidth than fp32. +Particularly +well-studied are π that use two predetermined floating- +point bitwidths (which are different) and optionally vary +the rest of the format from tensor to tensor. +We call +C : TS × {lo, hi} → FP a precision-candidate assignment +if C(t, lo) has the same bitwidth for all t ∈ TS, the same +holds for hi, and the bitwidth for lo is smaller than that for hi. +We also define Π(C) = +∆ {π : TS → FP | ∀t ∈ TS. π(t) ∈ +{C(t, lo), C(t, hi)}} to be the set of precision assignments +that conform to C. +Among various precision assignments in Π(C), two have +received the most attention: +the uniform assignment +πunif,C (Micikevicius et al., 2018) and the operator-based +assignment πop,C (Sun et al., 2019). The former assigns +low-precision formats to all tensors uniformly1, and the +latter to (most of) the input tensors of GEMM operators (in +both forward and backward passes): +πunif,C(t) = +∆ C(t, lo) for all t ∈ TS, +πop,C(t) = +∆ +� +� +� +� +� +� +� +C(t, lo) if t ∈ {vi, θi, dv i+1} for some i +and fi is a GEMM operator +(but not the first/last one) +C(t, hi) otherwise, +(2) +where a GEMM operator refers to a general matrix multi- +plication operator which arises in, e.g., fully-connected or +convolutional layers. A particular variant πop′,C of πop,C +has received much attention as well (Kalamkar et al., 2019; +PyTorch, 2022), which assigns low-precision formats to +(most of) the input and output tensors of GEMM operators: +it is defined as πop,C except that {vi, θi, dv i+1} in Eq. (2) +is replaced by {vi, θi, vi+1, dv i, dθi, dv i+1}. For several +choices of C, these assignments have been shown to produce +training accuracy similar to that by πfp32 on many datasets +and models (see §1–2). +1For simplicity we define πunif,C without the common +exceptions for tensors near v1 and/or vm+1. + +Training with Mixed-Precision Floating-Point Assignments +3.2 +Memory-Accuracy Tradeoff Problem +We now introduce the following problem based on §3.1, +to address the limitation of existing approaches for +low-precision floating-point training discussed in §1: +Problem 3.1 (Memory-accuracy tradeoff). Given training +data {(xi, yi)}, a model and loss network M and L, a +precision-candidate assignment C, and a lower bound +r ∈ [0, 1] on the low-precision ratio, find π ∈ Π(C) that +maximizes acc(π) subject to ratiolo(π) ≥ r. +Here acc(π) denotes the accuracy of the model M when +trained with π on {(xi, yi)}, and ratiolo(π) denotes the +low-precision ratio of π, i.e., the portion of the tensors +represented in low-precision under π, among all tensors +appearing in a gradient computation:2 +ratiolo(π) = +∆ size({t ∈ TS | π(t) = C(t, lo)}) +size(TS) +∈ [0, 1] +where size(T) = +∆ +� +t∈T size(t) denotes the total size +(i.e., number of elements) of all tensors in T ⊆ TS. For +instance, ratiolo(πhi) = 0 and ratiolo(πlo) = 1 for the +all-high-precision assignment πhi and the all-low-precision +assignment πlo. The problem asks for a precision assign- +ment that maximizes training accuracy under a memory +constraint, which is expressed as a fraction of the memory +required to train the model using πhi. +3.3 +NP-Hardness of the Problem +We prove that the memory-accuracy tradeoff problem from +§3.2 is NP-hard by showing that there is a polynomial-time +reduction from the knapsack problem to this problem: +Theorem 3.2. Problem 3.1 is NP-hard. +Proof sketch. Recall the knapsack problem: given n items +with weights wi ∈ N and profits pi ∈ N (i ∈ [n]), find a +subset of the items that maximizes the total profit while its +total weight does not exceed a given threshold W ∈ N. +Given an instance (w, p, W) of the knapsack problem, we +construct an instance of Problem 3.1 such that we get the +following (informal) correspondence between the two: wi +corresponds to the size of the parameter tensor θi; pi to the +i-th component of the input data; W to the lower bound r +on the low-precision ratio (in an inverse way); and selecting +the i-th item corresponds to assigning a high-precision +format to the tensor θi (and related tensors), which roughly +decreases the low-precision ratio by wi while increasing +the accuracy of the model (after training) by pi. Based +2As explained in §1, the low-precision ratio is a proxy for +the reduction in memory as well as training time (because the +low-precision ratio increases as the model aggregate decreases). +Note that it is not always possible to simply measure training time, +as some floating-point bitwidths of interest (e.g., 8-bit) are not +supported natively by current hardware. +on this informal correspondence, we formally prove that +an optimal solution to the above instance of Problem 3.1 +can be converted in linear time to an optimal solution to +the given knapsack problem (w, p, W). That is, we have +a linear-time reduction from the knapsack problem (which +is NP-hard (Karp, 1972)) to Problem 3.1 which is therefore +NP-hard. For a detailed proof, see Appendix A. +Intuitively, the proof relies on two aspects of Problem 3.1: +the size of the search space (i.e., |Π(C)|) is exponential in +the size of the problem (especially |TS|), and some values +representable in a high-precision format underflow to 0 in +a lower-precision format. Note that underflows are relevant +in low-precision training: they frequently arise in practice, +degrading the results of training (Micikevicius et al., 2018). +The NP-hardness result indicates that it is unlikely any +polynomial-time algorithm solves the problem exactly. +4 +ALGORITHM +In this section, we propose a novel (heuristic) algorithm for +the memory-accuracy tradeoff problem (§4.1), and a new +technique to handle overflows arising in training (§4.2). +4.1 +Precision Demotion for Saving Memory +Consider an input to the memory-accuracy trade-off prob- +lem (Problem 3.1): a model and loss network M = (f1, +· · · , fn) and L = (fn+1, · · · , fm), a precision-candidate +assignment C, and a lower bound r on the low-precision +ratio. Given the input, our algorithm works in two steps. +Tensor grouping. We first group tensors in TS such that +each group consists of all the tensors between two “adjacent” +GEMM operators (see below for details). This grouping re- +duces the search space over precision assignments, from all +of Π(C) to a subset in which the same precision is assigned +to the tensors in the same group. This specific grouping +strategy is based on two observations: a majority of floating- +point operations are carried out in GEMM operators, and it +is standard (e.g., in PyTorch) to use the same precision for a +forward tensor and its corresponding backward tensor. +Formally, we group tensors as follows. Let fk and fk′ +(k < k′) be GEMM operators that are “adjacent”, i.e., there +is no GEMM operator in {fk+1, · · · , fk′−1}. For each such +(fk, fk′), we create a group {vi, dv i, θj, dθj | i ∈ (k, k′] ∩ +[m + 1], j ∈ (k, k′] ∩ [n]}. After that, we create two more +groups for the remaining tensors: one for the tensors near v1 +and the other for tensors near vm+1. As a result, we obtain +a set of disjoint groups of tensors {T1, T2, · · · } ⊆ 2TS. +Precision demotion. Given the groups of tensors, T1, T2, +· · · , we construct a precision assignment π as follows: ini- +tialize π to the all-high-precision assignment and update +π by demoting the precision of all tensors in a group to +low precision, one group at a time, until the low-precision + +Training with Mixed-Precision Floating-Point Assignments +ratio of π becomes greater than r. We demote the pre- +cision of groups in decreasing order of their sizes (i.e., +the total number of elements in tensors); that is, the pre- +cision of a larger size group is demoted earlier. Formally, +let {T ′ +1, T ′ +2, · · · } be the reordering of {T1, T2, · · · } such +that size(T ′ +1) ≥ size(T ′ +2) ≥ · · · . After initializing π by +π(t) = C(t, hi) for all t, we iterate over i ∈ N and update π +to π(t) = C(t, lo) for all t ∈ T ′ +i, until ratiolo(π) ≥ r is first +satisfied. The resulting π is the output of our algorithm. +The intuition behind using group size as the priority order +for precision demotion is based on the fact that it is actually +optimal in a very simplified setting. Suppose that an input +x to the model M stores a quantity of information I and +the forward computation of M is nothing but a process of +extracting the information in the input into a small number +of values, i.e., the tensor vn+1. Assume that passing through +each group Oi = {fk+1, · · · , fk′} of operators (corre- +sponding to the group Ti of tensors) reduces the amount of +information by a factor αi ∈ (0, 1), and using low precision +on the group Ti further reduces the amount of information +by a constant factor β ∈ (0, 1) for all i. Then, the amount +of information left in vn+1 becomes I × (α1α2 · · · ) × βl, +where l is the number of groups in low precision. In this +simplified setting, maximizing the amount of information in +vn+1 is equivalent to minimizing the number of groups in +low precision, which is achieved precisely by demoting the +largest groups first (when there is a constraint on the low- +precision ratio). We show empirically (§5.4) that using the +decreasing size order in precision demotion indeed produces +better precision assignments than using other orders. +4.2 +Precision Promotion for Handling Overflows +While our algorithm in §4.1 exerts a constraint on memory +usage, it places no explicit constraint on training accuracy, +and so not surprisingly for some models and datasets +the resulting precision assignment causes training to +diverge—accuracy decreases significantly and remains low +after some point. We observe that when training begins to +diverge (and a bit before that), many overflows occur in the +rounding function of some tensors ˆvi, i.e., an input tensor to +the function rndπ(vi)(·) in Eq. (1) contains many elements +whose magnitude is larger than the maximum representable +number of the format π(vi) (Figure 6(a-b); §5.4). This +rapid increase in overflows in individual tensors is a signal +that training may diverge. +Precision promotion. Based on this observation, after each +gradient computation we update the current precision as- +signment π by promoting to high precision (i.e., C(t, hi)) +any forward tensor t whose overflow ratio is greater than a +given threshold Θ ∈ (0, 1); this updated precision assign- +ment is used in the next gradient computation. Here the +overflow ratio of t ∈ TS denotes the number of overflows +arising in the rounding function of ˆt in Eq. (1), divided by +the number of elements in ˆt. We show empirically (§5.4) +that training always converges using this technique and the +additional memory cost of promotion is small (in our exper- +iments, < 3% of the maximum model aggregate). For the +experiments, we use Θ = 0.01; in fact we found that a wide +range of values for Θ (0.1, 0.01, and 0.001) all work well. +Note that this technique is not specific to our algorithm and +can also be applied to other precision assignment methods. +We apply precision promotion only to forward tensors for +two reasons. First, dynamic loss scaling (Micikevicius +et al., 2018; Sun et al., 2019; Nvidia, 2019; PyTorch, 2022) +already handles overflows in backward tensors, but not in +forward tensors: loss scaling multiplies the backward loss +tensor dv m+1 by a constant before performing backward +computation, to scale up all backward tensors; the dynamic +version adjusts the constant during training in a way that +avoids overflows in backward tensors. Note that dynamic +loss scaling does not affect forward tensors at all. Second, +we cannot use a similar idea to handle overflows in forward +tensors, because forward tensors are not linear in the input +tensor v1 whereas backward tensors are linear in the back- +ward loss tensor dv m+1 (by the linearity of differentiation). +Precision promotion incurs little if any computational over- +head: checking whether a single rounding operation over- +flows is cheap, and we only apply rounding functions to the +output tensor of an arithmetic-intensive operator (e.g., con- +volution and batch normalization), amortizing the cost of the +overflow checks over a large number of other operations. +5 +EXPERIMENTS +In this section, we evaluate our precision assignment +technique (developed in §4) on standard training tasks to +answer three research questions: +• Does our technique explore the tradeoff between +memory and accuracy and achieve a better tradeoff than +existing (fixed) precision assignments (§5.3)? +• Are the two main components of our technique, +precision demotion/promotion of larger/overflowing +tensors, important for good performance (§5.4)? +• How can we choose the parameter r in our technique +(i.e., a lower bound on the low-precision ratio) (§5.5)? +5.1 +Implementation +We have implemented our precision assignment technique +using PyTorch (Paszke et al., 2019). Given a model and loss +network, and a dataset, our implementation takes as param- +eters a precision-candidate assignment C and a lower bound +r on the low-precision ratio; it then automatically assigns +precisions to tensors (appearing in training) according to +our technique and uses those assigned precisions in gradient +computations. To make these procedures automatic, our +implementation works as follows: + +Training with Mixed-Precision Floating-Point Assignments +• For each PyTorch class for a primitive operator (e.g., +torch.nn.Conv2d), our implementation provides its +wrapped version (e.g., mpa.nn.Conv2d) which records +auxiliary information for our technique (e.g., floating- +point format of input/output tensors) and applies proper +rounding functions in forward/backward computations +based on the auxiliary information. Models should now +use the wrapped classes instead of the original ones. +• Our implementation first constructs a computation graph +(of a given model and loss network) dynamically by +running a forward computation on a minibatch of input +data. The computation graph and other information (e.g., +each tensor’s size) are recorded in the wrapped classes. +• Using the auxiliary information just recorded, our +implementation then constructs a precision assignment +according to §4.1, uses it in gradient computations, and +updates it after each gradient computation according +to §4.2. We record the precision assignment also in the +wrapped classes to automatically apply proper rounding +functions in gradient computations. +As no current hardware natively supports low-precision +formats used in the experiments (e.g., 8-bit floats) and +their operations, we simulate them with 32-bit floats +and 32-bit operations followed by rounding functions as +described in Eq. (1). We implement the rounding functions +based on the QPyTorch library (Zhang et al., 2019); a few +extensions are required though, e.g., to support exponent +bias and signal overflows for dynamic loss scaling. We +automatically apply these rounding functions after each +primitive operator, by using PyTorch’s hook feature (e.g., +nn.Module.register_*hook). +5.2 +Experiment Setups +Datasets and models. As benchmarks for our experiments, +we use the image classification task and three datasets for the +task: CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), and +ImageNet (Russakovsky et al., 2015); we choose them since +they have been widely used in recent works on low-precision +training as a standard choice (Wang et al., 2018; Sakr & +Shanbhag, 2019; Rajagopal et al., 2020; Chmiel et al., 2021). +For the task and datasets, we use four well-known models: +SqueezeNet (Iandola et al., 2016), ShuffleNet-v2 (Ma et al., +2018), MobileNet-v2 (Sandler et al., 2018), and ResNet- +18 (He et al., 2016); they are chosen since models with +relatively few weights, such as these, are generally known +to be more difficult to train with low precision than those +with more weights (Sun et al., 2019). We considered other +tasks (e.g., language modeling) and related models (e.g., +RNN/transformer-based models) but did not include them +in our experiments because substantial additional implemen- +tation effort orthogonal to our main contributions would be +required: these models use some PyTorch operators that do +not support per-tensor precision assignments, so applying +our technique to these models requires significant modifica- +tions to PyTorch internals. +Precision-candidate and precision assignments. For the +experiments, we use the precision-candidate assignment +C studied in (Sun et al., 2019), which uses 16-bit (and +8-bit) floats for high (and low) precision; in particular, +C(t, hi) = fp(6, 9, 0) for all (forward/backward) tensors +t, and C(t, lo) = fp(4, 3, 4) for all forward tensors t and +fp(5, 2, 0) otherwise. We choose this particular C because it +uses sub-32-bit floating-point formats for both low and high +precision and the precision assignment πop,C was shown to +achieve accuracy comparable to 32-bit training (Sun et al., +2019). The three floating-point formats used in C allow +subnormals but no infinities and NaNs, which are rounded +to the largest or smallest representable numbers. While no +current hardware is available for the latter two 8-bit formats, +they will be supported natively on NVIDIA’s forthcoming +H100 GPU (Andersch et al., 2022). Because our technique +is parameterized by a precision-candidate assignment, it is +easily applied to other assignments as well. +We evaluate our technique by varying its parameter r +(i.e., a lower bound on low-precision ratio) over deciles +r ∈ {0, 0.1, 0.2, · · · , 1}. We write πours,r to denote the +precision assignment chosen by our technique (described +in §4) for a given r; e.g., πours,0 is the all-high-precision +assignment, and πours,1 is the all-low-precision assignment +equipped with our precision promotion technique (§4.2). +By following (Sun et al., 2019), all precision assignments +(including πours,r) in our experiments use high precision +(i.e., 16 bits) for all backward weight tensors (i.e., ˆ +dθj). +Other setups and compute time. All experiments were +performed on NVIDIA V100 GPUs; total compute time for +all experiments was 1,008 GPU-days. We train all models in +a standard way: we apply dynamic loss scaling (a standard +technique used in low-precision floating-point training; see +§4.2 for details) except for 32-bit training, and use standard +settings (e.g., learning rate); see Appendix B for details. +Due to random variations in training, we perform four runs +of training for each configuration and report the average and +the range of measured quantities. +5.3 +Comparison with Existing Precision Assignments +To compare our technique with existing precision assign- +ments for floating-point training, we train each model +with the following precision assignments: all-32-bit πfp32, +uniform πunif (Micikevicius et al., 2018), operator-based +πop (Sun et al., 2019), its variant πop′ (Kalamkar et al., +2019; PyTorch, 2022), and ours πours,r (see §3.1 and §5.2 +for their definitions). +We choose πunif, πop, and πop′ +as baselines because existing precision assignments for +floating-point training fall into one of the three assignments +(or their variants) (see §1–2). + +Training with Mixed-Precision Floating-Point Assignments +Figure 3: Results of training ShuffleNet-v2 on ImageNet with πfp32, πunif (Micikevicius et al., 2018), πop (Sun et al., +2019), πop′ (Kalamkar et al., 2019), and πours,r. Left: Each line shows the average training trajectory for each precision +assignment; πours,r is colored from navy to yellow (darker for smaller r). Right: Each point shows the memory-accuracy +tradeoff of each precision assignment; a red-dashed line shows the accuracy of πfp32; and shaded areas show the variation +among four training runs. In the right figure, top-right points are better than bottom-left ones. Observe that there are •s +above and to the right of +and +, respectively. ⋆ is missing as its y-value is too small. +(a) CIFAR-10, SqueezeNet +(b) CIFAR-100, SqueezeNet +(c) CIFAR-100, SqueezeNet† +(d) CIFAR-10, ShuffleNet-v2 +(e) CIFAR-100, ShuffleNet-v2 +(f) CIFAR-100, ShuffleNet-v2† +(g) CIFAR-10, MobileNet-v2 +(h) CIFAR-100, MobileNet-v2 +(i) CIFAR-100, MobileNet-v2† +(j) CIFAR-10, ResNet-18 +(k) CIFAR-100, ResNet-18 +(l) CIFAR-100, ResNet-18† +Figure 4: Memory-accuracy tradeoffs of πunif (Micikevicius et al., 2018), πop (Sun et al., 2019), πop′ (Kalamkar et al., +2019), and πours,r for four models and their smaller variants on CIFAR-10 and CIFAR-100. The variant models have width +multiplier 0.25 and are marked by †. Top-right points are better than bottom-left ones. In all but three plots, there are •s +above and to the right of +and +, respectively; even in the three plots (g,h,k), •s have almost the same tradeoffs to +and +. In half of all plots, ⋆ has much smaller y-values than other points. The training trajectories for the above plots +and the results of other smaller models are in Appendix C.1. + +92 +90 +88 +fp32 +op +86 +op +unif +80 +ours +78 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio72 +70 +68 +fp32 +op +66 +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio66 +test accuracy (%) +64 +X +62 +fp32 +60 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio94 +(%) +92 +test accuracy ( +90 +fp32 +88 +op +op +unif +80 +ours +78 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio74 +(%) +72 +test accuracy ( +70 +fp32 +68 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio68 +accuracy (%) +66 +64 +fp32 +62 +op +test a +op +unif +40 +ours +38 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio96 +test accuracy (%) +94 +92 +fp32 +op +90 +op +unif +76 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio80 +(%) +78 +test accuracy ( +76 +fp32 +74 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio70 +68 +66 +fp32 +op +64 +op +unif +52 +ours +50 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio60 +test accuracy (%) +40 +fp32 +op +20 +op +unif +ours +0 +0 +20 +40 +60 +80 +epoch66 +test accuracy (%) +64 +62 +fp32 +op +60 +op +unif +34 +ours +32 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio94 +(%) +92 +test accuracy ( +90 +fp32 +88 +op +op +unif +72 +ours +70 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio72 +(%) +70 +test accuracy ( +68 +66 +fp32 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio56 +test accuracy (%) +54 +52 +fp32 +op +50 +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments +We train the four models mentioned in §5.2 on CIFAR-10 +and CIFAR-100, and ShuffleNet-v2 on ImageNet. We also +train smaller variants of the four models (which are more +difficult to train with low precision) on CIFAR-100. We +obtain these variant models by following (Sun et al., 2019), +i.e., by applying a well-known approach for model reduction +that uses a parameter called the width multiplier (Howard +et al., 2017): each variant model reduces the number of +channels in most tensors by a width multiplier; we use three +values {0.5, 0.25, 0.1} for the width multiplier. We train +just one model on ImageNet due to the large amount of +computation involved: for each model, 44 training runs (11 +choices for r and 4 runs for each choice) are required for +πours,r and each run on ImageNet takes nearly a half day +with 16 GPUs. We use ShuffleNet-v2 for ImageNet since +the model shows interesting memory-accuracy tradeoffs +when trained on the (smaller) CIFAR datasets. +ImageNet. Figure 3 presents training results of ShuffleNet- +v2 on ImageNet: its left graph plots the average training +trajectory for each precision assignment, and its right graph +shows how each precision assignment trades off between +memory and accuracy, where memory is represented (in- +versely) by the low-precision ratio of the assignment and ac- +curacy is the best test accuracy of the model during training. +Each point in the right graph shows the average accuracy +of four runs of training, while the shaded area shows the +variation in accuracy among those four training runs. +Figure 3 shows three points. +First, as the parameter r +increases, the average accuracy drop of πours,r from πfp32 +increases (up to 5%). In contrast, πunif and πop′ have a +much larger average accuracy drop (more than 30%), as +some training runs diverge when πunif and πop′ are used. +Second, the tradeoff given by πours,r is better (i.e., closer +to Pareto-optimal) than by πop: πours,r for r ∈ {0.3, 0.4} +has both higher accuracy and larger low-precision ratio (i.e., +memory reduction) than πop. In particular, πours,0.4 has +1.6× the memory reduction of πop. Third, πours,r provides +options that πop cannot (which has an accuracy drop of +>1%). If we want accuracy closer to πfp32, say within 0.5%, +we can use πours,0.2 with 2.6% more memory than πop. If +we can tolerate a larger accuracy loss, say ≈ 3%, then we +can use πours,0.7 with 2.9× the memory reduction of πop. +CIFAR-10/100. Figure 4 presents the memory-accuracy +tradeoffs of precision assignments for the four models on +CIFAR-10 and CIFAR-100, and their smaller variants (with +width multiplier 0.25) on CIFAR-100. The results for other +smaller variants are similar and included in Appendix C.1. +The conclusions from Figure 3 hold for Figure 4: πours,r +provides a range of options by varying r and exhibits a +better tradeoff than πunif, πop, and πop′ in almost all cases. +We give a detailed comparison as follows. First, in half of +all 12 plots, πunif shows a similar tradeoff to πours,1. But +in the remaining half, πunif has an accuracy drop much +larger than all other precision assignments including πours,r, +since using πunif often makes training diverge while using, +e.g., πours,1 does not do so. Second, in all but two plots, +πours,r shows a strictly better tradeoff than πop: πours,r has +noticeably larger (> 2×) memory reduction than πop while +maintaining similar accuracy. Even in the two plots, πours,r +has a tradeoff very close to πop. Note that in three plots, +πop has an accuracy drop of >1% while πours,r provides +several options that have smaller accuracy drops and more +memory savings at the same time. Third, πours,r shows a +strictly better (or similar) tradeoff than πop′ in all but two +(or two) plots. Note that πop′ has accuracy smaller than πop +in all but one plots. Also it has an accuracy drop of >1% in +half of all plots, and sometimes makes training even diverge +(in one plot here and three other plots in Appendix C.1). +5.4 Ablation Study: Precision Demotion and Promotion +Precision demotion. +To evaluate the decision to use +precision demotion in decreasing-size order, we train +the four models on CIFAR-100 with πours,r, πours[inc],r +(which demotes tensor groups in increasing-size order) and +πours[rand],r (which demotes tensor groups in random or- +der). For πours[rand], three different random orders are used +in each case. The results, presented in Figure 5 (and Ap- +pendix C.2), show that the order of precision demotion has a +significant impact on the resulting memory-accuracy trade- +off, and that decreasing order provides the best results in +nearly all cases. Increasing order consistently shows the +worst results, suggesting our intuition (given in §4.1) for +choosing decreasing order has some basis in reality. +Precision promotion. To understand whether precision pro- +motion of overflowing tensors is important to our technique, +we train ShuffleNet-v2 on ImageNet using πours[no-promo],r +which does not promote tensors. The results, presented in +Figure 6(a), show that several training trajectories diverge +in early epochs and fail to recover afterwards. Figure 6(b) +plots the top-5 tensor overflow ratios for the highlighted +trajectory in Figure 6(a). The overflow ratios first spike +about when divergence occurs around epoch 11. A closer +look shows that the spike in overflow ratio occurs shortly +before divergence, and starts first in a few tensors and then +propagates to others. These observations indicate that an +excessive number of overflows in a few tensors are the cause +of the training divergence. +Finally, Figure 6(c-d) shows that precision promotion is +effective at preventing the divergence of training while +sacrificing only a small amount of memory reduction. The +figure shows ShuffleNet-v2 on ImageNet trained using our +technique with and without precision promotion. Figure 6(c) +shows that without precision promotion large accuracy +drops occur due to divergence, whereas with precision pro- +motion training converges. Figure 6(d) shows that the total + +Training with Mixed-Precision Floating-Point Assignments +(a) SqueezeNet +(b) ShuffleNet-v2 +(c) MobileNet-v2 +Figure 5: Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for three models on CIFAR-100. Observe +that •s are above and to the right of other points in nearly all cases. The results of ResNet-18 are in Appendix C.2. +(a) +(b) +(c) +(d) +Figure 6: Training ShuffleNet-v2 on ImageNet with πours,r and πours[no-promo],r. (a) Training trajectories of πours[no-promo],r +for different r; colors denote r values (darker for smaller r). (b) Top-5 overflow ratios of tensors at each epoch, for the +highlighted trajectory in (a); the largest ratio is blue and the fifth largest red. (c) Memory-accuracy tradeoffs of πours,r +and πours[no-promo],r. (d) Low-precision ratio when training ends vs. when training starts, for πours,r and πours[no-promo],r. +The results on CIFAR-10 are in Appendix C.2. +size of tensors promoted to high precision is small for all r +values. See Appendix C.2 for similar results for CIFAR-10. +5.5 +Choosing the value of r +The time and space savings of our method are most signif- +icant when a model is regularly retrained, which commonly +occurs when new data is periodically incorporated into +an existing model. Assuming that new data has a similar +distribution to existing data, we can choose a single r (a +parameter in our method) by conducting one set of exper- +iments where we train with πfp32 and πours,r for different r +and then choose the r value that maximizes model aggregate +savings while still having an acceptable drop in accuracy. +To simulate this scenario, we create five datasets ImageNet- +200-i (i ∈ [5]) as follows, so that each of them contains +different but similar data: randomly select 1/5 of the classes +in ImageNet (which has 1000 classes in total), and split the +training data of each class evenly into five new datasets. +For each ImageNet-200-i, we train ShuffleNet-v2 with +πfp32 and πours,r and present the results in Figure 7. Based +on the tradeoff results of πours,r, we can choose r = 0.4 if +we desire an average of < 1% accuracy drop from πfp32, and +we can choose r = 0.9 if an average ≈ 3% accuracy drop +is tolerable. We make two more observations: the tradeoff +result of πours,r is similar across all five datasets even +though each dataset is different, and for each r the variance +Figure 7: +Memory-accuracy tradeoffs of πours,r for +ShuffleNet-v2 on ImageNet-200-i (i ∈ [5]). +in the accuracy of πours,r from different datasets and runs of +training is similar to that of πfp32. Thus we expect that on +a new but similar dataset, πours,r would have an accuracy +drop similar to Figure 7 with acceptable variance. +6 +LIMITATIONS AND FUTURE WORK +Our work has the same limitation present in prior works +on low-precision floating-point training: as 8-bit floats and +operations are not handled natively in hardware, but rather +simulated in software, we cannot directly measure the poten- +tial speedup of our method, though we do expect speedups +to be proportional to the reduction in the model aggregate. +We leave it as future work to perform such experiments +on future hardware (e.g., NVIDIA’s H100) that natively +supports more low-precision formats. Another direction +for future work is to integrate our method into systems for +automatically optimizing deep learning computations (e.g., +(Jia et al., 2019; Unger et al., 2022)) to accelerate training. + +72 +accuracy (%) +70 +68 +test +ours[inc] +ours[rand] +66 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio74- +test accuracy (%) +72 +70 +ours[inc] +68 +ours[rand] +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio74 +(%) +accuracy( +72 +70 +test +ours[inc] +ours[rand] +68 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio80 +60 +40 +20 +0 +20 +40 +60 +80 +epoch1.0 +0.8 +overflow ratio +0.6 +0.4 +top-1 +top-2 +top-3 +0.2 +top-4 +top-5 +0.0 +0 +20 +40 +60 +80 +epoch80 +(%) +60 +accuracy ( +40 +test +20 +ours[no-promo] +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio1.0 +low-prec. ratio (end) +0.8 +0.6 +0.2 +ours[no-promo] +ours +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +low-prec. ratio (start)64 +62 +茶 +60 +58 +fp32 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments +REFERENCES +Andersch, +M., +Palmer, +G., +Krashinsky, +R., +Stam, +N., +Mehta, +V., +Brito, +G., +and +Ramaswamy, +S. +NVIDIA +Hopper +Architecture +In-Depth. +https://developer.nvidia.com/blog/ +nvidia-hopper-architecture-in-depth/, +2022. +Banner, R., Hubara, I., Hoffer, E., and Soudry, D. 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QPyTorch: +A Low-Precision Arithmetic Simulation Framework. +arXiv:1910.04540, 2019. +Zhang, X., Liu, S., Zhang, R., Liu, C., Huang, D., Zhou, +S., Guo, J., Guo, Q., Du, Z., Zhi, T., and Chen, Y. +Fixed-Point Back-Propagation Training. In CVPR, pp. +2327–2335, 2020. + +Training with Mixed-Precision Floating-Point Assignments +Zhou, S., Ni, Z., Zhou, X., Wen, H., Wu, Y., and Zou, +Y. +DoReFa-Net: +Training Low Bitwidth Convolu- +tional Neural Networks with Low Bitwidth Gradients. +arXiv:1606.06160, 2016. + +Training with Mixed-Precision Floating-Point Assignments +A +PROBLEM: DEFERRED PROOF +Theorem 3.2. Problem 3.1 is NP-hard. +Proof. We prove the NP-hardness of Problem 3.1 (the +memory-accuracy tradeoff problem) by reducing the +knapsack problem (which is NP-hard) to the tradeoff +problem. +More precisely, we prove that the knapsack +problem can be solved in polynomial time if we assume +an oracle for the tradeoff problem. +Recall the knapsack problem: given n items with weights +wi ∈ N and profits pi ∈ N (i ∈ [n]), and given a threshold +W ∈ N, decide which items to choose such that the total +profit of the chosen items is maximized while their total +weight does not exceed W. That is, find α ∈ {0, 1}n that +maximizes � +i∈[n] αipi subject to � +i∈[n] αiwi ≤ W. This +problem is well-known to be NP-hard (Karp, 1972). +Given an instance of the knapsack problem (w, p, W), we +construct an instance of the tradeoff problem as follows: +• Notations. +The following construct uses a constant +k ∈ N and floating-point formats fphi, fplo ∈ FP (one +for high precision and the other for low precision). +Below we will specify the conditions they should satisfy, +and show that some k, fphi, and fplo indeed satisfy the +conditions. We write rndhi(·) and rndlo(·) as shorthand +for rndfphi(·) and rndfplo(·). +• Training setups. We consider a very simple setting for +training: the gradient descent algorithm with a learning +rate η = 2−l (l ∈ N) is applied for just one epoch; all +parameters are initialized to 0 and their master copies +are represented in fphi; and the negative loss of a model +on training data (i.e., −L(fθ(x), y) using notations to +be described below) is used as the accuracy of the model. +Here l ∈ N can be any natural number. +• Model and loss networks. A model network M and +a loss network L are given as Figure 8, where M has n +parameter tensors θi ∈ Rwi of size wi (i ∈ [n]). For an +input-output pair (x, y) ∈ Rn × R, M and L compute a +predicted output fθ(x) ∈ R and a loss L(fθ(x), y) ∈ R +as follows (assuming that no rounding functions are +applied): +fθ(x) = +∆ � +i∈[n] +� +j∈[wi] +θi,jxi, +L(fθ(x), y) = +∆ 2−k|fθ(x) − y|. +Roughly speaking, M is (a variant of) a linear classifier +and L is a ℓ1-loss (scaled by 2−k). +• Training data. Training data consists of a single input- +output pair (x, y) ∈ Rn × R that satisfies the following: +xi = rndlo( +� +pi/wi), +y < −2−(k+l) � +i∈[n] +wix2 +i +for all i ∈ [n]. Here y can take any value as long as +it satisfies the above inequality. Note that xi can be +different from +� +pi/wi since the latter value may not +be representable in fplo. +• Precision-candidate +assignment. +A +precision- +candidate assignment C : TS×{hi, lo} → FP is given as: +C(t, hi) = +∆ fphi, +C(t, lo) = +∆ fplo +for all t ∈ TS. +That is, for all tensors, fphi is used as a high-precision +format and fphi as a low-precision format. Here fphi and +fplo should satisfy the following: +ehi ≥ elo, +mhi ≥ mlo, +(3) +|rndlo(s) − s| < |s| · err +for all s ∈ S1, +(4) +rndlo(s) = 0 +for all s ∈ S2, +(5) +rndhi(s) = s +for all s ∈ S2 ∪ S3, +(6) +where ehi and mhi (and elo and mlo) denote the number +of exponent bits and mantissa bits of fphi (and fplo), and +err = +∆ 1/(6n · maxi∈[n]pi), +S1 = +∆ { +� +pi/wi | i ∈ [n]}, +S2 = +∆ {2−k} ∪ {2−kxi | i ∈ [n]}, +S3 = +∆ {2−(k+l)xi | i ∈ [n]}. +Eq. (4) says that the relative error of representing each +s ∈ S1 in fplo should be less than err. Eq. (5) says that +each s ∈ S2 should underflow to 0 when represented +in fplo. Eq. (6) says that each s ∈ S2 ∪ S3 should be +representable in fphi. +• Low-precision ratio. A lower bound r ∈ [0, 1] on the +low-precision ratio is given as: +r = +∆ max +� +0, 1 − 2W + 1 +size(TS) +� +∈ [0, 1]. +So r decreases linearly as W increases. +We make three points on the above construction. +• First, each part of the knapsack problem (w, p, W) is +used in the following parts of the construction: wi is +used mainly in the size of the parameter tensor θi; pi +in the input xi; and W in the lower bound r. +• Second, there exist k ∈ N and fphi, fplo ∈ FP that sat- +isfy Eqs. (3)–(6). This can be shown as follows: first, +by taking sufficiently many exponent and mantissa bits +for fplo, we can make Eq. (4) satisfied; next, by taking a +sufficiently large k, we can make Eq. (5) satisfied; finally, +by taking sufficiently many exponent and mantissa bits +for fphi, we can make Eq. (3) and Eq. (6) satisfied (since +xi is representable in fplo and 2−(k+l) is a power of two). +• Third, some well-known models (e.g., ShuffleNet-v2) +have a similar structure to M in that they apply the fol- +lowing operations as a subroutine: split a tensor into mul- +tiple tensors, apply some operators to each split tensor, +and combine the resulting tensors into a single tensor. + +Training with Mixed-Precision Floating-Point Assignments +⫶ +𝑥! !∈ # +.∑!∈ # ∑$∈ %! 𝑣#&!,$. +2() 𝑣*#&+ − 𝑦 +𝑣,,+ +𝜃+,$ ⋅ 𝑣+ .$∈ %" +𝑣,,# +𝜃+,$ .$∈ %" +𝜃#,$ ⋅ 𝑣# .$∈ %# +𝜃#,$ .$∈ %# +𝑦 +𝑣,∈ℝ# +𝑣+∈ℝ +𝑣#∈ℝ +𝜃#∈ℝ%# +𝜃+∈ℝ%" +𝑣*#∈ℝ%# +𝑣#&+∈ℝ%" +𝑣*#&+∈ℝ +𝑣*#&*∈ℝ +𝑦∈ℝ +⫶ +⫶ +split +conv +conv +sum +loss +ℳ +ℒ +Figure 8: The model network M and the loss network L used in the proof of Theorem 3.2. +We now prove that the knapsack problem (w, p, W) can +be solved in polynomial time, if an oracle to the above +tradeoff problem is given. Suppose that π ∈ Π(C) is an +optimal solution to the above tradeoff problem (given by +the oracle). Define an item selection α ∈ {0, 1}n for the +knapsack problem as: +αi = +∆ +� +1 +if π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi +0 +otherwise +for each i ∈ [n]. Note that α can be constructed from π in +linear time. Thus, it suffices to show that α is an optimal +solution to the knapsack problem (w, p, W), which is +equivalent to the following two claims: +• Claim 1: We have � +i∈[n] αiwi ≤ W. +• Claim 2: For any α′ ∈ {0, 1}n with � +i∈[n] α′ +iwi ≤ W, +we have � +i∈[n] α′ +ipi ≤ � +i∈[n] αipi. +We now prove each claim as follows. +Proof of Claim 1. If α = (0, · · · , 0), then the claim clearly +holds. Suppose that α ̸= (0, · · · , 0). Then, we have +1 − +1 + 2 � +i∈[n] αiwi +size(TS) +≥ ratiolo(π) +≥ r ≥ 1 − 1 + 2W +size(TS). +Here the first inequality uses α ̸= (0, · · · , 0) and the defini- +tion of α and M; the second inequality uses the fact that π is +a valid solution to the above tradeoff problem; and the third +inequality uses the definition of r. Hence, the claim holds. +Proof of Claim 2. Suppose that the claim does not hold. +Then, there exists α′ ∈ {0, 1}n such that +� +i∈[n] +α′ +iwi ≤ W, +� +i∈[n] +α′ +ipi > +� +i∈[n] +αipi. +Define a precision assignment π′ ∈ Π(C) as: +π′(dv 2n+1) = +∆ fphi, +π′(dθi) = +∆ π′(dv n+i) = +∆ fphi +for all i ∈ [n] with α′ +i = 1, +π′(t) = +∆ fplo +for all other t ∈ TS. +Then, we have ratiolo(π′) ≥ r by � +i∈[n] α′ +iwi ≤ W and +the definition of π′, M, and r. Hence, it suffices to show +acc(π) < acc(π′), because this would contradict to the fact +that π is an optimal solution. +To show acc(π) < acc(π′), we prove the following two +lemmas: the first lemma gives a closed form of acc(π) and +acc(π′), and the second lemma shows that � +i∈[n] βiwix2 +i +is close to � +i∈[n] βipi (where the former summation +appears in acc(π) and acc(π′)). +Lemma A.1. The following hold: +acc(π) = 2−ky + 2−(2k+l) � +i∈[n] +αiwix2 +i , +acc(π′) = 2−ky + 2−(2k+l) � +i∈[n] +α′ +iwix2 +i . +Proof. We prove the equation for acc(π) only, since the +equation for acc(π′) can be proved similarly. +First, we show that for all i ∈ [n] and j ∈ [wi], +ˆ +dθi,j = αi · 2−kxi. +(7) +Pick any i ∈ [n] and j ∈ [wi]. Note that by the definition +of M, we have +ˆ +dθi,j = rndπ(dθi) +� +rndπ(dv n+i)(rndπ(dv 2n+1)(2−k)) +· rndvi(rndv0(xi)) +� += rndπ(dθi) +� +rndπ(dv n+i)(rndπ(dv 2n+1)(2−k)) · xi +� +, +where the second equality uses Eq. (3) and that xi is +representable in fplo. We prove Eq. (7) by case analysis +on αi. Suppose αi = 1. Then, by the definition of αi, + +Training with Mixed-Precision Floating-Point Assignments +π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi. From this, we +get the desired equation: +ˆ +dθi,j = rndhi +� +rndhi(rndhi(2−k)) · xi +� += rndhi(2−k · xi) = 2−kxi, +where the last two equalities use Eq. (6). Suppose now +αi = 0. Then, by the definition of αi, at least one of π(dθi), +π(dv n+i), and π(dv 2n+1) is fplo. If π(dv n+i) = fplo or +π(dv 2n+1) = fplo, we get the desired equation: +ˆ +dθi,j = rndπ(dθi) +� +rndlo(2−k) · xi +� += rndπ(dθi)(0 · xi) = 0, +where the first equation uses Eq. (3) and Eq. (6), and the +second equation uses Eq. (5). The remaining case is when +π(dv n+i) = π(dv 2n+1) = fphi and π(dθi) = fplo. We get +the desired equation in this case as well: +ˆ +dθi,j = rndlo +� +rndhi(rndhi(2−k)) · xi +� += rndlo(2−k · xi) = 0, +where the second equality uses Eq. (6), and the last equality +uses Eq. (5). Hence, we have proved Eq. (7). +Next, let θi be the i-th parameter tensor before training +starts, and θ′ +i be the corresponding tensor after training ends +(i ∈ [n]). Then, by the definition of the tradeoff problem +constructed above, we have θi,j = 0 and +θ′ +i,j = θi,j − rndhi(2−l · ˆ +dθi,j) += 0 − rndhi(2−l · (αi · 2−kxi)) += αi · (−2−(k+l)xi), +where the second equality uses Eq. (7) and the third equality +uses Eq. (6). Using this equation, we finally obtain the +conclusion of this lemma: +acc(π) = −L(fθ′(x), y) += −2−k���y − +� +i∈[n] +� +j∈[wi] +θ′ +i,jxi +��� += −2−k���y − +� +i∈[n] +� +j∈[wi] +αi · (−2−(k+l)xi) · xi +��� += −2−k���y + +� +i∈[n] +αi · 2−(k+l)wix2 +i +��� += 2−k� +y + +� +i∈[n] +αi · 2−(k+l)wix2 +i +� += 2−ky + 2−(2k+l) � +i∈[n] +αiwix2 +i , +where the first two equalities use the definition of accuracy, +and the second last equality uses the definition of y. This +concludes the proof of the lemma. +■ +Lemma A.2. For any β ∈ {0, 1}n, +��� +� +i∈[n] +βiwix2 +i − +� +i∈[n] +βipi +��� < 1 +2. +Proof. We first show that for any i ∈ [n], +|wix2 +i − pi| < 1 +2n. +Pick any i ∈ [n]. By Eq. (4) and the definition of xi, we have +���xi − +� pi +wi +��� < +� pi +wi +· +1 +6n · maxj∈[n] pj +≤ +� pi +wi +· +1 +6npi +. +From this, we have +� pi +wi +� +1 − +1 +6npi +� +< xi < +� pi +wi +� +1 + +1 +6npi +� +, +pi +wi +� +1 − +1 +6npi +�2 +< x2 +i < pi +wi +� +1 + +1 +6npi +�2 +. +From this, we obtain the desired result: +|wix2 +i − pi| < pi +�� +1 + +1 +6npi +�2 +− 1 +� += pi +� +1 +3npi ++ +1 +(6npi)2 +� +< pi +� +1 +3npi ++ +1 +6npi +� += pi · +1 +2npi += 1 +2n, +where the second inequality uses 6npi > 1 (as n, pi ∈ N). +Using this result, we can show the conclusion as follows: +��� +� +i∈[n] +βiwix2 +i − +� +i∈[n] +βipi +��� = +��� +� +i∈[n] +βi(wix2 +i − pi) +��� +≤ +� +i∈[n] +|βi| · |wix2 +i − pi| +< +� +i∈[n] +1 +2n = 1 +2, +where the last inequality uses |βi| ≤ 1. This completes the +proof of the lemma. +■ +Using the two lemmas, we prove acc(π) < acc(π′) as fol- +lows. First, by Lemma A.2 and � +i∈[n] αipi < � +i∈[n] α′ +ipi, +we have +� +i∈[n] +αiwix2 +i < +� +i∈[n] +αipi + 1 +2 +≤ +� +i∈[n] +α′ +ipi − 1 +2 < +� +i∈[n] +α′ +iwix2 +i , +where the second inequality comes from αi, α′ +i ∈ {0, 1} +and pi ∈ N. From this, and by Lemma A.1, we obtain +acc(π) < acc(π′) as desired. This concludes the proof of +Claim 2, thereby finishing the proof of the theorem. + +Training with Mixed-Precision Floating-Point Assignments +Remark A.3. In the proof of Theorem 3.2, we proved +the NP-hardness of Problem 3.1 by making use of only a +few limited aspects of the problem. For instance, we used +the fact that some values representable in a high-precision +format round to zero in a low-precision format; on the other +hand, many other values representable in a high-precision +format round to non-zero values in a low-precision format, +and this indeed occurs in practical training (even more +frequently than underflows). Also, we used a simple setting +for training in which a gradient descent algorithm is applied +for one epoch, training data consist of one input-output pair, +and test data is the same as training data; on the other hand, +in practical training, a gradient descent algorithm is applied +for many epochs, training data consists of many input-output +pairs, and test data is different from training data. +Problem 3.1 is general enough so that it embraces all the +aforementioned aspects of floating-points and training, +including those that are not considered in the proof of +Theorem 3.2. Since those aspects are likely to make the +problem even more difficult, we conjecture that the problem +would be more intractable than being NP-hard. +B +EXPERIMENTS: DEFERRED DETAILS +The datasets we use have the following licenses: +• CIFAR-10 and CIFAR-100: These datasets are under +the MIT license. +• ImageNet: This dataset can be used “only for non- +commercial research and educational purposes.” For +more details, see its webpage (Stanford Vision Lab, +2020). +The implementations of models we use have the following +licenses: +• SqueezeNet for CIFAR-10 and CIFAR-100: We adapt +an implementation of the model in a public GitHub +repository (Pathak, 2020), whose license information +is not available. +• ShuffleNet-v2, +MobileNet-v2, +and ResNet-18 for +CIFAR-10 and CIFAR-100: We adapt an implementation +of these models in a public GitHub repository (kuangliu, +2021), which is under the MIT license. +• ShuffleNet-v2 for ImageNet and ImageNet-200-i: We +adapt an implementation of the model in the torchvision +library (PyTorch, 2022b), which is under the BSD +3-Clause license. +The details of how we train models are as follows: +• Four models on CIFAR-10 and CIFAR-100: We train +the four models with a standard setup (kuangliu, 2021). +In particular, we run the (non-Nesterov) SGD optimizer +for 200 epochs with minibatch size of 128 (over 1 +GPU), learning rate of 0.1, momentum of 0.9, weight +decay of 5 × 10−4, and the cosine annealing scheduler +for learning rate. For dynamic loss scaling, we use +initial scale of 216, growth factor of 2, back-off factor +of 0.5, and growth interval of 1 epoch, as suggested in +PyTorch (PyTorch, 2022a). +• ShuffleNet-v2 on ImageNet: We train the model with the +default setup given in PyTorch’s GitHub repository (Py- +Torch, 2022c), except that we use larger minibatch size +and learning rate as in (Kalamkar et al., 2019; PyTorch, +2022d; Krizhevsky, 2014; Goyal et al., 2017) to reduce +the wall-clock time of training. In particular, we run +the (non-Nesterov) SGD optimizer for 90 epochs with +minibatch size of 1024 (over 16 GPUs), learning rate +of 0.4, momentum of 0.9, weight decay of 10−4, and +the cosine annealing scheduler for learning rate. For +dynamic loss scale, we use initial scale of 216, growth +factor of 2, back-off factor of 0.5, and growth interval +of 0.5 epoch, as suggested in PyTorch (PyTorch, 2022a). +• ShuffleNet-v2 on ImageNet-200-i: We train the model +with the same settings for ImageNet except that we use +the default values for minibatch size and learning rate +given in (PyTorch, 2022c), i.e., minibatch size of 256 +(over 4 GPUs) and learning rate of 0.1. +C +EXPERIMENTS: DEFERRED RESULTS +C.1 +Comparison with Existing Precision Assignments +Figure 9 presents results omitted in Figure 4: training +results of smaller variant models (which have width +multiplier 0.5 or 0.1) on CIFAR-100 with πfp32, πunif, +πop, πop′, and πours,r. The figure shows similar results +to Figure 4: the results for the variant models with width +multiplier 0.5 (and 0.1) are similar to those for the original +models (and the variant models with width multiplier 0.25). +Figures 10 and 11 show the average training trajectories +for the configurations presented in Figures 4 and 9. +C.2 Ablation Study: Precision Demotion and Promotion +Figure 12 presents results omitted in Figure 5: training re- +sults of ResNet-18 on CIFAR-100 with πours,r, πours[inc],r, +and πours[rand],r. +The figure shows similar results to +Figure 5 except that it shows smaller differences in memory- +accuracy tradeoff between the three precision assignments. +Figure 13 presents results omitted in Figure 6: training +results of four models on CIFAR-10 with πours,r and +πours[no-promo],r. +The figure shows similar results to +Figure 6 except that the training of ResNet-18 on CIFAR-10 +does not diverge even with πours[no-promo],r for all r values. + +Training with Mixed-Precision Floating-Point Assignments +(a) CIFAR-100, SqueezeNet‡ +(b) CIFAR-100, SqueezeNet¶ +(c) CIFAR-100, ShuffleNet-v2‡ +(d) CIFAR-100, ShuffleNet-v2¶ +(e) CIFAR-100, MobileNet-v2‡ +(f) CIFAR-100, MobileNet-v2¶ +(g) CIFAR-100, ResNet-18‡ +(h) CIFAR-100, ResNet-18¶ +Figure 9: Continued from Figure 4. Memory-accuracy tradeoffs of πunif (Micikevicius et al., 2018), πop (Sun et al., 2019), +πop′ (Kalamkar et al., 2019), and πours,r for smaller variants of four models on CIFAR-100. The variant models have width +multiplier 0.5 (marked by ‡) or 0.1 (marked by ¶). Top-right points are better than bottom-left ones. In all but one plots, +there are •s above and to the right of +and +, respectively; even in the one plot (g), •s have almost the same tradeoffs to +and +. In three of all plots, ⋆ has much smaller y-values than other points; ⋆ is missing in (h) as its y-value is too small. + +66 +test accuracy (%) +64 +62 +fp32 +60 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio44 +accuracy (%) +40 +36 +fp32 +32 +op +test a +op +unif +24 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio70 +test accuracy (%) +68 +66 +fp32 +64 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio56 +52 +X +48 +44 +fp32 +op +op +unif +24 +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio72 +test accuracy (%) +70 +68 +fp32 +66 +op +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio56 +test accuracy (%) +52 +48 +fp32 +44 +op +op +unif +36 +ours +32 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio76 +test accuracy (%) +74 +72 +fp32 +op +70 +op +unif +2 +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio60 +test accuracy (%) +56 +52 +fp32 +op +48 +op +unif +40 ++ +ours +36 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratioTraining with Mixed-Precision Floating-Point Assignments +(a) CIFAR-10, SqueezeNet +(b) CIFAR-100, SqueezeNet +(c) CIFAR-100, SqueezeNet† +(d) CIFAR-10, ShuffleNet-v2 +(e) CIFAR-100, ShuffleNet-v2 +(f) CIFAR-100, ShuffleNet-v2† +(g) CIFAR-10, MobileNet-v2 +(h) CIFAR-100, MobileNet-v2 +(i) CIFAR-100, MobileNet-v2† +(j) CIFAR-10, ResNet-18 +(k) CIFAR-100, ResNet-18 +(l) CIFAR-100, ResNet-18† +Figure 10: Training trajectories for the configurations shown in Figure 4. Each line shows the average training trajectory +for each precision assignment. πours,r is colored from navy to yellow (darker for smaller r). + +100 +(%) +90 +test accuracy ( +80 +fp32 +op +70 +op' +unif +ours +60 +0 +50 +100 +150 +200 +epoch70 +accuracy (%) +60 +fp32 +50 +op +test a +op' +unif +40 +ours +0 +50 +100 +150 +200 +epoch60 +test accuracy (%) +50 +40 +fp32 +op +op' +30 +unif +ours +20 +0 +50 +100 +150 +200 +epoch100 +(%) +90 +test accuracy ( +80 +fp32 +op +op' +70 +unif +ours +60 +0 +50 +100 +150 +200 +epoch80 +test accuracy (%) +70 +60 +fp32 +op +50 +op' +unif +ours +40 +0 +50 +100 +150 +200 +epoch70 +test accuracy (%) +60 +50 +fp32 +op +40 +op' +unif +ours +30 +0 +50 +100 +150 +200 +epoch100 +(%) +90 +test accuracy ( +80 +fp32 +op +op' +70 +unif +ours +60 +0 +50 +100 +150 +200 +epoch80 +test accuracy (%) +70 +60 +fp32 +op +50 +op' +unif +ours +40 +0 +50 +100 +150 +200 +epoch70 +test accuracy (%) +60 +50 +fp32 +op +40 +op' +unif +ours +30 +0 +50 +100 +150 +200 +epoch100 +(%) +90 +test accuracy ( +80 +fp32 +op +70 +op' +unif +ours +60 +0 +50 +100 +150 +200 +epoch80 +accuracy (%) +70 +fp32 +60 +op +test a +,do +unif +50 +ours +0 +50 +100 +150 +200 +epoch70 +accuracy (%) +60 +fp32 +50 +op +test a +op' +unif +40 +ours +0 +50 +100 +150 +200 +epochTraining with Mixed-Precision Floating-Point Assignments +(a) CIFAR-100, SqueezeNet‡ +(b) CIFAR-100, SqueezeNet¶ +(c) CIFAR-100, ShuffleNet-v2‡ +(d) CIFAR-100, ShuffleNet-v2¶ +(e) CIFAR-100, MobileNet-v2‡ +(f) CIFAR-100, MobileNet-v2¶ +(g) CIFAR-100, ResNet-18‡ +(h) CIFAR-100, ResNet-18¶ +Figure 11: Training trajectories for the configurations shown in Figure 9. Each line shows the average training trajectory +for each precision assignment. πours,r is colored from navy to yellow (darker for smaller r). + +70 +test accuracy (%) +60 +50 +fp32 +op +40 +op' +unif +ours +30 +0 +50 +100 +150 +200 +epoch50 +test accuracy (%) +40 +30 +fp32 +op +20 +op' +unif +ours +10 +0 +50 +100 +150 +200 +epoch70 +accuracy (%) +60 +fp32 +50 +op +test +op' +unif +40 +ours +0 +50 +100 +150 +200 +epoch60 +50 +40 +fp32 +op +30 +op +unif +ours +20 +0 +50 +100 +150 +200 +epoch70 +accuracy (%) +60 +fp32 +50 +op +test a +op' +unif +40 +ours +0 +50 +100 +150 +200 +epoch60 +50 +40 +fp32 +op +30 +op' +unif +ours +20 +0 +50 +100 +150 +200 +epoch80 +70 +60 +fp32 +op +50 +,do +unif +ours +40 +0 +50 +100 +150 +200 +epoch70 +test accuracy (%) +60 +50 +fp32 +op +40 +op' +unif +ours +30 +0 +50 +100 +150 +200 +epochTraining with Mixed-Precision Floating-Point Assignments +(a) ResNet-18 +Figure 12: Continued from Figure 5. Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for ResNet-18 +on CIFAR-100. Observe that •s are above and to the right of other points in nearly all cases. +(a) CIFAR-10, SqueezeNet +(b) CIFAR-10, ShuffleNet-v2 +(c) CIFAR-10, MobileNet-v2 +(d) CIFAR-10, ResNet-18 +Figure 13: Continued from Figure 6. Training four models on CIFAR-10 with πours,r and πours[no-promo],r. Column 1: +Training trajectories of πours[no-promo],r for different r; colors denote r values (darker for smaller r). Column 2: Top-5 +overflow ratios of tensors at each epoch, for the highlighted trajectory in (a); the largest ratio is blue and the fifth largest +red. Column 3: Memory-accuracy tradeoffs of πours,r and πours[no-promo],r. Column 4: Low-precision ratio when training +ends vs. when training starts, for πours,r and πours[no-promo],r. + +82 +(%) +80 +test accuracy ( +78 +76 +ours[inc] +ours[rand] +ours +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio100 +test accuracy (%) +80 +60 +40 +20 +0 +0 +50 +100 +150 +200 +epoch1.0 +0.8 +overflow ratio +0.6 +0.4 +top-1 +top-2 +top-3 +0.2 +top-4 +top-5 +0.0 +0 +50 +100 +150 +200 +epoch100 +(%) +80 +accuracy ( +60 +40 +test +20 +ours[no-promo] +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio1.0 +ratio (end) +0.8 +0.6 +low-prec. +0.2 +ours[no-promo] +ours +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +low-prec. ratio (start)100 +test accuracy (%) +80 +60 +40 +20 +0 +0 +50 +100 +150 +200 +epoch1.0 +0.8 +overflow ratio +0.6 +0.4 +top-1 +top-2 +top-3 +0.2 +top-4 +top-5 +0.0 +0 +50 +100 +150 +200 +epoch100 +(%) +80 +accuracy ( +60 +40 +test +20 +ours[no-promo] +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio1.0 +ratio (end) +0.8 +0.6 +low-prec. +0.2 +ours[no-promo] +ours +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +low-prec. ratio (start)100 +test accuracy (%) +80 +60 +40 +20 +0 +0 +50 +100 +150 +200 +epoch1.0 +0.8 +overflow ratio +0.6 +0.4 +top-1 +top-2 +top-3 +0.2 +top-4 +top-5 +0.0 +0 +50 +100 +150 +200 +epoch100 +(%) +80 +accuracy ( +60 +40 +test +20 +ours[no-promo] +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio1.0 +ratio (end) +0.8 +0.6 +low-prec. +0.2 +ours[no-promo] +ours +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +low-prec. ratio (start)100 +test accuracy (%) +80 +60 +40 +20 +0 +0 +50 +100 +150 +200 +epoch0.10 +0.08 +overflow ratio +0.06 +0.04 +top-1 +top-2 +top-3 +0.02 +top-4 +top-5 +0.00 +0 +50 +100 +150 +200 +epoch100 +(%) +80 +accuracy ( +60 +40 +test +20 +ours[no-promo] +ours +0 +0.00 +0.25 +0.50 +0.75 +1.00 +low-prec. ratio1.0 +low-prec. ratio (end) +0.8 +0.6 +0.2 +ours[no-promo] +ours +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +low-prec. ratio (start)Training with Mixed-Precision Floating-Point Assignments +REFERENCES (FOR APPENDIX) +Goyal, P., Dollár, P., Girshick, R. B., Noordhuis, P., +Wesolowski, L., Kyrola, A., Tulloch, A., Jia, Y., and He, +K. Accurate, Large Minibatch SGD: Training ImageNet +in 1 Hour. arXiv:1706.02677, 2017. +Kalamkar, D. D., Mudigere, D., Mellempudi, N., Das, +D., Banerjee, K., Avancha, S., Vooturi, D. T., Jammala- +madaka, N., Huang, J., Yuen, H., Yang, J., Park, J., +Heinecke, A., Georganas, E., Srinivasan, S., Kundu, +A., Smelyanskiy, M., Kaul, B., and Dubey, P. +A +Study of BFLOAT16 for Deep Learning Training. +arXiv:1905.12322, 2019. +Karp, R. M. Reducibility Among Combinatorial Problems. +In Complexity of Computer Computations, pp. 85–103, +1972. +Krizhevsky, +A. +One weird trick for parallelizing +convolutional neural networks. arXiv:1404.5997, 2014. +kuangliu. +https://github.com/kuangliu/ +pytorch-cifar, 2021. +Pathak, +G. +https://github.com/gsp-27/ +pytorch_Squeezenet, 2020. +PyTorch. +Documentation +of +torch.amp. +https://pytorch.org/docs/stable/amp. +html#gradient-scaling, 2022a. +PyTorch. +https://github.com/pytorch/ +vision/tree/main/torchvision/models, +2022b. +PyTorch. +https://github.com/pytorch/ +vision/tree/main/references/ +classification, 2022c. +PyTorch. +https://github.com/pytorch/ +vision/tree/main/references/ +classification#resnext, 2022d. +Stanford Vision Lab. +https://image-net.org/ +download.php, 2020. + diff --git a/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/load_file.txt b/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ecbd67d95bac7eab038bdc270e091526c31da8f3 --- /dev/null +++ b/EdFRT4oBgHgl3EQfBDfd/content/tmp_files/load_file.txt @@ -0,0 +1,2049 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf,len=2048 +page_content='TRAINING WITH MIXED-PRECISION FLOATING-POINT ASSIGNMENTS Wonyeol Lee 1 Rahul Sharma 2 Alex Aiken 1 ABSTRACT When training deep neural networks, keeping all tensors in high precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 32-bit or even 16-bit floats) is often wasteful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' However, keeping all tensors in low precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 8-bit floats) can lead to unacceptable accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Hence, it is important to use a precision assignment—a mapping from all tensors (arising in training) to precision levels (high or low)—that keeps most of the tensors in low precision and leads to sufficiently accurate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' We provide a technique that explores this memory-accuracy tradeoff by generating precision assignments that (i) use less memory and (ii) lead to more accurate models at the same time, compared to the precision assignments considered by prior work in low-precision floating-point training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Our method typically provides > 2× memory reduction over a baseline precision assignment while preserving training accuracy, and gives further reductions by trading off accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Compared to other baselines which sometimes cause training to diverge, our method provides similar or better memory reduction while avoiding divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' 1 INTRODUCTION In deep neural network training, floating-point formats are usually used to represent tensors and it is worthwhile to use the smallest bitwidth format that gives acceptable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' For example, it is common to replace tensors using 32-bit floats with tensors that use 16-bit floats (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' The benefits are easy to under- stand: computations using lower-precision floats not only use less memory but are also faster (due to improved vec- tor parallelism, locality, and reduced data movement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' The downside is that there is generally some loss of training accu- racy, and in the worst case training may not even converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' For such low-precision floating-point training, the most common approaches use two floating-point formats—one for lower-precision floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 8-bit floats) and the other for higher-precision floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 16-bit floats)—and assign one of the two formats to each tensor (including weights, activations, and their gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' The precision assignments studied in previous work fall into one of two assignment schemes (which both have several variants): the uniform assignment uses low precision for almost all tensors (often excepting those in the first and/or last few layers) (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 2018), while the operator-based assignment limits low precision to the input tensors of certain operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', convolutions) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Prior work has shown that both precision assignment schemes (with well-chosen low-bitwidth floating-point formats) can match the accuracy of 32-bit-float training 1Stanford University, USA 2Microsoft Research, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'} +page_content=' Correspondence to: Wonyeol Lee 0 is computed by minimizing the +negative log-likelihood (NLL) on a validation dataset. +2.1.6 +Ensemble Variants: +Additionally, ensemble variants of the methods SVI, MCDO and TTA are implemented and tested. +2.2 +Uncertainty Metrics +Given the predictions generated by the previously introduced methods, the literature developed multiple metrics to +estimate the predictive uncertainty, which we compare in this work. +3 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +2.2.1 +Confidence: +A commonly used method, especially in the field of calibration [Guo et al., 2017], is to take the maximum of the +softmax, which is also called the “confidence” of the prediction. The core idea is that decisions that are considered +certain are far away from the decision boundary, while uncertain decisions lie close to the boundary at 1/N, where N is +the number of classes. +2.2.2 +Entropy: +Another often used metric [Mobiny et al., 2019, Band et al., 2021] is to take the entropy of the prediction as a measure +of uncertainty. The entropy of the model’s output probability is computed as +H(y|x, D) = − +� +c ∈ C +p(y = c|x, D) log p(y = c|x, D) +(3) +Since the maximum possible entropy varies with the number of classes, we compute the normed entropy as +Hnorm = +H +Hmax +with Hmax = − +N +� +i=1 +1 +N log +� 1 +N +� += log N +(4) +When using the normed entropy, the uncertainty is high when Hnorm → 1 and uncertainty is low when Hnorm → 0. +Confidence (conf) and entropy behave very similarly in a binary classification setting. Given two predictions yi and yj +the following relation holds between them: +H(yi) < H(yj) ⇐⇒ conf(yi) > conf(yj) +(5) +2.2.3 +Variance: +Since all included uncertainty estimation methods generate a distribution of predictions, the variance of the distribution +can be used as a measure of uncertainty [Lakshminarayanan et al., 2017, Abdar et al., 2021]. If all predictors agree on a +result, the variance is zero, whereas a high variance indicates high uncertainty. +2.3 +Evaluation Settings +This section covers the different evaluation settings and the performance metrics used in each setting. +2.3.1 +Classification: +For the tumor classification on tile-level, we compute accuracy and balanced accuracy. Given the tumor class as the +positive class, the balanced accuracy is defined as the arithmetic mean between the true positive rate TPR = TP/P +and the true negative rate TNR = TN/N: +Balanced Accuracy = 1 +2 +�TP +P + TN +N +� +(6) +,where P is the total number of positive samples and N is the total number of negative samples. +2.3.2 +Calibration: +For measuring calibration, we utilize the Expected Calibration Error (ECE) [Guo et al., 2017, Nixon et al., 2019]. Given +a prediction for every data point in the dataset, the output probability or "confidence" for each sample should on average +match the correctness of the prediction. In other words, we expect a prediction that has a confidence value of 60% to be +correct in about 60% of the cases. To validate this intuition, the predictions are split into a predetermined number M of +bins B of equal confidence range. Then the absolute difference between the average confidence and accuracy within +each bin is summed up: +ECE = +M +� +m=1 +|Bm| +n +���conf(Bm) − acc(Bm) +��� +(7) +Here, |Bm| denotes the number of samples in the m-th bin and n is the total number of samples. +4 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +2.3.3 +Reject Option: +In a clinical setting, the model should be able to refer predictions with high uncertainty to human practitioners for +evaluation. Models suitable for this task should assign higher uncertainties to their wrong predictions than to their +correct predictions, thereby allowing to cut off a large number of false predictions, by thresholding at a certainty level. +To compare multiple models on this ability, we compute the accuracy-reject curve [Nadeem et al., 2009], plotting the +achieved accuracy against the percentage of rejected data points in the dataset. +2.3.4 +Label Noise: +Medical annotations are often subject to an unquantified amount of label noise [Joskowicz et al., 2019, Jensen et al., +2019, Karimi et al., 2020], which may deteriorate the performance of supervised machine learning approaches. To our +knowledge, the previously described methods have not been compared in their robustness to label noise in the medical +domain. We evaluate the effect by creating multiple datasets with increasing levels and different types of label noise +and evaluate the methods under these changing conditions. +3 +Experiment setup +3.1 +Dataset and data processing +We conduct our experiments on the lesion-level annotated slides of the Cameylon17 dataset [Bándi et al., 2019]. This +part of the Camelyon17 dataset consists of 50 WSIs of breast lymph node tissue, with the goal of detecting metastatic +tissue. The slides were obtained from five different clinics in the Netherlands, using three different scanners, which +serves as an ideal setting to estimate the influence of domain shift between clinics on the model performance. +To create a distribution shift between the in-distribution (ID) and out-of-distribution (OOD) domain, we split the dataset +into three groups, using the centers 0, 1 and 3 with the 3DHistech scanner as ID data, where the training occurs, while +using the 10 slides of centers 2 (Hamamatsu scanner) and 4 (Philips scanner) as OOD data each. By splitting the data +between the centers, we induce differences in location and acquisition process between the datasets. We then further +split the ID data into the training, validation and test set. We sort the slides of each center by the area of annotated +tumor cells they contain and use the two median slides as test set for each center. The training and validation sets are +generated by a randomized 75%/25% split of the tiles. +The tiles themselves are generated following [Khened et al., 2021] with median filtering and Otsu’s thresholding of the +HSV saturation component of the WSI image, followed by finally applying opening and closing dilation. After that, +tiles of the size 256 × 256 are extracted. The tumor regions on the slide are indicated by polygonal annotations. From +the annotations, a tumor coverage per tile is computed. Tiles with more than 25% tumor coverage are counted as tumor +tiles and all tiles with 0% tumor coverage are counted as non-tumor tiles. Tumor tiles that are covered by the tumor +annotation by less than 25% are discarded for our standard training, to minimize the risk of label noise that can arise +due to high inter-observer variability [Jensen et al., 2019, Joskowicz et al., 2019, Karimi et al., 2020]. +3.2 +Training setup and hyperparameter tuning +We use a ResNet-34 [He et al., 2016] with a batch size of 128 and a learning rate of 0.001 for all our experiments. As +optimizer we utilize Adam [Kingma and Ba, 2017], with a reduction of the learning rate by a factor of 10 if the validation +loss, which is chosen as the cross-entropy loss, does not decrease for 3 epochs. For data augmentations, we follow +[Tellez et al., 2019] applying random crops to size 224 × 224, random 90° rotations and color jitter (brightness:±20%, +contrast:±30%, hue:±10%, saturation:±10%). The inputs are normalized with the mean and variance of the training +data. The best model is chosen by balanced accuracy on the validation set. During training, we balance the training set +by samples per class, but we do not balance the validation set. +For the Deep Ensemble architecture, we choose n = 5 members following recent literature [Linmans et al., 2020, +Thagaard et al., 2020]. For MCDO, we place a dropout layer after each ResNet block, with dropout probability p = 0.3. +This is in contrast to [Linmans et al., 2020, Thagaard et al., 2020], who only place a dropout layer before the last layer, +observing no improvement in performance. For inference during testing, we use 10 SVI-, MCDO- and TTA- samples. +Following [Wenzel et al., 2020], we use an additional hyperparameter for weighting the influence of the SVI prior +(Kullback-Leibler-Divergence to the normal distribution) on the training. We tune this hyperparameter as well as the +dropout probability, the dropout layer placement and the learning rate with the python library Optuna [Akiba et al., +2019]. All our experiments are conducted with PyTorch 1.11 on Nvidia GPUs with CUDA 11.3. +5 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +4 +Results +We trained each method 5 times with different seeds, reporting their average or median performance. +4.1 +Tile Classifier Performance +ID centers +Center 2 +Center 4 +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +Value +Accuracy +ID centers +Center 2 +Center 4 +0.725 +0.750 +0.775 +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +Balanced Accuracy +ID centers +Center 2 +Center 4 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +ECE +Method +ResNet +ResNet Ensemble +MCDO +MCDO Ensemble +TTA +TTA Ensemble +SVI +SVI Ensemble +Figure 1: Accuracy, balanced accuracy and estimated calibration error (ECE) of the proposed methods on the in- +distribution (ID) centers and out-of-distribution centers 2 and 4. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rejection rate +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Balanced Accuracy +ID centers +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rejection rate +OOD (center 2) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rejection rate +OOD (center 4) +Method +ResNet +ResNet Ensemble +MCDO +MCDO Ensemble +TTA +TTA Ensemble +SVI +SVI Ensemble +Figure 2: (Balanced) Accuracy-Reject Curves on ID and OOD centers. The x-axis contains the proportion of rejected +inputs over the dataset, which is plotted against the reached Balanced Accuracy on the remaining images. The order of +the excluded points was determined by the predictive confidence (see 2.2.1). We plot the mean curves over all 5 trials, +with the shaded area denoting the 95% confidence interval. +Figure 1 shows the accuracy, balanced accuracy and ECE on the ID data, as well as on the OOD centers 2 and 4. +Detailed numeric results for all our experiments can be found in the appendix in Table 2. +We start by analyzing the accuracy values on the three test data splits and between all evaluated uncertainty methods. +On the ID test data, we can see that we achieve very high accuracies of about 98% for the classification of tumor and +non-tumor tiles. Between the evaluated uncertainty methods, we only observe small differences in accuracy on the ID +data. As expected, the ensemble variants outperform their non-ensemble counterparts and TTA performs better than the +baseline. However, SVI achieves a slightly lower accuracy (97.4%) than the baseline ResNet, while MCDO can only +slightly increase upon the baseline. +On OOD centers 2 and 4, we observe a decrease in accuracy compared to the ID test data, with the decrease on center 4 +being smaller than the decrease observed on center 2. This behavior indicates a larger domain shift on center 2, than on +center 4. We can observe an increase in performance with almost all uncertainty methods compared to the baseline +ResNet, except for SVI and MCDO on center 2, where the results show a large spread in observed values. Especially +the ensemble variants (ResNet Ensemble, MCDO Ensemble, TTA Ensemble and SVI Ensemble) consistently achieve +higher accuracies than the baseline ResNet classifier. +6 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +Table 1: Mean and standard deviation of the area under the curve of the accuracy-reject curves (AUARC) presented in +Figure 2. We compare the confidence of a single prediction, the confidence of the mean prediction for ensembling-like +approaches and the variance over multiple predictions. +Unc. Measure +Unc. Method +ID Centers +OOD (Center 2) +OOD (Center 4) +Confidence +ResNet +0.906±0.017 +0.893±0.031 +0.954±0.010 +Confidence +ResNet Ensemble +0.918±0.007 +0.932±0.010 +0.972±0.004 +MCDO +0.886±0.030 +0.914±0.012 +0.971±0.010 +MCDO Ensemble +0.905±0.008 +0.924±0.012 +0.971±0.007 +TTA +0.932±0.005 +0.897±0.025 +0.970±0.004 +TTA Ensemble +0.933±0.002 +0.884±0.044 +0.963±0.009 +SVI +0.895±0.027 +0.931±0.025 +0.972±0.012 +SVI Ensemble +0.896±0.006 +0.949±0.003 +0.981±0.002 +Variance +ResNet Ensemble +0.914±0.007 +0.928±0.011 +0.972±0.004 +MCDO +0.877±0.033 +0.911±0.011 +0.968±0.012 +MCDO Ensemble +0.898±0.010 +0.915±0.011 +0.969±0.008 +TTA +0.926±0.005 +0.877±0.031 +0.965±0.005 +TTA Ensemble +0.926±0.002 +0.861±0.053 +0.959±0.010 +SVI +0.890±0.026 +0.929±0.023 +0.970±0.013 +SVI Ensemble +0.888±0.007 +0.942±0.004 +0.981±0.001 +Since we have to deal with a significant class imbalance between tumor and non-tumor tiles, we further investigate +balanced accuracy in the center plot of Figure 1. Compared to the previous evaluation of accuracy, we observe lower +values concerning balanced accurac/y for all centers. On the ID centers, the baseline ResNet achieves a balanced +accuracy of 82.9%, while TTA and TTA Ensemble significantly improve performance (86.9% each), with MCDO +even reaching a slightly lower balanced accuracy than the baseline. On the out-of-distribution centers all methods +perform better than the baseline ResNet. Of all the methods TTA, TTA Ensemble and SVI Ensemble lead to the largest +increases, while MCDO and MCDO Ensemble barely outperformed the pure ResNet Ensemble. Noticeable here is the +high performance of SVI Ensemble on the OOD data, which could already be observed in terms of accuracy. SVI alone +on the other hand again leads to a very high spread in observed values. +When computing the balanced accuracy, a higher weight is attributed to the under-represented tumor class. The lower +values in balanced accuracy compared to standard accuracy indicate that our model performs worse in classifying tumor +tiles than non-tumor tiles. Surprisingly, we see higher values in balanced accuracy on OOD center 4 compared to the +other two test splits. To explain this behavior, we added an overview of the dataset distribution in the appendix (see +B). The center 4 dataset has a larger ratio of tumor tiles to non-tumor tiles because it contains one slide with two large +tumor metastases, much larger than the tumor metastases on all other WSIs. Therefore, the balanced accuracy metric is +heavily influenced by the large increase in tumor tiles that lie inside the area of annotated tumor regions. As we will see +in Section 4.3, these tumor tiles lead to higher confidences and better classification results, than tumor tiles that lie at +the border of the annotation. +In the plot on the right-hand side of Figure 1, we evaluate model calibration in terms of ECE (see Section 2.3.2). On +the ID test data, we observe low ECE values across all evaluated methods, which means that the output confidence +values represent a good estimate of the correctness of the predictions. ResNet Ensemble (0.0145), as well as the MCDO +approaches (0.0135 and 0.0137), produce better-calibrated results than the baseline, while all TTA and SVI methods +slightly worsen the calibration. As shown in [Ovadia et al., 2019], we expect higher ECE values for the domain shift +towards OOD centers 2 and 4. On center 2 this is also reflected in our results, as we see higher ECE values and higher +interquartile ranges for all evaluated methods compared to the ID test data. Here the relative order remains mostly +the same, with TTA and TTA Ensemble now performing better than the ResNet, while the ResNet Ensemble and the +MCDO methods still perform best. Center 4 on the other hand shows lower ECE values, comparable to the values on +the ID test data. This indicates that with the applied augmentations, we were able to mostly compensate for the domain +shift between ID and OOD data, which was already visible in our accuracy results. Here the MCDO methods and the +ResNet Ensemble perform better than the baseline method, while the situation of SVI and TTA methods is reversed to +that of center 2. +In C, we analyze the influence of data augmentations on network calibration by comparing the results to training with +basic augmentations consisting of random crops and flips. There we see higher ECE values for both OOD centers 2 and +4 compared to the ID test data. In the evaluation in the appendix we also include Temperature Scaling as a reference +method for improving network calibration. +7 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +To summarize the results, we find that data augmentations applied during training have a larger impact on the achieved +accuracies and the calibration compared to utilizing one of our evaluated uncertainty methods and as such a large focus +should be placed on them. Amongst the methods evaluated in Figure 1, ensembles lead to slightly better-calibrated +results than the baseline, while TTA and SVI behave inconsistently and can have a negative impact on the resulting +ECE values. In terms of achieved accuracies, the ensemble approaches consistently perform best. The SVI method +leads to inconsistent results across the evaluated metrics, therefore no direct conclusion is possible. TTA does not +require architectural changes and no extra computational resources during training and leads to a large increase in +observed accuracies. If however larger training capacity is available, a TTA Ensemble or even an SVI Ensemble can be +recommended. The basic ResNet Ensemble, while not the best in most evaluations, performed the most consistent, +always outperforming the baseline method. +4.2 +Reject Option +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +Tiles with lowest uncertainty +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +1 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +1 +1 +0 +1 +0 +0 +1 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +Tiles with highest uncertainty +Figure 3: Least and most uncertain tiles from the in-distribution test set computed by the confidence of a ResNet +Ensemble. The numbers represent the ground-truth labels: ’1’ indicates tumor and ’0’ indicates a non-tumor tile. +We compare the uncertainty methods in terms of their ability to detect mispredictions. For that we compute the +(Balanced) Accuracy-Reject Curves (see Section 2.3.3). In Figure 2, we compare the methods and their performance +with an increasing ratio of rejected tiles, for the test slides on the ID centers and OOD centers 2 and 4, respectively. To +determine the most uncertain tiles that are rejected, we choose the confidence measure (see Section 2.2.1). For each +split and method, we see a mostly linear increase of balanced accuracy when rejecting a fraction of the most uncertain +tiles. On the ID test data, TTA and the TTA Ensemble consistently achieve the highest balanced accuracies. For the +OOD centers, the SVI Ensemble leads to the highest accuracy-reject-curves, while the baseline ResNet performs worst. +From our evaluated uncertainty methods, the MCDO methods averaged over all three datasets led to the worst results. +While most methods show increasing performance on center 2 when rejecting the most uncertain tiles, TTA and TTA +Ensemble show an unexpected stagnation of performance when rejecting more than 20% of uncertain tiles. +In Table 1 we show the area under the curve for the accuracy-reject curves (AUARC) across all datasets and compare +different uncertainty measures. On the ID data, ResNet Ensemble, TTA and TTA Ensemble lead to an increase of the +AUARC over the baseline value, while the MCDO and SVI methods lead to a decrease of said metric. On center 2 and +center 4, all methods (other than TTA Ensemble on center 2) lead to higher AUARC values than the baseline. From our +experiments, not one single method can be identified, that performs best between all data splits. However, the ResNet +Ensemble consistently beats the basic ResNet, while TTA and TTA Ensemble perform at least as well as the baseline or +better. +When comparing the scores computed from using the confidence of the predictions or the variance across individual +predictions, the relative order of methods remains the same across all datasets. Furthermore, every method performs +worse in terms of AUARC, when the uncertainty is calculated by the variance across individual predictions instead of +the confidence. Although the disagreement of ensembling methods measured in terms of the variance is often claimed +to be a good indicator of uncertainty, in our experiments it consistently performs worse when compared to using the +confidence for detecting mispredictions. As using the confidence for misprediction detection by rejection is equivalent +to using the entropy in the binary case (see Section 2.2.2), our experiments strongly suggest utilizing confidence or +entropy over the variance of predictions. +8 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +In Figure 3 we present a collection of the most certain and most uncertain tiles within the ID test data, computed with +a ResNet Ensemble. We observe that the neural network appears to be most confident on tumor tiles (label 1), that +cover the whole tile and possess a similar cell structure. For the most uncertain tiles on the right side, no comparable +structure among the tiles is observable. These tiles both contain tumor and non-tumor tissue and often lie at border +regions between tissue types, for example, the border to fat-tissue, as can be identified by the white areas in the H&E +stained slides. Many uncertain tiles seem to lie at the border of annotated tumor regions, that we suspect to have a larger +degree of label noise, and in regions that have a different appearance than the majority of healthy tissue. +4.3 +Label Noise +Image +Annotation +Non-Tumor +Tumor +Tumor Confidence +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 4: WSI (patient_017_node_2) with ground-truth annotations and model predictions. The tumor confidence +decreases in areas near the border of the annotation, while the uncertainty thereby increases. This result is consistent +over the whole dataset. +25% +0% +Uniform +Border +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +Value +Accuracy +25% +0% +Uniform +Border +0.76 +0.78 +0.80 +0.82 +0.84 +0.86 +0.88 +Balanced Accuracy +25% +0% +Uniform +Border +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +ECE +Method +ResNet +ResNet Ensemble +MCDO +MCDO Ensemble +TTA +TTA Ensemble +SVI +SVI Ensemble +Figure 5: Performance of the proposed methods on different datasets under label noise. We compare the performance +on the original dataset (25%), a dataset with a 0% tumor coverage threshold for tumor tiles (0%), applying uniform +noise to the tile labels (Uniform) and randomly flipping labels of tiles which are located at the border of the annotation +(Border). +For translating our tile-level observations to slide-level, we stitch the tile-level predictions back to a tumor confidence +map on slide-level that we show on the example of one slide in Figure 4. +When observing the generated confidence maps, we can see lower tumor confidences at the border of annotated tumor +regions. This goes along with the effect of inter-observer variability, where the border of the annotated tumor area is +expected to vary between observers. +Building on these observations, in our label noise experiments we investigate the effects of imprecise tumor annotations +in the border area of annotated tumor regions. To this end, we define three supplementary datasets by introducing +different types of label noise to the annotations. As we suspect the annotations to be rough themselves, we first create +a dataset by setting the inclusion threshold by tumor coverage for tumor tiles to 0%. We previously excluded every +tile that was covered by less than 25% by tumor annotations (see Section 3.1) as we already suspected a high chance +of noisy annotations on these tiles. The other two datasets are created by applying random label noise to the training +split of the 0% threshold dataset. First, we apply uniform label noise to the whole slide, with a 25% chance of flipping +the tile class. As this type of annotation noise does not reflect real-world inter-observer variability, we next apply +9 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +label flipping to the border regions of the annotation. We flip the labels of the tumor tiles, which lie at the border of +the annotation polygon and thereby are not fully covered by annotated tumor cells. We set the chance of this event +occurring to 25% per tile. +In Figure 5 we show the results of the label noise experiments. Detailed results can be found in Table 3. In terms of +accuracy, the ensemble methods outperform their non-ensemble counterparts by a significant margin. Only on the +Uniform dataset, ResNet and MCDO perform very similar to their ensemble counterparts (∼ 95.6% accuracy for each +method). When viewing the balanced accuracy metric, TTA and TTA Ensemble exceed every other method by a large +margin. SVI and SVI Ensemble are the least robust methods when exposed to label noise, with SVI often performing +significantly worse than our single baseline ResNet. +We can conclude that TTA and ensembling approaches are not only robust to domain shifts and image corruptions +[Ovadia et al., 2019] but in a similar manner also to label noise, in our case in histopathological images. SVI and +MCDO, however, are not fit to deal with label noise often leading to only slightly improved or even worse results. +TTA however does not perform well in terms of calibration error. Here MCDO outperforms TTA and SVI, which +produced the overall worst calibrated predictions, in contrast to recent literature [Ashukha et al., 2020, Ayhan et al., +2020]. We can see a large increase in ECE on the dataset with uniform label noise and a slight increase in the +miscalibration on the other two datasets with label noise compared to our baseline dataset. Except for the original +dataset, no trend of ensembling methods decreasing calibration error is visible. Ensembling does not seem to improve +calibration when confronted with larger quantities of label noise, contrary to the setting of domain shift (Figure 1) +where ensembling decreased the calibration error. +Finally, we observe that the results for the split with the 0% tumor coverage threshold are worse than the results for our +originally selected dataset, containing a tumor coverage of at least 25% per tile and the dataset with added noise in the +border regions. We suspect this behavior to be due to a large amount of inherent label noise in the annotations of the +Camelyon17 dataset. +5 +Discussion +In the previous section, we have gone through an extensive comparison of the most prominent methods for uncertainty +estimation under domain shift on histopathological WSIs. In this section, we want to discuss the observations that we +have made and we want to formulate recommendations for other researchers that try to integrate uncertainty estimation +into digital pathology. Our results show that mispredictions can be detected reliably and that the right methods can +increase the robustness to domain shift and label noise, while also providing better-calibrated predictions. +Among the methods for uncertainty estimation, ensembles lead to the most reliable uncertainty estimates and additionally +improve classification performance and network calibration. When training an ensemble of multiple networks is too +expensive, TTA has, specifically on ID data, shown to improve performance when discarding the most uncertain +tiles, while requiring no change in the used architectures. However, if a good calibration is required, TTA can not be +recommended. On the other hand, the often used MCDO methodology did not lead to significant improvements in our +experiments and can not be recommended. Flipout SVI leads to inconsistent results, being very susceptible to label +noise and underperforming in the task of misprediction detection on ID data. +As expected, combining MCDO, TTA and SVI with ensembling leads to further improvements in classification +performance, however, it also entails a steep increase in computational requirements, which might not be possible in +some medical environments. +In terms of misprediction detection on ID data, all methods provide a reliable improvement in classification performance, +with a growing rejection rate making misprediction detection a feasible scenario in a clinical setting. The largest +increase for every method is at the beginning of the accuracy-reject-curve, with the slope decreasing rapidly around the +20% reject rate threshold. From our results, it is plausible that in many cases it could be enough to only reject around +20% of most uncertain predictions to receive a significant accuracy boost. Our experiments reconfirm the strong impact +of choosing appropriate data augmentations, as has been reported in [Tellez et al., 2019, Stacke et al., 2021]. Choosing +the right set of augmentations is a critical factor for OOD performance. +We compared multiple uncertainty metrics, determining the most uncertain tiles in terms of minimum confidence, +maximum entropy and maximum variance. From our results, we recommend using the confidence metric as it is defined +for every method and performed slightly better in our experiments. It would be of high interest to compare these metrics +in a histopathological multi-class setting to generate a stronger recommendation. +By sorting the tile predictions by uncertainty, we observe visually recognizable differences between the most certain +and most uncertain tiles, which are consistent over all methods. On the WSIs, the networks make especially confident +10 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +predictions for tiles that lie inside tumor metastases, while being the most uncertain in border regions between different +tissue types, which can also be seen when visualizing the uncertainty on slide level. Our label noise experiments +strengthen this point, by raising the suspicion that the Camelyon17 dataset contains noisy labels in the border area +of annotated tumor regions. The inclusion of the likely noisy labels from the dataset immediately leads to a drop in +classifier performance. Practitioners should therefore not only concern themselves with the possibility of domain shift +but should also put a large focus on extracting pure ground truth labels, dropping labels that are potentially misclassified. +In our experiments, the ensemble approaches and especially TTA showed increased resistance to label noise. +Further studies could extend our experiments to a broader range of datasets, as our scope is limited by the availability of +tile-level annotated histopathological data in the Camelyon datasets. Moreover, the evaluation of uncertainty on slide +level remains an open problem. Additional methods could be compared, for example, integrating multi-head ensembles +[Linmans et al., 2020] or deterministic uncertainty methods [Postels et al., 2022], which offer a sampling-free and fast +alternative. Our label noise experiments could be extended to more methods and more realistic scenarios. +6 +Conclusion +Deployment of AI-based diagnostic systems in the safety-critical area of histopathology demand uncertainty-aware +machine learning algorithms, which generate trust in the model’s predictions. +To this end, we compared multiple uncertainty estimation methods and uncertainty metrics across domain shift and label +noise scenarios in their performance, calibration and ability to detect mispredictions in the histopathological setting. Our +results show that on in-domain data, ensembles and TTA are well-performing methods, while under domain shift the +relative order and gain of methods is harder to determine. Existing methods are well-capable to detect mispredictions +and reject inputs they are unlikely to classify correctly. Furthermore, ensembles generally are better calibrated than +their singular counterpart. +As label noise can be a large problem in medical data, practitioners should put great care into identifying potentially +mislabeled inputs, as they can lead to large performance decreases. In terms of robustness to label noise, ensembles +and TTA also perform best. When comparing the uncertainty metrics of confidence, variance and entropy, we found +no significant difference between them and suggest using the predictive confidence, as it is easy to implement and a +reliable metric to detect mispredictions. +The ensemble performed the most consistent over all our experiments, TTA can definitely be recommended in addition +to ensembling or even by itself, as it is compute-efficient and easy to implement without requiring architectural changes +or retraining. MCDO performed worse than TTA, SVI or the ResNet ensemble on most measures, only providing good +results in terms of predictive calibration. SVI, on the other hand, could compete with TTA in some scenarios, but is +harder to implement and train and in our experiments lead to a higher variability between results. +While no single benchmark can give all-encompassing results and insights, we hope that our evaluation gives guidance +for the utilization of uncertainty methods in the area of histopathology. Our published code is designed to be easily +reproducible and extendable to further studies. +Acknowledgments +The research is funded by the Ministerium für Soziales und Integration, Baden Württemberg, Germany. +11 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +References +Andre Esteva, Brett Kuprel, Roberto A. Novoa, Justin Ko, Susan M. Swetter, Helen M. Blau, and Sebastian Thrun. +Dermatologist-level classification of skin cancer with deep neural networks. Nature, 542(7639):115–118, February +2017. ISSN 1476-4687. doi:10.1038/nature21056. +H. A. Haenssle, C. Fink, R. Schneiderbauer, F. Toberer, T. 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PMLR, June 2022. +14 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +A +Detailed Results +Table 2: Detailed results for tile classifier performance on Camelyon17. We report median and interquartile range as +shown in the boxplots. +Split +Method +Accuracy ↑ +Balanced Accuracy ↑ +ECE ↓ +ID +ResNet +0.9762 0.0010 +0.8291 0.0073 +0.0181 0.0020 +ResNet Ensemble +0.9798 0.0015 +0.8474 0.0105 +0.0145 0.0011 +MCDO +0.9777 0.0053 +0.8244 0.0380 +0.0135 0.0001 +MCDO Ensemble +0.9794 0.0006 +0.8363 0.0026 +0.0137 0.0010 +TTA +0.9788 0.0013 +0.8689 0.0076 +0.0214 0.0026 +TTA Ensemble +0.9805 0.0003 +0.8690 0.0043 +0.0207 0.0003 +SVI +0.9743 0.0009 +0.8372 0.0385 +0.0213 0.0073 +SVI Ensemble +0.9787 0.0007 +0.8452 0.0076 +0.0196 0.0007 +OOD (Center 2) +ResNet +0.9166 0.0146 +0.7926 0.0355 +0.0546 0.0177 +ResNet Ensemble +0.9327 0.0032 +0.8359 0.0046 +0.0378 0.0091 +MCDO +0.9103 0.0207 +0.8110 0.0293 +0.0529 0.0241 +MCDO Ensemble +0.9319 0.0032 +0.8367 0.0268 +0.0366 0.0203 +TTA +0.9306 0.0077 +0.8357 0.0200 +0.0412 0.0196 +TTA Ensemble +0.9333 0.0016 +0.8488 0.0056 +0.0504 0.0249 +SVI +0.9173 0.0775 +0.8254 0.0636 +0.0965 0.1000 +SVI Ensemble +0.9367 0.0141 +0.8687 0.0123 +0.0791 0.0281 +OOD (Center 4) +ResNet +0.9401 0.0080 +0.8800 0.0174 +0.0238 0.0061 +ResNet Ensemble +0.9500 0.0005 +0.9045 0.0079 +0.0139 0.0009 +MCDO +0.9513 0.0032 +0.9145 0.0129 +0.0190 0.0045 +MCDO Ensemble +0.9517 0.0039 +0.9134 0.0110 +0.0142 0.0009 +TTA +0.9448 0.0036 +0.9225 0.0041 +0.0294 0.0065 +TTA Ensemble +0.9489 0.0021 +0.9208 0.0028 +0.0249 0.0030 +SVI +0.9495 0.0117 +0.9222 0.0227 +0.0176 0.0049 +SVI Ensemble +0.9577 0.0007 +0.9244 0.0054 +0.0169 0.0004 +15 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +Table 3: Detailed results for our experiments under label noise. We report median and interquartile range as shown in +the boxplots. +Experiment +Method +Accuracy ↑ +Balanced Accuracy ↑ +ECE ↓ +25% +ResNet +0.9762 0.0010 +0.8291 0.0073 +0.0181 0.0020 +ResNet Ensemble +0.9798 0.0015 +0.8474 0.0105 +0.0145 0.0011 +MCDO +0.9777 0.0053 +0.8244 0.0380 +0.0135 0.0001 +MCDO Ensemble +0.9794 0.0006 +0.8363 0.0026 +0.0137 0.0010 +TTA +0.9788 0.0013 +0.8689 0.0076 +0.0214 0.0026 +TTA Ensemble +0.9805 0.0003 +0.8690 0.0043 +0.0207 0.0003 +SVI +0.9745 0.0019 +0.8453 0.0219 +0.0240 0.0066 +SVI Ensemble +0.9787 0.0007 +0.8452 0.0076 +0.0196 0.0007 +0% +ResNet +0.9612 0.0004 +0.8323 0.0124 +0.0531 0.0046 +ResNet Ensemble +0.9658 0.0007 +0.8299 0.0110 +0.0460 0.0061 +MCDO +0.9628 0.0048 +0.8055 0.0229 +0.0387 0.0165 +MCDO Ensemble +0.9677 0.0012 +0.8217 0.0181 +0.0491 0.0030 +TTA +0.9637 0.0011 +0.8598 0.0105 +0.0558 0.0038 +TTA Ensemble +0.9654 0.0005 +0.8596 0.0029 +0.0570 0.0014 +SVI +0.9596 0.0050 +0.8243 0.0115 +0.0586 0.0129 +SVI Ensemble +0.9635 0.0008 +0.8306 0.0062 +0.0571 0.0020 +Uniform +ResNet +0.9563 0.0037 +0.8383 0.0215 +0.2678 0.0012 +ResNet Ensemble +0.9564 0.0039 +0.8325 0.0143 +0.2698 0.0013 +MCDO +0.9563 0.0034 +0.8074 0.0143 +0.2633 0.0024 +MCDO Ensemble +0.9557 0.0073 +0.7934 0.0228 +0.2644 0.0057 +TTA +0.9483 0.0047 +0.8294 0.0204 +0.2712 0.0013 +TTA Ensemble +0.9500 0.0003 +0.8358 0.0092 +0.2719 0.0012 +SVI +0.9299 0.0072 +0.8010 0.0014 +0.2765 0.0059 +SVI Ensemble +0.9317 0.0057 +0.8019 0.0030 +0.2770 0.0011 +Border +ResNet +0.9679 0.0016 +0.7879 0.0119 +0.0468 0.0015 +ResNet Ensemble +0.9739 0.0005 +0.7878 0.0060 +0.0463 0.0015 +MCDO +0.9706 0.0065 +0.7790 0.0182 +0.0420 0.0025 +MCDO Ensemble +0.9747 0.0018 +0.7659 0.0134 +0.0405 0.0042 +TTA +0.9719 0.0029 +0.8047 0.0098 +0.0523 0.0053 +TTA Ensemble +0.9746 0.0024 +0.8107 0.0038 +0.0500 0.0059 +SVI +0.9718 0.0058 +0.7922 0.0144 +0.0439 0.0074 +SVI Ensemble +0.9738 0.0001 +0.7997 0.0002 +0.0532 0.0017 +B +Dataset statistics +16 + +Benchmarking Uncertainty in Histopathology +MEHRTENS ET AL., 2022 +0 +10 +20 +30 +40 +50 +Slide index +0 +10000 +20000 +30000 +40000 +50000 +Number of tiles +Non-Tumor +Tumor +Figure 6: Tumor vs. Non-Tumor ratio for all lesion-level annotated slides. Slides are sorted by center with 10 slides +belonging to each center. This means slides with index [21, 30] belong to OOD center 2 and slides with index [41, 50] +belong to OOD center 4. All other slides are part of the in-distribution dataset consisting of center 0, 1 and 3 which use +the same slide scanner. +C +Effects of data augmentations on network calibration +Basic augmentations +Strong augmentations +ID centers +OOD (center 2) +OOD (center 4) +0.05 +0.10 +0.15 +0.20 +0.25 +ECE +Method +ResNet +Temp Scaling +ResNet Ensemble +MCDO +MCDO Ensemble +ID centers +OOD (center 2) +OOD (center 4) +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +ECE +Method +ResNet +Temp Scaling +ResNet Ensemble +MCDO +MCDO Ensemble +Table 4: Expected calibration error between data splits for training with two different settings of augmentations. For +basic augmentations, we see increasing calibration error accross methods on both OOD centers. The plot on the right +represents the augmentations used in the main part. As uncertainty methods we include MCDO and Deep Ensembles +and we additionally include Temperature Scaling as another method to improve network calibration. +17 + diff --git a/I9AzT4oBgHgl3EQfH_tx/content/tmp_files/load_file.txt b/I9AzT4oBgHgl3EQfH_tx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7eaeb903a2dafda7302c8a7eb7c5f8798f49fa5f --- /dev/null +++ b/I9AzT4oBgHgl3EQfH_tx/content/tmp_files/load_file.txt @@ -0,0 +1,1407 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf,len=1406 +page_content='BENCHMARKING COMMON UNCERTAINTY ESTIMATION METHODS WITH HISTOPATHOLOGICAL IMAGES UNDER DOMAIN SHIFT AND LABEL NOISE PREPRINT Hendrik Mehrtens1, Alexander Kurz1, Tabea-Clara Bucher, Titus J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Brinker2 Division of Digital Biomarkers for Oncology German Cancer Research Center (DKFZ) Heidelberg, Germany {hendrikalexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='mehrtens},{alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='kurz},{tabea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='bucher}@dkfz-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='de titus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='brinker@nct-heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='de September 2022 ABSTRACT In the past years, deep learning has seen an increase of usage in the domain of histopathological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' However, while these approaches have shown great potential, in high-risk environments deep learning models need to be able to judge their own uncertainty and be able to reject inputs when there is a significant chance of misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In this work, we conduct a rigorous evaluation of the most commonly used uncertainty and robustness methods for the classification of Whole-Slide- Images under domain shift using the H&E stained Camelyon17 breast cancer dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Although it is known that histopathological data can be subject to strong domain shift and label noise, to our knowledge this is the first work that compares the most common methods for uncertainty estimation under these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In our experiments, we compare Stochastic Variational Inference, Monte-Carlo Dropout, Deep Ensembles, Test-Time Data Augmentation as well as combinations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We observe that ensembles of methods generally lead to higher accuracies and better calibration and that Test-Time Data Augmentation can be a promising alternative when choosing an appropriate set of augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Across methods, a rejection of the most uncertain tiles leads to a significant increase in classification accuracy on both in-distribution as well as out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Furthermore, we conduct experiments comparing these methods under varying conditions of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We observe that the border regions of the Camelyon17 dataset are subject to label noise and evaluate the robustness of the included methods against different noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Lastly, we publish our code framework to facilitate further research on uncertainty estimation on histopathological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Keywords Deep learning, Uncertainty estimation, Robustness, Histopathology, Domain shift, Label noise 1 Introduction Deep Neural Networks (DNNs) have been shown to be of equal or even superior performance in studies on many medical tasks [Esteva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017, Haenssle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2018, Hekler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019], compared to human practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Nonetheless, they have rarely been adopted in clinical praxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' One commonly given reason [Begoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Kompa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021, van der Laak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021] is their inability to provide well-calibrated estimates of their predictive uncertainty [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017], thereby prohibiting the practitioner from judging the reliability of the system’s decision, which is a necessary condition in areas of high uncertainty like medical decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 1Contributed equally 2Corresponding author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='01054v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='IV] 3 Jan 2023 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 Ideally, a well-calibrated deep learning system should be able to judge and communicate a correct estimate of its certainty in each prediction, not only informing the human practitioner of its momentary reliability but also enabling the system to automatically reject inputs [Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021] and refer them to humans for inspection, thereby enhancing the reliability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Another problem in practice is the high vulnerability of DNN-based systems to "domain shifts", which are differences between the data distributions in the training and deployment setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In the general machine learning literature, [Hendrycks and Dietterich, 2019, Ovadia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019] noticed the vulnerability of deep neural networks to even slight artificial perturbations of images, which result in drastic deterioration of classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In the area of digital pathology, these kinds of shifts are very common [Stacke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' They arise due to different data-acquisition processes across clinics, for example, due to different staining procedures or different scanners but can also be caused by changes in the distribution of patient characteristics (gender, age, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The unpredictable behavior of models in new data regimes limits the ability to put confidence in a system’s decision, especially in high-risk environments, an example being decisions that affect a patient’s treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Traditional methods for improving the generalization of a machine learning system use augmentations of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This technique is currently heavily investigated in the field of digital histopathology [Tellez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' However, as the topics of uncertainty quantification and generalization under domain shift have received attention in the whole field of machine learning, in recent years many new techniques have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' All of these methods, however, have mostly been evaluated in the general setting of computer vision and object detection and not in the setting of digital pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Whole-Slide-Images (WSIs) are a challenging domain for deep learning, due to the large size of the images, typically in the dimension of Gigapixels, and due to the sparsely available labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Additionally, as the evaluation of a WSI is error-prone, with high inter-observer variability [Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020], the labels are subject to a high degree of label noise, making the integration of predictive certainty a vital component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' [Linmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020] investigated the ability of multi-headed ensemble networks to detect out-of-distribution (OOD) inputs in WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' [Thagaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020] compared Deep Ensembles [Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017] and Monte-Carlo Dropout (MCDO) [Gal and Ghahramani, 2016] in their ability to estimate an uncertainty on tile-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To our knowledge, there has been no thorough and reproducible investigation of these methods with respect to histopathological data, especially in the context of domain shift towards other clinics and the general effect of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As the estimation of uncertainty and robustness to domain shifts and label noise is of utmost importance in digital pathology, we evaluate the performance of the most popular approaches, comparable to the benchmark of [Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021] on retina scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To this end, we carefully compare the robustness of Stochastic Variational Inference (SVI) [Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2015], MCDO [Gal and Ghahramani, 2016], Deep Ensembles [Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017] and Test-Time Data Augmentation (TTA) [Ayhan and Berens, 2018] and combinations thereof under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We study the impact of the reject option by uncertainty on the performance of the model, observing a significant increase in accuracy in high certainty predictions, and compare multiple metrics of estimating the final certainty from the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Finally, we simulate the effects of label noise in the edge regions of the tumor annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For our experiments, we use the Camelyon17 dataset [Bándi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019] of breast cancer tissue, which consists of 50 slides with annotated tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The slides were obtained in five different clinics, using three different scanners, which enables us to evaluate the different approaches under a realistic domain shift scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our main contributions are: A systematic comparison of the most popular uncertainty estimation methods under domain shift in terms of predictive accuracy and network calibration with respect to histopathological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' A detailed analysis to what extent rejecting uncertain tiles improves classification performance and an analysis of the distribution of the rejected tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' An investigation of the influence of label noise on the classification of WSIs and the robustness of the included uncertainty estimation methods against it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The release of an easily extendible code repository1 to facilitate further research on the applicability of uncertainty estimation for deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2 Methods In this section, we describe the methods, uncertainty measures and evaluation settings used in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='com/DBO-DKFZ/uncertainty-benchmark 2 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Uncertainty Estimation Methods In uncertainty estimation, we want to compute the posterior predictive distribution of the output y, given an input x and training data D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The distribution can be formalized in terms of Bayesian Model Averaging over the model’s parameters w as p(y|x, D) = � p(y|x, w)p(w|D) dw (1) For neural networks, the posterior predictive distribution is analytically intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In recent years several methods have been proposed that approximate the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In the following, we briefly introduce the most prominent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Stochastic Variational Inference (SVI): [Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2015] and [Graves, 2011] approximate the posterior distribution by placing a Gaussian distribution over every parameter of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As an objective for the approximation, the estimated lower-bound (ELBO) is minimized [Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We use the Flipout formulation [Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2018] of SVI for stabilizing the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 Monte-Carlo Dropout (MCDO): [Gal and Ghahramani, 2016] show that the dropout operation, originally intended as a regularization method to stabilize neural network training, can be used to approximate the true posterior of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For the approximation, multiple forward passes of the same input, with activated dropout layers, are aggregated during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The distribution over the predictions obtained through this method can be seen as samples from an approximation of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3 Deep Ensemble: A Deep Ensemble [Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017] consists of multiple, in architecture identical, neural networks that are trained from different random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The mean of all ensemble members serves as the prediction during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' [Ovadia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Ashukha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020] show that Deep Ensembles outperform many other methods in terms of calibration and robustness under domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 Test-Time Data Augmentations (TTA): In contrast to Deep Ensembles, which employ multiple models during inference, TTA uses the same model multiple times by augmenting the input in different ways during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Generally, the same augmentations as at training time are applied during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' [Ayhan and Berens, 2018, Ashukha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020] show the good performance of TTA in terms of robustness and calibration, which can come close to the performance of a Deep Ensemble while requiring less training time, as only one model is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='5 Temperature Scaling: In their work, [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017] show that introducing a single scaling factor T on the network’s output can improve network calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' More specifically, the output logits zi are divided by T before applying the softmax function to the i-th input sample: ˆpi = max c ∈ C σSM(zi/T)(c) (2) ,where c is one class in the set of all possible classes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The optimal temperature T > 0 is computed by minimizing the negative log-likelihood (NLL) on a validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6 Ensemble Variants: Additionally, ensemble variants of the methods SVI, MCDO and TTA are implemented and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 Uncertainty Metrics Given the predictions generated by the previously introduced methods, the literature developed multiple metrics to estimate the predictive uncertainty, which we compare in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 3 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Confidence: A commonly used method, especially in the field of calibration [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017], is to take the maximum of the softmax, which is also called the “confidence” of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The core idea is that decisions that are considered certain are far away from the decision boundary, while uncertain decisions lie close to the boundary at 1/N, where N is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 Entropy: Another often used metric [Mobiny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Band et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021] is to take the entropy of the prediction as a measure of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The entropy of the model’s output probability is computed as H(y|x, D) = − � c ∈ C p(y = c|x, D) log p(y = c|x, D) (3) Since the maximum possible entropy varies with the number of classes, we compute the normed entropy as Hnorm = H Hmax with Hmax = − N � i=1 1 N log � 1 N � = log N (4) When using the normed entropy, the uncertainty is high when Hnorm → 1 and uncertainty is low when Hnorm → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Confidence (conf) and entropy behave very similarly in a binary classification setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Given two predictions yi and yj the following relation holds between them: H(yi) < H(yj) ⇐⇒ conf(yi) > conf(yj) (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3 Variance: Since all included uncertainty estimation methods generate a distribution of predictions, the variance of the distribution can be used as a measure of uncertainty [Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017, Abdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' If all predictors agree on a result, the variance is zero, whereas a high variance indicates high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3 Evaluation Settings This section covers the different evaluation settings and the performance metrics used in each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Classification: For the tumor classification on tile-level, we compute accuracy and balanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Given the tumor class as the positive class, the balanced accuracy is defined as the arithmetic mean between the true positive rate TPR = TP/P and the true negative rate TNR = TN/N: Balanced Accuracy = 1 2 �TP P + TN N � (6) ,where P is the total number of positive samples and N is the total number of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 Calibration: For measuring calibration, we utilize the Expected Calibration Error (ECE) [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2017, Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Given a prediction for every data point in the dataset, the output probability or "confidence" for each sample should on average match the correctness of the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In other words, we expect a prediction that has a confidence value of 60% to be correct in about 60% of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To validate this intuition, the predictions are split into a predetermined number M of bins B of equal confidence range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Then the absolute difference between the average confidence and accuracy within each bin is summed up: ECE = M � m=1 |Bm| n ���conf(Bm) − acc(Bm) ��� (7) Here, |Bm| denotes the number of samples in the m-th bin and n is the total number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 4 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3 Reject Option: In a clinical setting, the model should be able to refer predictions with high uncertainty to human practitioners for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Models suitable for this task should assign higher uncertainties to their wrong predictions than to their correct predictions, thereby allowing to cut off a large number of false predictions, by thresholding at a certainty level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To compare multiple models on this ability, we compute the accuracy-reject curve [Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2009], plotting the achieved accuracy against the percentage of rejected data points in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 Label Noise: Medical annotations are often subject to an unquantified amount of label noise [Joskowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020], which may deteriorate the performance of supervised machine learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To our knowledge, the previously described methods have not been compared in their robustness to label noise in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We evaluate the effect by creating multiple datasets with increasing levels and different types of label noise and evaluate the methods under these changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 3 Experiment setup 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Dataset and data processing We conduct our experiments on the lesion-level annotated slides of the Cameylon17 dataset [Bándi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This part of the Camelyon17 dataset consists of 50 WSIs of breast lymph node tissue, with the goal of detecting metastatic tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The slides were obtained from five different clinics in the Netherlands, using three different scanners, which serves as an ideal setting to estimate the influence of domain shift between clinics on the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To create a distribution shift between the in-distribution (ID) and out-of-distribution (OOD) domain, we split the dataset into three groups, using the centers 0, 1 and 3 with the 3DHistech scanner as ID data, where the training occurs, while using the 10 slides of centers 2 (Hamamatsu scanner) and 4 (Philips scanner) as OOD data each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' By splitting the data between the centers, we induce differences in location and acquisition process between the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We then further split the ID data into the training, validation and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We sort the slides of each center by the area of annotated tumor cells they contain and use the two median slides as test set for each center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The training and validation sets are generated by a randomized 75%/25% split of the tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The tiles themselves are generated following [Khened et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021] with median filtering and Otsu’s thresholding of the HSV saturation component of the WSI image, followed by finally applying opening and closing dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' After that, tiles of the size 256 × 256 are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The tumor regions on the slide are indicated by polygonal annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' From the annotations, a tumor coverage per tile is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Tiles with more than 25% tumor coverage are counted as tumor tiles and all tiles with 0% tumor coverage are counted as non-tumor tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Tumor tiles that are covered by the tumor annotation by less than 25% are discarded for our standard training, to minimize the risk of label noise that can arise due to high inter-observer variability [Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Joskowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Karimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 Training setup and hyperparameter tuning We use a ResNet-34 [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2016] with a batch size of 128 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='001 for all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As optimizer we utilize Adam [Kingma and Ba, 2017], with a reduction of the learning rate by a factor of 10 if the validation loss, which is chosen as the cross-entropy loss, does not decrease for 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For data augmentations, we follow [Tellez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019] applying random crops to size 224 × 224, random 90° rotations and color jitter (brightness:±20%, contrast:±30%, hue:±10%, saturation:±10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The inputs are normalized with the mean and variance of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The best model is chosen by balanced accuracy on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' During training, we balance the training set by samples per class, but we do not balance the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For the Deep Ensemble architecture, we choose n = 5 members following recent literature [Linmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020, Thagaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For MCDO, we place a dropout layer after each ResNet block, with dropout probability p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This is in contrast to [Linmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020, Thagaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020], who only place a dropout layer before the last layer, observing no improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For inference during testing, we use 10 SVI-, MCDO- and TTA- samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Following [Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020], we use an additional hyperparameter for weighting the influence of the SVI prior (Kullback-Leibler-Divergence to the normal distribution) on the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We tune this hyperparameter as well as the dropout probability, the dropout layer placement and the learning rate with the python library Optuna [Akiba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' All our experiments are conducted with PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='11 on Nvidia GPUs with CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 5 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 4 Results We trained each method 5 times with different seeds, reporting their average or median performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 Tile Classifier Performance ID centers Center 2 Center 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='98 Value Accuracy ID centers Center 2 Center 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='925 Balanced Accuracy ID centers Center 2 Center 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='14 ECE Method ResNet ResNet Ensemble MCDO MCDO Ensemble TTA TTA Ensemble SVI SVI Ensemble Figure 1: Accuracy, balanced accuracy and estimated calibration error (ECE) of the proposed methods on the in- distribution (ID) centers and out-of-distribution centers 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 Rejection rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='00 Balanced Accuracy ID centers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 Rejection rate OOD (center 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 Rejection rate OOD (center 4) Method ResNet ResNet Ensemble MCDO MCDO Ensemble TTA TTA Ensemble SVI SVI Ensemble Figure 2: (Balanced) Accuracy-Reject Curves on ID and OOD centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The x-axis contains the proportion of rejected inputs over the dataset, which is plotted against the reached Balanced Accuracy on the remaining images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The order of the excluded points was determined by the predictive confidence (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We plot the mean curves over all 5 trials, with the shaded area denoting the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Figure 1 shows the accuracy, balanced accuracy and ECE on the ID data, as well as on the OOD centers 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Detailed numeric results for all our experiments can be found in the appendix in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We start by analyzing the accuracy values on the three test data splits and between all evaluated uncertainty methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the ID test data, we can see that we achieve very high accuracies of about 98% for the classification of tumor and non-tumor tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Between the evaluated uncertainty methods, we only observe small differences in accuracy on the ID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As expected, the ensemble variants outperform their non-ensemble counterparts and TTA performs better than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' However, SVI achieves a slightly lower accuracy (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4%) than the baseline ResNet, while MCDO can only slightly increase upon the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On OOD centers 2 and 4, we observe a decrease in accuracy compared to the ID test data, with the decrease on center 4 being smaller than the decrease observed on center 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This behavior indicates a larger domain shift on center 2, than on center 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We can observe an increase in performance with almost all uncertainty methods compared to the baseline ResNet, except for SVI and MCDO on center 2, where the results show a large spread in observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Especially the ensemble variants (ResNet Ensemble, MCDO Ensemble, TTA Ensemble and SVI Ensemble) consistently achieve higher accuracies than the baseline ResNet classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 6 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 Table 1: Mean and standard deviation of the area under the curve of the accuracy-reject curves (AUARC) presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We compare the confidence of a single prediction, the confidence of the mean prediction for ensembling-like approaches and the variance over multiple predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Measure Unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Method ID Centers OOD (Center 2) OOD (Center 4) Confidence ResNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='906±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='893±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='954±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='010 Confidence ResNet Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='918±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='932±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='972±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='004 MCDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='886±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='914±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='971±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='010 MCDO Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='905±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='924±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='971±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='007 TTA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='932±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='897±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='970±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='004 TTA Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='933±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='884±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='963±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='009 SVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='895±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='931±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='972±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='012 SVI Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='896±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='949±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='981±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='002 Variance ResNet Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='914±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='928±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='972±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='004 MCDO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='877±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='911±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='968±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='012 MCDO Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='898±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='915±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='969±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='008 TTA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='926±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='877±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='965±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='005 TTA Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='926±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='861±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='959±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='010 SVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='890±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='929±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='970±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='013 SVI Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='888±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='942±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='981±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='001 Since we have to deal with a significant class imbalance between tumor and non-tumor tiles, we further investigate balanced accuracy in the center plot of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Compared to the previous evaluation of accuracy, we observe lower values concerning balanced accurac/y for all centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the ID centers, the baseline ResNet achieves a balanced accuracy of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='9%, while TTA and TTA Ensemble significantly improve performance (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='9% each), with MCDO even reaching a slightly lower balanced accuracy than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the out-of-distribution centers all methods perform better than the baseline ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Of all the methods TTA, TTA Ensemble and SVI Ensemble lead to the largest increases, while MCDO and MCDO Ensemble barely outperformed the pure ResNet Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Noticeable here is the high performance of SVI Ensemble on the OOD data, which could already be observed in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' SVI alone on the other hand again leads to a very high spread in observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When computing the balanced accuracy, a higher weight is attributed to the under-represented tumor class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The lower values in balanced accuracy compared to standard accuracy indicate that our model performs worse in classifying tumor tiles than non-tumor tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Surprisingly, we see higher values in balanced accuracy on OOD center 4 compared to the other two test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To explain this behavior, we added an overview of the dataset distribution in the appendix (see B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The center 4 dataset has a larger ratio of tumor tiles to non-tumor tiles because it contains one slide with two large tumor metastases, much larger than the tumor metastases on all other WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Therefore, the balanced accuracy metric is heavily influenced by the large increase in tumor tiles that lie inside the area of annotated tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As we will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3, these tumor tiles lead to higher confidences and better classification results, than tumor tiles that lie at the border of the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In the plot on the right-hand side of Figure 1, we evaluate model calibration in terms of ECE (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the ID test data, we observe low ECE values across all evaluated methods, which means that the output confidence values represent a good estimate of the correctness of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' ResNet Ensemble (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0145), as well as the MCDO approaches (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0135 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0137), produce better-calibrated results than the baseline, while all TTA and SVI methods slightly worsen the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As shown in [Ovadia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019], we expect higher ECE values for the domain shift towards OOD centers 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On center 2 this is also reflected in our results, as we see higher ECE values and higher interquartile ranges for all evaluated methods compared to the ID test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Here the relative order remains mostly the same, with TTA and TTA Ensemble now performing better than the ResNet, while the ResNet Ensemble and the MCDO methods still perform best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Center 4 on the other hand shows lower ECE values, comparable to the values on the ID test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This indicates that with the applied augmentations, we were able to mostly compensate for the domain shift between ID and OOD data, which was already visible in our accuracy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Here the MCDO methods and the ResNet Ensemble perform better than the baseline method, while the situation of SVI and TTA methods is reversed to that of center 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In C, we analyze the influence of data augmentations on network calibration by comparing the results to training with basic augmentations consisting of random crops and flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' There we see higher ECE values for both OOD centers 2 and 4 compared to the ID test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In the evaluation in the appendix we also include Temperature Scaling as a reference method for improving network calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 7 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 To summarize the results, we find that data augmentations applied during training have a larger impact on the achieved accuracies and the calibration compared to utilizing one of our evaluated uncertainty methods and as such a large focus should be placed on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Amongst the methods evaluated in Figure 1, ensembles lead to slightly better-calibrated results than the baseline, while TTA and SVI behave inconsistently and can have a negative impact on the resulting ECE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In terms of achieved accuracies, the ensemble approaches consistently perform best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The SVI method leads to inconsistent results across the evaluated metrics, therefore no direct conclusion is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' TTA does not require architectural changes and no extra computational resources during training and leads to a large increase in observed accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' If however larger training capacity is available, a TTA Ensemble or even an SVI Ensemble can be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The basic ResNet Ensemble, while not the best in most evaluations, performed the most consistent, always outperforming the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='Reject Option ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='Tiles with highest uncertainty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='Figure 3: Least and most uncertain tiles from the in-distribution test set computed by the confidence of a ResNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The numbers represent the ground-truth labels: ’1’ indicates tumor and ’0’ indicates a non-tumor tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We compare the uncertainty methods in terms of their ability to detect mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For that we compute the (Balanced) Accuracy-Reject Curves (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In Figure 2, we compare the methods and their performance with an increasing ratio of rejected tiles, for the test slides on the ID centers and OOD centers 2 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To determine the most uncertain tiles that are rejected, we choose the confidence measure (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For each split and method, we see a mostly linear increase of balanced accuracy when rejecting a fraction of the most uncertain tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the ID test data, TTA and the TTA Ensemble consistently achieve the highest balanced accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For the OOD centers, the SVI Ensemble leads to the highest accuracy-reject-curves, while the baseline ResNet performs worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' From our evaluated uncertainty methods, the MCDO methods averaged over all three datasets led to the worst results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' While most methods show increasing performance on center 2 when rejecting the most uncertain tiles, TTA and TTA Ensemble show an unexpected stagnation of performance when rejecting more than 20% of uncertain tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In Table 1 we show the area under the curve for the accuracy-reject curves (AUARC) across all datasets and compare different uncertainty measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the ID data, ResNet Ensemble, TTA and TTA Ensemble lead to an increase of the AUARC over the baseline value, while the MCDO and SVI methods lead to a decrease of said metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On center 2 and center 4, all methods (other than TTA Ensemble on center 2) lead to higher AUARC values than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' From our experiments, not one single method can be identified, that performs best between all data splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' However, the ResNet Ensemble consistently beats the basic ResNet, while TTA and TTA Ensemble perform at least as well as the baseline or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When comparing the scores computed from using the confidence of the predictions or the variance across individual predictions, the relative order of methods remains the same across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Furthermore, every method performs worse in terms of AUARC, when the uncertainty is calculated by the variance across individual predictions instead of the confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Although the disagreement of ensembling methods measured in terms of the variance is often claimed to be a good indicator of uncertainty, in our experiments it consistently performs worse when compared to using the confidence for detecting mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As using the confidence for misprediction detection by rejection is equivalent to using the entropy in the binary case (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2), our experiments strongly suggest utilizing confidence or entropy over the variance of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 8 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 In Figure 3 we present a collection of the most certain and most uncertain tiles within the ID test data, computed with a ResNet Ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We observe that the neural network appears to be most confident on tumor tiles (label 1), that cover the whole tile and possess a similar cell structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For the most uncertain tiles on the right side, no comparable structure among the tiles is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' These tiles both contain tumor and non-tumor tissue and often lie at border regions between tissue types, for example, the border to fat-tissue, as can be identified by the white areas in the H&E stained slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Many uncertain tiles seem to lie at the border of annotated tumor regions, that we suspect to have a larger degree of label noise, and in regions that have a different appearance than the majority of healthy tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='3 Label Noise Image Annotation Non-Tumor Tumor Tumor Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='0 Figure 4: WSI (patient_017_node_2) with ground-truth annotations and model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The tumor confidence decreases in areas near the border of the annotation, while the uncertainty thereby increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This result is consistent over the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 25% 0% Uniform Border 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='98 Value Accuracy 25% 0% Uniform Border 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='88 Balanced Accuracy 25% 0% Uniform Border 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='25 ECE Method ResNet ResNet Ensemble MCDO MCDO Ensemble TTA TTA Ensemble SVI SVI Ensemble Figure 5: Performance of the proposed methods on different datasets under label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We compare the performance on the original dataset (25%), a dataset with a 0% tumor coverage threshold for tumor tiles (0%), applying uniform noise to the tile labels (Uniform) and randomly flipping labels of tiles which are located at the border of the annotation (Border).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For translating our tile-level observations to slide-level, we stitch the tile-level predictions back to a tumor confidence map on slide-level that we show on the example of one slide in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When observing the generated confidence maps, we can see lower tumor confidences at the border of annotated tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This goes along with the effect of inter-observer variability, where the border of the annotated tumor area is expected to vary between observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Building on these observations, in our label noise experiments we investigate the effects of imprecise tumor annotations in the border area of annotated tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To this end, we define three supplementary datasets by introducing different types of label noise to the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As we suspect the annotations to be rough themselves, we first create a dataset by setting the inclusion threshold by tumor coverage for tumor tiles to 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We previously excluded every tile that was covered by less than 25% by tumor annotations (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1) as we already suspected a high chance of noisy annotations on these tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The other two datasets are created by applying random label noise to the training split of the 0% threshold dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' First, we apply uniform label noise to the whole slide, with a 25% chance of flipping the tile class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As this type of annotation noise does not reflect real-world inter-observer variability, we next apply 9 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 label flipping to the border regions of the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We flip the labels of the tumor tiles, which lie at the border of the annotation polygon and thereby are not fully covered by annotated tumor cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We set the chance of this event occurring to 25% per tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In Figure 5 we show the results of the label noise experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Detailed results can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In terms of accuracy, the ensemble methods outperform their non-ensemble counterparts by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Only on the Uniform dataset, ResNet and MCDO perform very similar to their ensemble counterparts (∼ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='6% accuracy for each method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When viewing the balanced accuracy metric, TTA and TTA Ensemble exceed every other method by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' SVI and SVI Ensemble are the least robust methods when exposed to label noise, with SVI often performing significantly worse than our single baseline ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We can conclude that TTA and ensembling approaches are not only robust to domain shifts and image corruptions [Ovadia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019] but in a similar manner also to label noise, in our case in histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' SVI and MCDO, however, are not fit to deal with label noise often leading to only slightly improved or even worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' TTA however does not perform well in terms of calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Here MCDO outperforms TTA and SVI, which produced the overall worst calibrated predictions, in contrast to recent literature [Ashukha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020, Ayhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We can see a large increase in ECE on the dataset with uniform label noise and a slight increase in the miscalibration on the other two datasets with label noise compared to our baseline dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Except for the original dataset, no trend of ensembling methods decreasing calibration error is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Ensembling does not seem to improve calibration when confronted with larger quantities of label noise, contrary to the setting of domain shift (Figure 1) where ensembling decreased the calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Finally, we observe that the results for the split with the 0% tumor coverage threshold are worse than the results for our originally selected dataset, containing a tumor coverage of at least 25% per tile and the dataset with added noise in the border regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We suspect this behavior to be due to a large amount of inherent label noise in the annotations of the Camelyon17 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 5 Discussion In the previous section, we have gone through an extensive comparison of the most prominent methods for uncertainty estimation under domain shift on histopathological WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In this section, we want to discuss the observations that we have made and we want to formulate recommendations for other researchers that try to integrate uncertainty estimation into digital pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our results show that mispredictions can be detected reliably and that the right methods can increase the robustness to domain shift and label noise, while also providing better-calibrated predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Among the methods for uncertainty estimation, ensembles lead to the most reliable uncertainty estimates and additionally improve classification performance and network calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When training an ensemble of multiple networks is too expensive, TTA has, specifically on ID data, shown to improve performance when discarding the most uncertain tiles, while requiring no change in the used architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' However, if a good calibration is required, TTA can not be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the other hand, the often used MCDO methodology did not lead to significant improvements in our experiments and can not be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Flipout SVI leads to inconsistent results, being very susceptible to label noise and underperforming in the task of misprediction detection on ID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As expected, combining MCDO, TTA and SVI with ensembling leads to further improvements in classification performance, however, it also entails a steep increase in computational requirements, which might not be possible in some medical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In terms of misprediction detection on ID data, all methods provide a reliable improvement in classification performance, with a growing rejection rate making misprediction detection a feasible scenario in a clinical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The largest increase for every method is at the beginning of the accuracy-reject-curve, with the slope decreasing rapidly around the 20% reject rate threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' From our results, it is plausible that in many cases it could be enough to only reject around 20% of most uncertain predictions to receive a significant accuracy boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our experiments reconfirm the strong impact of choosing appropriate data augmentations, as has been reported in [Tellez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2019, Stacke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Choosing the right set of augmentations is a critical factor for OOD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' We compared multiple uncertainty metrics, determining the most uncertain tiles in terms of minimum confidence, maximum entropy and maximum variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' From our results, we recommend using the confidence metric as it is defined for every method and performed slightly better in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' It would be of high interest to compare these metrics in a histopathological multi-class setting to generate a stronger recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' By sorting the tile predictions by uncertainty, we observe visually recognizable differences between the most certain and most uncertain tiles, which are consistent over all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' On the WSIs, the networks make especially confident 10 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 predictions for tiles that lie inside tumor metastases, while being the most uncertain in border regions between different tissue types, which can also be seen when visualizing the uncertainty on slide level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our label noise experiments strengthen this point, by raising the suspicion that the Camelyon17 dataset contains noisy labels in the border area of annotated tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The inclusion of the likely noisy labels from the dataset immediately leads to a drop in classifier performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Practitioners should therefore not only concern themselves with the possibility of domain shift but should also put a large focus on extracting pure ground truth labels, dropping labels that are potentially misclassified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In our experiments, the ensemble approaches and especially TTA showed increased resistance to label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Further studies could extend our experiments to a broader range of datasets, as our scope is limited by the availability of tile-level annotated histopathological data in the Camelyon datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Moreover, the evaluation of uncertainty on slide level remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Additional methods could be compared, for example, integrating multi-head ensembles [Linmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2020] or deterministic uncertainty methods [Postels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022], which offer a sampling-free and fast alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our label noise experiments could be extended to more methods and more realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 6 Conclusion Deployment of AI-based diagnostic systems in the safety-critical area of histopathology demand uncertainty-aware machine learning algorithms, which generate trust in the model’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' To this end, we compared multiple uncertainty estimation methods and uncertainty metrics across domain shift and label noise scenarios in their performance, calibration and ability to detect mispredictions in the histopathological setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our results show that on in-domain data, ensembles and TTA are well-performing methods, while under domain shift the relative order and gain of methods is harder to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Existing methods are well-capable to detect mispredictions and reject inputs they are unlikely to classify correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Furthermore, ensembles generally are better calibrated than their singular counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As label noise can be a large problem in medical data, practitioners should put great care into identifying potentially mislabeled inputs, as they can lead to large performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' In terms of robustness to label noise, ensembles and TTA also perform best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' When comparing the uncertainty metrics of confidence, variance and entropy, we found no significant difference between them and suggest using the predictive confidence, as it is easy to implement and a reliable metric to detect mispredictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The ensemble performed the most consistent over all our experiments, TTA can definitely be recommended in addition to ensembling or even by itself, as it is compute-efficient and easy to implement without requiring architectural changes or retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' MCDO performed worse than TTA, SVI or the ResNet ensemble on most measures, only providing good results in terms of predictive calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' SVI, on the other hand, could compete with TTA in some scenarios, but is harder to implement and train and in our experiments lead to a higher variability between results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' While no single benchmark can give all-encompassing results and insights, we hope that our evaluation gives guidance for the utilization of uncertainty methods in the area of histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Our published code is designed to be easily reproducible and extendable to further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Acknowledgments The research is funded by the Ministerium für Soziales und Integration, Baden Württemberg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' 11 Benchmarking Uncertainty in Histopathology MEHRTENS ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=', 2022 References Andre Esteva, Brett Kuprel, Roberto A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Novoa, Justin Ko, Susan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Swetter, Helen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Blau, and Sebastian Thrun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Dermatologist-level classification of skin cancer with deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Nature, 542(7639):115–118, February 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' ISSN 1476-4687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1038/nature21056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Haenssle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Fink, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Schneiderbauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Toberer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Buhl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Blum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Kalloo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Ben Hadj Hassen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Thomas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Enk, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Uhlmann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Reader study level-I and level-II Groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Christina Alt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Monika Arenbergerova,' metadata={'source': 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+page_content=' Naira Braghiroli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Ralph Braun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Kristina Buder-Bakhaya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Timo Buhl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Horacio Cabo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Leo Cabrijan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Naciye Cevic,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' ISBN 978-3-030-59710-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='1007/978-3-030-59710-8_80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Non-Tumor ratio for all lesion-level annotated slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' Slides are sorted by center with 10 slides belonging to each center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' This means slides with index [21, 30] belong to OOD center 2 and slides with index [41, 50] belong to OOD center 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' All other slides are part of the in-distribution dataset consisting of center 0, 1 and 3 which use the same slide scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' C Effects of data augmentations on network calibration Basic augmentations Strong augmentations ID centers OOD (center 2) OOD (center 4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='25 ECE Method ResNet Temp Scaling ResNet Ensemble MCDO MCDO Ensemble ID centers OOD (center 2) OOD (center 4) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content='16 ECE Method ResNet Temp Scaling ResNet Ensemble MCDO MCDO Ensemble Table 4: Expected calibration error between data splits for training with two different settings of augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' For basic augmentations, we see increasing calibration error accross methods on both OOD centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' The plot on the right represents the augmentations used in the main part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfH_tx/content/2301.01054v1.pdf'} +page_content=' As uncertainty methods we include MCDO and Deep Ensembles and we additionally include Temperature Scaling as another method to 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Traffic Law Documents +Vuong T. Pham1,2,4 +a, Hien D. Nguyen3,4 +b,*, Thinh Le3,4, Binh Nguyen2,4 +c, Hung Q. Ngo5 +d +1Faculty of Information Technology, Sai Gon University, Ho Chi Minh City, Vietnam +2Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam +3University of Information Technology, Ho Chi Minh City, Vietnam +4Vietnam National University, Ho Chi Minh City, Vietnam +4Technological University Dublin, Dublin, Ireland +vuong.pham@sgu.edu.vn, hiennd@uit.edu.vn, 19520285@gm.uit.edu.vn, ngtbinh@hcmus.edu.vn, hung.ngo@tudublin.ie +Keywords: +knowledge base, searching system, traffic law, law on road traffic, legal document +Abstract: +In this paper, an ontology-based approach is used to organize the knowledge base of legal documents in road +traffic law. This knowledge model is built by the improvement of ontology Rela-model. In addition, several +searching problems on traffic law are proposed and solved based on the legal knowledge base. The intelligent +search system on Vietnam road traffic law is constructed by applying the method. The searching system can +help users to find concepts and definitions in road traffic law. Moreover, it can also determine penalties and +fines for violations in the traffic. The experiment results show that the system is efficient for users' typical +searching and is emerging for usage in the real-world. +1. INTRODUCTION +Nowadays, transportation is a need for everyone. +Almost every adult has a vehicle - the traffic is +increasingly complicated, especially road traffic. In +Vietnam, there are more than three million traffic law +violations, with more than 14,500 traffic accidents in +2020 (National Traffic Safety Committee, 2020). +Some cases have resulted in injuries or deaths. The +reason for those cases is that people have low +awareness of the rules of traffic law. +Ontology is an effective approach to representing +knowledge (Jakus et al., 2013). This model has been +used to organize knowledge in education and +healthcare (Do et al., 2018). Moreover, several +studies adopt ontologies to represent the knowledge + +* Corresponding author +a + https://orcid.org/0000-0002-3879-9677 +b + https://orcid.org/0000-0002-8527-0602 +c + https://orcid.org/0000-0001-5249-9702 +d + https://orcid.org/0000-0001-8246-8392 +d + https://orcid.org/0000-0001-5249-9702 +d + https://orcid.org/0000-0001-8246-8392 +of legal documents, while other studies use ontology +to organize legal knowledge (Valente and Breuker, +1992, Fawei et al., 2019). However, they did not +mention the traffic law for searching its content and +determining penalties for violations. +This paper proposes a method for building the +knowledge base for Vietnam road traffic law +(Vietnam National Assembly, 2008, Vietnam +Government 2019). This method is applied to +construct a search system in this law. The designed +system supports users in finding the content of the law +related to their queries, and it can determine penalties +for violations in road traffic via this law. In addition, +the system helps to raise people's awareness about +traffic law. + +iDiDThe primary value of the designed system is the +ability to search for penalties and fines for road traffic +offenses based on the keywords of the inputted query. +Therefore, the system's knowledge base is organized +as a relational ontology, which includes concepts, +entities, their relations, and the rules of Vietnam Law +on road traffic. In order to do that, the knowledge +domain about road traffic law is collected and +classified into knowledge components: concepts, +relations, and rules. +The following section presents related works for +constructing relational ontology, especially in the law +domain, and several search systems on legal +documents. Section 3 proposed an improved model of +Rela-model to represent the knowledge of the road +traffic code in Vietnam. Section 4 builds an +architecture and searching problems of an intelligent +querying system on Vietnam traffic code. The +designed system can support finding the content of +law related to the query and penalties for road traffic +offenses. The last section concludes the results of this +paper and gives some future works. +2. RELATED WORK +There are many studies to organize legal +knowledge. For example, Valente and Breuker (1992) +stated three approaches for the legal knowledge base: +the logic approach, the case-based approach, and the +pragmatic approach. Those approaches are used to +build legal ontologies and documents for data- +retrieving systems (Sator et al., 2011). +Ontology LIDO for Legal Informatics Document +is built based on the standard CEN Metalex (Sartor et +al., 2019). It represents legal actions that affect the +document, the legal temporal events, the structure of +the legal resource, and the semantic structure of +organization of legal documentss. +Ngo et al. (2021) proposed a method of data +augmentation based on legal domain knowledge for +the legal textual entailment. This method is used to +design a system for Vietnamese legal text processing. +Nguyen et al. (2022c) also proposed a training data +augmentation +procedure and an unsupervised +embedding learning method to retrieve the legal +document. However, those proposed methods only +show the articles of a specified query and does not use +legal knowledge to explain its results clearly. +Pham et al. (2019) built an ontology-L for +representing the Law of Public Investment and +designed a consultant system for estimating the costs +of a project based on this law. In addition, an +intelligent chatbot was designed to tutor some +administrative procedures in printing licensing based +on the ontology Rela-Ops model (Nguyen et al., +2020a). However, those methods are challenging to +apply in searching the content of a law document +related to the working domain. +There are some legal search systems in Vietnam, +such as the National Database of Legal Documents +(2022) of the Ministry of Justice and law library +(2022). However, these systems generally only allow +users to search for documents or entities with +keywords. However, they cannot help users find a +deeper search for legal documents in the real world. +For example, in traffic law, users need to search for +penalties and fines for a violation based on rules in +the legal document. Therefore, the current systems +are not suitable for supporting users in practice. +This study tends to build an intelligent search +system based on the ontology of the Vietnam road +traffic code. This ontology can be used to represent +the content of the law code and to deduce based on +the inference rules extracted from the code. +3. + KNOWLEDGE BASE OF +VIETNAMESE TRAFFIC LAW +3.1 The structure of the Vietnamese +law on the road traffic +This section gives more details about the structure +of Vietnamese law on road traffic and the knowledge +model of the system. Through Vietnam National +Assembly (2015), the system of legal documents in +Vietnam has the following levels: +1. The highest validity is Constitution; +2. Codes/Laws and resolutions of National +Assembly; +3. Sub-law documents for instructing the detail of +the law established by National Assembly. +In general, a law document has a structure with +three parts: heading, content, and ending. The +heading shows the national name, the crest, number, +and sign of the document, enact place and date, type +and name of the document, and the basis of the +document. The content is a list of parts, chapters, +articles, clauses, and points. The ending is the signing +of the person that implements the document. +Inside the content, part is the highest level, then, +in order are chapters, sections, articles, clauses and +points. Through (Vietnam Ministry of Justice, 2011), +based on the type of document, there will be different +structures, for example some documents have + +chapters, articles, clauses, and points but there is no +section. Each part, section, or chapter defines a +different factor. Below chapter are articles and +clauses which are used to define concepts, principles, +penalties, or regulations. If a clause needs more than +a sentence to define it, there will be several points in +addition to it. +Concepts in legal documents have two parts, +concept names and their definitions. For the offences, +each principle, penalty, or regulation which are +defined in articles and clauses of the legal document, +they always have the subject (the person or +organization that participate or engage in the event), +a fact (or action) and penalties if there is any. +In particular, the Vietnamese traffic law has the +same structure as stated. Two legal documents +currently implement and have most effect in the +social are: Law on road traffic (Vietnam National +Assembly, 2008) which prescribes interpretation of +concepts, road traffic rules, regulations for vehicles +and +users +on +the +road traffic; Decree +of +Administrative of penalties for road traffic offences +and rail transport offences (Vietnam Government, +2019) (known as Decree 100) which states penalties +and fines for administrative violations of road traffic. +In addition, there is National Technical Regulation on +Traffic Signs and Signals (Vietnam Ministry of +Transport, 2019) to define and describe the road +traffic signs. +3.2 Knowledge model for road +traffic law +Ontology Rela-model is a useful ontology +representing the knowledge of relations. This model +includes three components about concepts, relations +between concepts (Nguyen et al., 2015). It is effective +to represent knowledge domains in education, +consultant the finance method based on the +investment +law. +Rela-model +includes +three +components which are used to represent concepts, +relations between concepts and inference rules of the +knowledge domain. +For representing the knowledge of a legal +document, Rela-model has been improved the +structure of its concept-component being suitable the +legal domain (Nguyen et al., 2022a). The knowledge +model for Vietnamese road traffic law is based on the +concepts or entities and their relations. Each relation +of them defines an action or event of road traffic. +Based on those relations and rules of law on road +traffic, the issues about retrieving the information of +offences and their penalties have been also proposed. +Definition 2.1: The knowledge model for +representing the legal domain of road traffic is +improved from ontology Rela-model, named Traffic- +Law model. This model consists of three components +as follows: +(C, R, Rules) +In which, C is the set of concepts or entities of road +traffic law, R is the set of relations between +concepts/facts, Rules represent the inference rules to +specify the relation between concepts or determine +offences and their penalties. The structure of Traffic- +Law model is summarized as Figure 1. + +Figure 1. The Traffic-Law model. + +Concepts +RULES +Relation +Rules representthe +The set of +inferencerulesto +specifytherelation +relations +between +betweenconceptsor +concepts/facts +Properties +determine offences and +their penalties. +Traffic-Law Model +Keywords +(C,R,RULES) +Concept +The set of +Keywords +concepts or +entities ofroad +Attributes +trafficlawSet C is the set of concepts and entities in road +traffic law. There are three kinds of concepts in C: +users and vehicles of road traffic; traffic signs and +signals; road infrastructure. Based on those kinds, +each concept c  C has the structure: +(Name, Meaning, Attrs, Keywords) +where, each element has the type and meaning for +specifying the corresponding concept as Table 1: +Table 1. Structure of a concept +Element +Type +Meaning +Name +Text +Name of the concepts +Meaning +Text +Meaning of the concepts. +Attrs +Dict +List of attributes of the +concepts. +Keywords +Set +Set +of +keywords +determined or related to the +concepts. +Example 1: The concepts “Electric motorcycle” +in (Vietnam Government, 2019) is described: +Element +Content +Name +Electric motorcycle +Meaning +“a two-wheel vehicle operated by +an electric engine with power not +exceeding 4 kW and maximum +speed not exceeding 50 km/h” +Attrs +Attrs = [kind, type, legal] + kind: road traffic vehicle + type: two-wheel vehicle + legal: [Article 3, Clause 1, Point +d, Decree No. 100/2019/ND-CP] +Keywords +Motorcycle; electric; two-wheel +vehicle +Set R is the set of relations between concepts in +set C. These relations determine a specific fact or an +action of the road traffic code. Each relation r  R has +the structure: +(Name, Conc, Meaning, Prop,Keywords) +where, each element has the type and meaning for +specifying the corresponding relation as Table 2: +Table 2. Structure of a relation +Element +Type +Meaning +Name +Text +Name of the relation. +Conc +List +List of parameters as concepts +of the relation. +Meaning +Text +Meaning of the relation. +Prop +Set +Set +of +properties +of +the +relation. +This +study +only +mentions two main properties +on a binary relation: transitive +and symmetric. +Keywords +Set +Keywords of the relation. +Example 2: The relation “comply” of two concepts +“car” (or “car-like vehicles”) and “traffic light”, +denoted comply (car, traffic light), means “Operators +of car and car-like vehicles failed to comply with the +traffic lights”. Its keywords are “comply”, “over”. +Set Rules is a set of inference rules. Those rules +deduce relations between concepts or determine +offences based on road traffic law. Each rule r  +Rules has the form +u(r)  v(r) +where, u(r) is the hypothesis facts of rule r and v(r) is +the result of rule r. +The Rules-set is classified two kinds of rules: +Rules = Ruleinfer  Ruleoffence +In which, Ruleinfer is the set of rules inferring the +relation between concepts, and Ruleoffence is the set of +rules determining offences and penalties. +3.3 Some problems for searching on +traffic law +Using the improved Rela-model, the knowledge +base for road traffic law has been organized. Based +on this knowledge base, the problems for searching +on the law document are studied. There are two +issues for searching on law, which are searching for +the concepts or definition of the law, especially the +law explanation, and determining offences and their +penalties and fines through the law document. To do +this, two searching problems need to be solved for +designing the intelligent searching system on the law +document: +Definition 2: The searching problems of an +intelligent searching system Traffic-Law model are: + Problem 1: Extracting the keywords from the +inputted query to search the concepts and +relations in the legal knowledge base related +to the keywords. + Problem 2: Retrieve the knowledge from the +knowledge base matching extracted concepts +and relations. +For solving Problem 1, the inputted query needs +to be classified. The input can be classified into two +kinds: query about meaning of a concept (“what is?”) +and query about the penalties & fines of an offence +(“how much”, “penalty”, “fines”). After that, from the +kind of the query, its main keywords are extracted. In +addition, some similar words for extracted keywords +are also achieved. The similar keywords can be +collected from legal document sources, experts (as +lawyers or legal lecturers), or from dictionaries. With +extracted keywords and determined similar words, +concepts related to those keywords are determined by + +using rules in Ruleinfer. The process also finds +inference rules used to deduce concepts and their +relations. +Algorithm 3.1: Given a law document d which +is already represented using ontology based on +Traffic-Law model. +Input: The knowledge base K = (C, R, Rules) +as Traffic-Law model. + Query q. +Output: A set of keywords, relations, and rules +retrieved from query q and knowledge K +Algorithm: +Step 1: Classify the query using Vietnamese NLP +toolkit +Step 2: Extract keywords from the query q and +find similarly words based on the knowledge +base K. +W :=keywords(q) +Step 3: Classify the kind of query based keywords +in W. +Step 4: Expands W with similar keywords +collected from legal sources. +Step 5: +G :={} // Set of concepts +P :={} // Set of rules + For each keyword w ∈ W do + Using Ruleinfer to search concepts and rules +related to w. + From found concepts, determine required +keywords and add them to G. + Add rules to P if not exists +Step 6: Return (G, P) are results of found +keywords and rules. +For solving Problem 2, after identifying the +concepts and relations, the article of legal documents +that states the offence is found by using rules in +Ruleoffence. Then, the information, penalties, and fines +of it are retrieved through the specified content of law +in the knowledge base. The process for solving this +problem is as follows: +Given the knowledge base K of road traffic law in +legal documents as Traffic-Law model. This +algorithm will determine the information, penalties, +or fines of an inputted query q. +Algorithm 3.2: Given a law document d which is +already represented using ontology based on Traffic- +Law model +Input: The knowledge base K = (C, R, Rules) as +Traffic-Law model, and a query q. +Output: Information, penalties, and fines of road +traffic offence for query q. + + +Algorithm: +Step 1: Retrieve set of keywords G from query q +based on Algorithm 3.1 + Concept := {c  C | c related to keyword in G} +Step 2: +Knowledge :={} +For each concept c  Concept do: +  Using rules in Ruleoffence to find the offence +in the knowledge base K. +  Retrieve the information, penalties, and +fines of the determined offence from the +specified law document. +  Update the results into Knowledge. +Step 3: Return Knowledge. +4. THE SEARCHING SYSTEM OF +VIETNAMESE LAW ON ROAD +TRAFFIC +4.1 Requirements of a searching +system on legal documents +The intelligent searching system on legal +documents needs to be supported the understanding +of users about the legal domain. In road traffic law, +moreover, the ability for solving of necessary issues +of the searching system, this system has some criteria +of intelligent software evaluation in searching +(Nguyen et al., 2020b, Giakoumakis and Xylomenos, +1996): +o Portability: This is the level of difficulty to +work with the same project with different machines. +o Installation: The requirements of software, +hardware for the simulator, and how straightforward +is the installation in a supported system. +o Usability: this criterion shows whether the +content is suitable and detailed with the current law +domain and whether it is updated and easily to use in +the practice. +o Understandability: this is one of the most +important characteristics of intelligent law searching +software quality. This system has to help users +understand the law content in legal documents. It can +influence users’ feelings about software and +reliability of software evolution in reuse or +maintenance. +Besides, the process for building this system is +worked through the constructing of a knowledge- +based system (Nguyen et al., 2022b). At first, the +databse of traffic regulations will be collected, and +orgnaized by Traffic-Law model as the knowledge +base of this system. After that, the searching +mechanism is designed through problems on traffic + +law searching and their alogrithms. Finally, the user +interface and testing of this system will be processed. +4.2 The dataset of traffic regulations +The traffic regulation dataset is a combination of +2 documents: +1. Vietnam National Assembly, Law on Road +Traffic (known as 23/2008/QH12). +2. The Decree of Administrative of penalties +for road traffic offences and rail transport +offences (Vietnam Government, 2019), +abbreviated as Decree 100. +From both documents, there are 175 articles +collected. The general structure of these documents +is: Chapter – Section – Article – Clause – Point. +Traffic-Law model is used as an ontology to represent +this knowledge. +By +default, +questions +about +Vietnamese +transportation are classified into many intents. There +intents include but not limited to: + Querying about concepts: These queries ask +definitions of concepts in the law. The system extracts +the apporiate article for the required concept. + Querying about penalties: These queries ask +about the penalty or fines for a traffic violation, such +as running the red light, driving contrariwise, etc. + Querying about procedures: The system give a +proceduce in traffic law, such as fine payment +procedure, the procedure for issuing driving licenses, +etc. + Querying about signs: This function support +user to retrieve the information of an inputted sign. +This function related to image processing. +However, because the scope of this study, only the +kinds of querying about concepts and penalties are +focused in this paper. There are 160 practical +collected queries related to road traffic regulation. +These queries will be augmented and used for training +query intent classification in Problem 1. +Table 3: Query Classification +Class +Meaning +Quantity +Concept +Require identifying the +meaning of a concept. +54 +Penalties +Require identifying the +fine of an offence. +83 +Out of +scope +Queries +that +do +not +belong to above kinds. +23 +Total +160 +4.3 The architecture of the searching +system on the traffic law +The architecture of the search system on traffic +law is presented in Figure 2. The system consists of +the user interface, the knowledge base, and the search +engine. +The knowledge of the road traffic codes is +collected from (Vietnam Government, 2019, Vietnam +National Assembly, 2008). These facts and entities of +those documents are organized as a knowledge base +by the improved ontology Rela-model and stored +inside a graph database. The similar words are +manually established via the collection of intellectual +experts and their experiences. +When a user inputs the query, the search engine +will execute the extract keywords tasks by Problem 1, +which are classifying the query, checking typo, +removing stop words, checking synonyms, and +checking equivalent keywords, to generate the query +values. From the extracted keywords, the similar +words will be determined through the knowledge base +Traffic-Law model. Those are used to search the +necessary knowledge by using inference rules of the +knowledge. In addition, their penalties and fines are +also retrieved by Problem 2. The result will be ranked +by the search engine before showing it in the user +interface. +4.4 Testing Results +Based on the knowledge base that has been +organized in Section 3 and the proposed architecture +in Section 4.2, an intelligent searching system on +Vietnam Road traffic law is designed. This section +presents some testing results of the system through +some kinds of inputted queries. +Example 3: The inputted query q1 = “What is +motorcycle?” +The system will extract keywords from the query +q1: “What is”, “motorcycle”. From that, it returns the +results as follows: +“Motorcycle means a motor vehicle that has two +or three wheels with a cylinder capacity of 50 cm3 or +higher, maximum speed over 50 km/h, and net weight +not exceeding 400 kg.” + + + +Figure 2. The architecture of an intelligent searching system on the Vietnam road traffic. +The word “what is” is used to classify the query +into the kind of declaring the meaning of a concept. +The keyword “motorcycle” helps to find the concept. +The result is retrieved from Article 3, Clause 3.31 of +National Technical Regulation on Traffic Signs and +Signals (Vietnam Ministry of Transport, 2019). +Example 4: The inputted query q2 = “The fines of +operator of motorbike driver who does not wear +helmet” + The keywords of the query q2 are “fines”, “not +wear”, helmet”, “operator of motorbike”. The word +“fines” is used to classify the query into stating +penalties and fines of offences. The word “operator +of motorbike” consists of “motorbike” that is similar +to the word “motorcycle”. The word “helmet” is in the +keywords of the concept “motorcycle helmet”. +Hence, the concepts of the query q2 are “operator of +motorcycle” and “motorcycle helmet”. The relational +keyword is “not wear”. With the concepts and +relation, the rules were used to match them and find +the result. +The result is returned: +“Through article 6, Decree 100/2019/ND-CP: +Penalties imposed upon operators of mopeds and +motorcycles (including electric motorcycles) and the +like violating road traffic rules. +2. A fine ranging from VND 200,000 to +VND 300,000 shall be imposed upon a vehicle +operator who commits any of the following +violations: +i) The operator or the passenger on the vehicle +does not wear a motorcycle helmet or does not wear +it properly;” +The designed system can do some common +searching on road traffic law. It is effective in finding +usual penalties and fines from road traffic law. This +system was tested on a set of 95 queries about the road +traffic codes. The results are shown in Table 4: +Table 4. Results for testing of queries +Kind +Quantity +Correct +Rate +Queries about +concepts / +definitions +54 +42 +78% +Queries about +penalties and +fines +83 +61 +73% +Total +137 +103 +75% +5. CONCLUSION AND FUTURE +WORK +This paper proposed an ontology-based model for +representing legal knowledge in the Vietnam road +traffic codes. This model is improved based on +ontology Rela-model in the structure of concepts, +relations, and inference rules. Through the designed +knowledge base, several searching issues on the +Vietnam road traffic codes are proposed, such as +extracting keywords and inferring the matched result +for inputted query. Moreover, the architecture of an +intelligent search system on road traffic law has been +constructed. This system can do several common +search queries, such as finding concepts/definitions in +the law and determining penalties for violations in the + +KnowledgeCollection +UserInterface +Use +Knowledgesources, +Lawyers,Experts +documents +Display +Provides +Extracts +Create +Events, +Keywords, +Decoderesult +entities, +equivalents +concepts +keywords +Traffic queries +SearchingResult +KnowledgeBase +Sendquery +Returnresult +SearchEngine +Traffic-LawModel +(C,R,RULES) +Importingkeywords +Classify,extract +KEYWORD +keywords,find +equivalent +Ranking +keywords +Result +Query +Knowiedgeresult +Results +Rankingroad traffic. At the moment, most knowledge is +collected by manual collection method. The next +work is the improvement of the collection method +within by using an automatic method. +In the future, the system can be involved other +legal aspects such as commercial law, civil law, etc. +Further, the system can be used to provide an e- +learning system for legal aspects. The abilities to use +AI to identify entities and concepts from an image or +use voice recognition to identify searching input are +also features considered to add more to the system. +ACKNOWLEDGMENT +This research was supported by The VNUHCM- +University of Information Technology's Scientific +Research Support Fund. +REFERENCES +Do, N., Nguyen, H., Selamat, A. 2018. Knowledge-Based +model +of +Expert +Systems +using +Rela- +model. International Journal of Software Engineering +and Knowledge Engineering 28(8), 1047 – 1090. +Fawei, B., Pan, J.Z, Kollingbaum, M., Wyner, A.Z. 2019. +A Semi-automated Ontology Construction for Legal +Question Answering. New Generation Computing +37(4), 453-478, 2019. +Giakoumakis, E.A., Xylomenos, G. 1996. Evaluation and +Selection +criteria +for +software +requirements +specification standards. Software Engineer Journal, +11(5), 307-319. +Jakus, G., et al. 2013. Concepts, Ontologies, and +Knowledge Representation. Springer Nature. +Law library. (2022). https://thuvienphapluat.vn/ +National database of Legal documents. 2022. +http://vbpl.vn/botuphap/Pages/Home.aspx +National Traffic Safety Committee. 2020. Final report the +traffic accidents in 2020. Vietnam Government. +Ngo, H., Nguyen, T., Nguyen, D., Pham, M. 2021. +AimeLaw at ALQAC 2021: Enriching Neural Network +Models with Legal-Domain Knowledge. In KSE 2021, +13th International Conference on Knowledge and +Systems Engineering, Nov. 2021. IEEE. +Nguyen, H.D., et al. 2015. A Mathematical Approach for +Representation Knowledge about Relations and Its +Application. In KSE 2015, 7th IEEE International +Conference on Knowledge and Systems Engineering, +Oct. 2015. IEEE. +Nguyen, H., Tran, D., Pham, H., Pham, V. 2020a. Design +an intelligent system to automatically tutor the method +for solving problems. International Journal of +Integrated Engineering 12(7), 211 – 223. +Nguyen, H., Do, N., Tran, N., Pham, H., Pham, V. 2020b. +Some criteria of the Knowledge Representation method +for an Intelligent Problem Solver in STEM education. +Applied +Computational +Intelligence +and +Soft +Computing 2020, Article ID 9834218. +Nguyen, T., Nguyen, H.D., Pham, V.T., et al. 2022a. Legal- +Onto: An Ontology-based model for Representing the +Knowledge of a Legal Document. In ENASE 2022, 17th +International Conference on Evaluation of Novel +Approaches to Software Engineering, April 2022. +Scitepress. +Nguyen, H. D., Do, N. V., Pham, V. T. 2022b. A +methodology for designing knowledge-based systems +and applications. In Applications of Computational +Intelligence in Multi-Disciplinary Research. Academic +Press, Elsevier. +Nguyen, D., et al. 2022c. An Unsupervised Learning +Method to improve Legal Document Retrieval task at +ALQAC 2022. In KSE 2022, 14th International +Conference on Knowledge and Systems Engineering, +Oct. 2022. IEEE. +Pham. H., Do, N., Nguyen, H. 2019. A Consulting System +for Estimating Costs of an Information Technology +Hardware Project based on Law of Public Investment. +In ICSSE 2019, 7th IEEE International Conference on +System Science and Engineering, July 2019. IEEE. +Valente, A., Breuker, J. 1992. A Model-Based Approach to +Legal Knowledge Engineering. In: Legal Knowledge +Based Systems: Information Technology & Law, +Grütters, +et +al. +(eds), +JURIX'92, +Koninklijke +Vermande, Lelystad, NL, 1992. +Sartor, G., Casanovas, P., Biasiotti, M., Fernandéz-Barrera, +M. 2011. Approaches to Legal Ontologies. Springer +Dordrecht Heidelberg, London, New York. +Vietnam National Assembly. 2008. Law on Road Traffic. +Law No. 23/2008/QH12. +Vietnam National Assembly. 2015. Law on Promulgation +of Legislative Documents. No. 80/2015/QH13. +Vietnam Government. 2019. Decree on Administrative +penalties for road traffic and rail transport offences. +No. 100/2019/ND-CP. +Vietnam Ministry of Justice. 2011. Circular of the Minister +about the formats, and techniques for legal documents +of the Government, the Prime Minister, Ministers, and +joint legal documents. No. 25/2011/TT-BTP. +Vietnam Ministry of Transport. 2019. National Technical +Regulation on Traffic Signs and Signals. QCVN +41:2019/BGTVT, 2019. + diff --git a/JdFIT4oBgHgl3EQfZSuO/content/tmp_files/load_file.txt b/JdFIT4oBgHgl3EQfZSuO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93ff19801995100c2ae2b87a3c0094466ffd7ef6 --- /dev/null +++ b/JdFIT4oBgHgl3EQfZSuO/content/tmp_files/load_file.txt @@ -0,0 +1,419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf,len=418 +page_content='Ontology-based Solution for Building an Intelligent Searching System on Traffic Law Documents Vuong T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Pham1,2,4 a, Hien D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Nguyen3,4 b,*, Thinh Le3,4, Binh Nguyen2,4 c, Hung Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Ngo5 d 1Faculty of Information Technology, Sai Gon University, Ho Chi Minh City, Vietnam 2Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam 3University of Information Technology, Ho Chi Minh City, Vietnam 4Vietnam National University, Ho Chi Minh City, Vietnam 4Technological University Dublin, Dublin, Ireland vuong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='pham@sgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='vn, hiennd@uit.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='ngo@tudublin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='ie Keywords: knowledge base, searching system, traffic law, law on road traffic, legal document Abstract: In this paper, an ontology-based approach is used to organize the knowledge base of legal documents in road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This knowledge model is built by the improvement of ontology Rela-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In addition, several searching problems on traffic law are proposed and solved based on the legal knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The intelligent search system on Vietnam road traffic law is constructed by applying the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The searching system can help users to find concepts and definitions in road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Moreover, it can also determine penalties and fines for violations in the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=" The experiment results show that the system is efficient for users' typical searching and is emerging for usage in the real-world." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' INTRODUCTION Nowadays, transportation is a need for everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Almost every adult has a vehicle - the traffic is increasingly complicated, especially road traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In Vietnam, there are more than three million traffic law violations, with more than 14,500 traffic accidents in 2020 (National Traffic Safety Committee, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Some cases have resulted in injuries or deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The reason for those cases is that people have low awareness of the rules of traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Ontology is an effective approach to representing knowledge (Jakus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This model has been used to organize knowledge in education and healthcare (Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Moreover, several studies adopt ontologies to represent the knowledge Corresponding author a https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0002-3879-9677 b https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0002-8527-0602 c https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0001-5249-9702 d https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0001-8246-8392 d https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0001-5249-9702 d https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='org/0000-0001-8246-8392 of legal documents, while other studies use ontology to organize legal knowledge (Valente and Breuker, 1992, Fawei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, they did not mention the traffic law for searching its content and determining penalties for violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This paper proposes a method for building the knowledge base for Vietnam road traffic law (Vietnam National Assembly, 2008, Vietnam Government 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This method is applied to construct a search system in this law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The designed system supports users in finding the content of the law related to their queries, and it can determine penalties for violations in road traffic via this law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=" In addition, the system helps to raise people's awareness about traffic law." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' iDiDThe primary value of the designed system is the ability to search for penalties and fines for road traffic offenses based on the keywords of the inputted query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=" Therefore, the system's knowledge base is organized as a relational ontology, which includes concepts, entities, their relations, and the rules of Vietnam Law on road traffic." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In order to do that, the knowledge domain about road traffic law is collected and classified into knowledge components: concepts, relations, and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The following section presents related works for constructing relational ontology, especially in the law domain, and several search systems on legal documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Section 3 proposed an improved model of Rela-model to represent the knowledge of the road traffic code in Vietnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Section 4 builds an architecture and searching problems of an intelligent querying system on Vietnam traffic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The designed system can support finding the content of law related to the query and penalties for road traffic offenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The last section concludes the results of this paper and gives some future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' RELATED WORK There are many studies to organize legal knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For example, Valente and Breuker (1992) stated three approaches for the legal knowledge base: the logic approach, the case-based approach, and the pragmatic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Those approaches are used to build legal ontologies and documents for data- retrieving systems (Sator et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Ontology LIDO for Legal Informatics Document is built based on the standard CEN Metalex (Sartor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' It represents legal actions that affect the document, the legal temporal events, the structure of the legal resource, and the semantic structure of organization of legal documentss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' (2021) proposed a method of data augmentation based on legal domain knowledge for the legal textual entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This method is used to design a system for Vietnamese legal text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' (2022c) also proposed a training data augmentation procedure and an unsupervised embedding learning method to retrieve the legal document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, those proposed methods only show the articles of a specified query and does not use legal knowledge to explain its results clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' (2019) built an ontology-L for representing the Law of Public Investment and designed a consultant system for estimating the costs of a project based on this law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In addition, an intelligent chatbot was designed to tutor some administrative procedures in printing licensing based on the ontology Rela-Ops model (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, those methods are challenging to apply in searching the content of a law document related to the working domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' There are some legal search systems in Vietnam, such as the National Database of Legal Documents (2022) of the Ministry of Justice and law library (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, these systems generally only allow users to search for documents or entities with keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, they cannot help users find a deeper search for legal documents in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For example, in traffic law, users need to search for penalties and fines for a violation based on rules in the legal document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Therefore, the current systems are not suitable for supporting users in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This study tends to build an intelligent search system based on the ontology of the Vietnam road traffic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This ontology can be used to represent the content of the law code and to deduce based on the inference rules extracted from the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' KNOWLEDGE BASE OF VIETNAMESE TRAFFIC LAW 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='1 The structure of the Vietnamese law on the road traffic This section gives more details about the structure of Vietnamese law on road traffic and the knowledge model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Through Vietnam National Assembly (2015), the system of legal documents in Vietnam has the following levels: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The highest validity is Constitution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Codes/Laws and resolutions of National Assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Sub-law documents for instructing the detail of the law established by National Assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In general, a law document has a structure with three parts: heading, content, and ending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The heading shows the national name, the crest, number, and sign of the document, enact place and date, type and name of the document, and the basis of the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The content is a list of parts, chapters, articles, clauses, and points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The ending is the signing of the person that implements the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Inside the content, part is the highest level, then, in order are chapters, sections, articles, clauses and points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Through (Vietnam Ministry of Justice, 2011), based on the type of document, there will be different structures, for example some documents have chapters, articles, clauses, and points but there is no section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Each part, section, or chapter defines a different factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Below chapter are articles and clauses which are used to define concepts, principles, penalties, or regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' If a clause needs more than a sentence to define it, there will be several points in addition to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Concepts in legal documents have two parts, concept names and their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For the offences, each principle, penalty, or regulation which are defined in articles and clauses of the legal document, they always have the subject (the person or organization that participate or engage in the event), a fact (or action) and penalties if there is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In particular, the Vietnamese traffic law has the same structure as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Two legal documents currently implement and have most effect in the social are: Law on road traffic (Vietnam National Assembly, 2008) which prescribes interpretation of concepts, road traffic rules, regulations for vehicles and users on the road traffic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Decree of Administrative of penalties for road traffic offences and rail transport offences (Vietnam Government, 2019) (known as Decree 100) which states penalties and fines for administrative violations of road traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In addition, there is National Technical Regulation on Traffic Signs and Signals (Vietnam Ministry of Transport, 2019) to define and describe the road traffic signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='2 Knowledge model for road traffic law Ontology Rela-model is a useful ontology representing the knowledge of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This model includes three components about concepts, relations between concepts (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' It is effective to represent knowledge domains in education, consultant the finance method based on the investment law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Rela-model includes three components which are used to represent concepts, relations between concepts and inference rules of the knowledge domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For representing the knowledge of a legal document, Rela-model has been improved the structure of its concept-component being suitable the legal domain (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The knowledge model for Vietnamese road traffic law is based on the concepts or entities and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Each relation of them defines an action or event of road traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Based on those relations and rules of law on road traffic, the issues about retrieving the information of offences and their penalties have been also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='1: The knowledge model for representing the legal domain of road traffic is improved from ontology Rela-model, named Traffic- Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This model consists of three components as follows: (C, R, Rules) In which, C is the set of concepts or entities of road traffic law, R is the set of relations between concepts/facts, Rules represent the inference rules to specify the relation between concepts or determine offences and their penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The structure of Traffic- Law model is summarized as Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The Traffic-Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Concepts RULES Relation Rules representthe The set of inferencerulesto specifytherelation relations between betweenconceptsor concepts/facts Properties determine offences and their penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Traffic-Law Model Keywords (C,R,RULES) Concept The set of Keywords concepts or entities ofroad Attributes trafficlawSet C is the set of concepts and entities in road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' There are three kinds of concepts in C: users and vehicles of road traffic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' traffic signs and signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' road infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Based on those kinds, each concept c \uf0ce C has the structure: (Name, Meaning, Attrs, Keywords) where, each element has the type and meaning for specifying the corresponding concept as Table 1: Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Structure of a concept Element Type Meaning Name Text Name of the concepts Meaning Text Meaning of the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Attrs Dict List of attributes of the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Keywords Set Set of keywords determined or related to the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Example 1: The concepts “Electric motorcycle” in (Vietnam Government, 2019) is described: Element Content Name Electric motorcycle Meaning “a two-wheel vehicle operated by an electric engine with power not exceeding 4 kW and maximum speed not exceeding 50 km/h” Attrs Attrs = [kind, type, legal] kind: road traffic vehicle type: two-wheel vehicle legal: [Article 3, Clause 1, Point d, Decree No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 100/2019/ND-CP] Keywords Motorcycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' electric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' two-wheel vehicle Set R is the set of relations between concepts in set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' These relations determine a specific fact or an action of the road traffic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Each relation r \uf0ce R has the structure: (Name, Conc, Meaning, Prop,Keywords) where, each element has the type and meaning for specifying the corresponding relation as Table 2: Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Structure of a relation Element Type Meaning Name Text Name of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Conc List List of parameters as concepts of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Meaning Text Meaning of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Prop Set Set of properties of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This study only mentions two main properties on a binary relation: transitive and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Keywords Set Keywords of the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Example 2: The relation “comply” of two concepts “car” (or “car-like vehicles”) and “traffic light”, denoted comply (car, traffic light), means “Operators of car and car-like vehicles failed to comply with the traffic lights”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Its keywords are “comply”, “over”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Set Rules is a set of inference rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Those rules deduce relations between concepts or determine offences based on road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Each rule r \uf0ce Rules has the form u(r) \uf0ae v(r) where, u(r) is the hypothesis facts of rule r and v(r) is the result of rule r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The Rules-set is classified two kinds of rules: Rules = Ruleinfer \uf0c8 Ruleoffence In which, Ruleinfer is the set of rules inferring the relation between concepts, and Ruleoffence is the set of rules determining offences and penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='3 Some problems for searching on traffic law Using the improved Rela-model, the knowledge base for road traffic law has been organized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Based on this knowledge base, the problems for searching on the law document are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' There are two issues for searching on law, which are searching for the concepts or definition of the law, especially the law explanation, and determining offences and their penalties and fines through the law document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' To do this, two searching problems need to be solved for designing the intelligent searching system on the law document: Definition 2: The searching problems of an intelligent searching system Traffic-Law model are: Problem 1: Extracting the keywords from the inputted query to search the concepts and relations in the legal knowledge base related to the keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Problem 2: Retrieve the knowledge from the knowledge base matching extracted concepts and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For solving Problem 1, the inputted query needs to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The input can be classified into two kinds: query about meaning of a concept (“what is?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=') and query about the penalties & fines of an offence (“how much”, “penalty”, “fines”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' After that, from the kind of the query, its main keywords are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In addition, some similar words for extracted keywords are also achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The similar keywords can be collected from legal document sources, experts (as lawyers or legal lecturers), or from dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' With extracted keywords and determined similar words, concepts related to those keywords are determined by using rules in Ruleinfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The process also finds inference rules used to deduce concepts and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='1: Given a law document d which is already represented using ontology based on Traffic-Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Input: The knowledge base K = (C, R, Rules) as Traffic-Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Output: A set of keywords, relations, and rules retrieved from query q and knowledge K Algorithm: Step 1: Classify the query using Vietnamese NLP toolkit Step 2: Extract keywords from the query q and find similarly words based on the knowledge base K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' W :=keywords(q) Step 3: Classify the kind of query based keywords in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Step 4: Expands W with similar keywords collected from legal sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Step 5: G :={} // Set of concepts P :={} // Set of rules For each keyword w ∈ W do Using Ruleinfer to search concepts and rules related to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' From found concepts, determine required keywords and add them to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Add rules to P if not exists Step 6: Return (G, P) are results of found keywords and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' For solving Problem 2, after identifying the concepts and relations, the article of legal documents that states the offence is found by using rules in Ruleoffence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Then, the information, penalties, and fines of it are retrieved through the specified content of law in the knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The process for solving this problem is as follows: Given the knowledge base K of road traffic law in legal documents as Traffic-Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This algorithm will determine the information, penalties, or fines of an inputted query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='2: Given a law document d which is already represented using ontology based on Traffic- Law model Input: The knowledge base K = (C, R, Rules) as Traffic-Law model, and a query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Output: Information, penalties, and fines of road traffic offence for query q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Algorithm: Step 1: Retrieve set of keywords G from query q based on Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='1 Concept := {c \uf0ce C | c related to keyword in G} Step 2: Knowledge :={} For each concept c \uf0ce Concept do: Using rules in Ruleoffence to find the offence in the knowledge base K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Retrieve the information, penalties, and fines of the determined offence from the specified law document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Update the results into Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Step 3: Return Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' THE SEARCHING SYSTEM OF VIETNAMESE LAW ON ROAD TRAFFIC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='1 Requirements of a searching system on legal documents The intelligent searching system on legal documents needs to be supported the understanding of users about the legal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In road traffic law, moreover, the ability for solving of necessary issues of the searching system, this system has some criteria of intelligent software evaluation in searching (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2020b, Giakoumakis and Xylomenos, 1996): o Portability: This is the level of difficulty to work with the same project with different machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' o Installation: The requirements of software, hardware for the simulator, and how straightforward is the installation in a supported system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' o Usability: this criterion shows whether the content is suitable and detailed with the current law domain and whether it is updated and easily to use in the practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' o Understandability: this is one of the most important characteristics of intelligent law searching software quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This system has to help users understand the law content in legal documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' It can influence users’ feelings about software and reliability of software evolution in reuse or maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Besides, the process for building this system is worked through the constructing of a knowledge- based system (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' At first, the databse of traffic regulations will be collected, and orgnaized by Traffic-Law model as the knowledge base of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' After that, the searching mechanism is designed through problems on traffic law searching and their alogrithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Finally, the user interface and testing of this system will be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='2 The dataset of traffic regulations The traffic regulation dataset is a combination of 2 documents: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Vietnam National Assembly, Law on Road Traffic (known as 23/2008/QH12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The Decree of Administrative of penalties for road traffic offences and rail transport offences (Vietnam Government, 2019), abbreviated as Decree 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' From both documents, there are 175 articles collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The general structure of these documents is: Chapter – Section – Article – Clause – Point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Traffic-Law model is used as an ontology to represent this knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' By default, questions about Vietnamese transportation are classified into many intents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' There intents include but not limited to: Querying about concepts: These queries ask definitions of concepts in the law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The system extracts the apporiate article for the required concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Querying about penalties: These queries ask about the penalty or fines for a traffic violation, such as running the red light, driving contrariwise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Querying about procedures: The system give a proceduce in traffic law, such as fine payment procedure, the procedure for issuing driving licenses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Querying about signs: This function support user to retrieve the information of an inputted sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This function related to image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' However, because the scope of this study, only the kinds of querying about concepts and penalties are focused in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' There are 160 practical collected queries related to road traffic regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' These queries will be augmented and used for training query intent classification in Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Table 3: Query Classification Class Meaning Quantity Concept Require identifying the meaning of a concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 54 Penalties Require identifying the fine of an offence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 83 Out of scope Queries that do not belong to above kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 23 Total 160 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='3 The architecture of the searching system on the traffic law The architecture of the search system on traffic law is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The system consists of the user interface, the knowledge base, and the search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The knowledge of the road traffic codes is collected from (Vietnam Government, 2019, Vietnam National Assembly, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' These facts and entities of those documents are organized as a knowledge base by the improved ontology Rela-model and stored inside a graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The similar words are manually established via the collection of intellectual experts and their experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' When a user inputs the query, the search engine will execute the extract keywords tasks by Problem 1, which are classifying the query, checking typo, removing stop words, checking synonyms, and checking equivalent keywords, to generate the query values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' From the extracted keywords, the similar words will be determined through the knowledge base Traffic-Law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Those are used to search the necessary knowledge by using inference rules of the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In addition, their penalties and fines are also retrieved by Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The result will be ranked by the search engine before showing it in the user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='4 Testing Results Based on the knowledge base that has been organized in Section 3 and the proposed architecture in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='2, an intelligent searching system on Vietnam Road traffic law is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This section presents some testing results of the system through some kinds of inputted queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Example 3: The inputted query q1 = “What is motorcycle?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The system will extract keywords from the query q1: “What is”, “motorcycle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' From that, it returns the results as follows: “Motorcycle means a motor vehicle that has two or three wheels with a cylinder capacity of 50 cm3 or higher, maximum speed over 50 km/h, and net weight not exceeding 400 kg.” Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The architecture of an intelligent searching system on the Vietnam road traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The word “what is” is used to classify the query into the kind of declaring the meaning of a concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The keyword “motorcycle” helps to find the concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The result is retrieved from Article 3, Clause 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='31 of National Technical Regulation on Traffic Signs and Signals (Vietnam Ministry of Transport, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Example 4: The inputted query q2 = “The fines of operator of motorbike driver who does not wear helmet” The keywords of the query q2 are “fines”, “not wear”, helmet”, “operator of motorbike”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The word “fines” is used to classify the query into stating penalties and fines of offences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The word “operator of motorbike” consists of “motorbike” that is similar to the word “motorcycle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The word “helmet” is in the keywords of the concept “motorcycle helmet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Hence, the concepts of the query q2 are “operator of motorcycle” and “motorcycle helmet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The relational keyword is “not wear”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' With the concepts and relation, the rules were used to match them and find the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The result is returned: “Through article 6, Decree 100/2019/ND-CP: Penalties imposed upon operators of mopeds and motorcycles (including electric motorcycles) and the like violating road traffic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' A fine ranging from VND 200,000 to VND 300,000 shall be imposed upon a vehicle operator who commits any of the following violations: i) The operator or the passenger on the vehicle does not wear a motorcycle helmet or does not wear it properly;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The designed system can do some common searching on road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' It is effective in finding usual penalties and fines from road traffic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This system was tested on a set of 95 queries about the road traffic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The results are shown in Table 4: Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Results for testing of queries Kind Quantity Correct Rate Queries about concepts / definitions 54 42 78% Queries about penalties and fines 83 61 73% Total 137 103 75% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK This paper proposed an ontology-based model for representing legal knowledge in the Vietnam road traffic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This model is improved based on ontology Rela-model in the structure of concepts, relations, and inference rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Through the designed knowledge base, several searching issues on the Vietnam road traffic codes are proposed, such as extracting keywords and inferring the matched result for inputted query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Moreover, the architecture of an intelligent search system on road traffic law has been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' This system can do several common search queries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' such as finding concepts/definitions in the law and determining penalties for violations in the KnowledgeCollection UserInterface Use Knowledgesources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Lawyers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='Experts documents Display Provides Extracts Create Events,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Keywords,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Decoderesult entities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' equivalents concepts keywords Traffic queries SearchingResult KnowledgeBase Sendquery Returnresult SearchEngine Traffic-LawModel (C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='RULES) Importingkeywords Classify,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='extract KEYWORD keywords,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content='find equivalent Ranking keywords Result Query Knowiedgeresult Results Rankingroad traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' At the moment, most knowledge is collected by manual collection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' The next work is the improvement of the collection method within by using an automatic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' In the future, the system can be involved other legal aspects such as commercial law, civil law, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} +page_content=' Further, the system can be used to provide an e- learning system for legal aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFIT4oBgHgl3EQfZSuO/content/2301.11252v1.pdf'} 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b/KNAyT4oBgHgl3EQf6Ppp/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7a7c6097b14bb4cca12970b5b3bd94c7c1439c9f920722c0eb092c73299874f3 +size 5767213 diff --git a/KNAzT4oBgHgl3EQfIPul/content/tmp_files/2301.01059v1.pdf.txt b/KNAzT4oBgHgl3EQfIPul/content/tmp_files/2301.01059v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2dcfe476d5727ac1dc1ad641227e89bb433c64c --- /dev/null +++ b/KNAzT4oBgHgl3EQfIPul/content/tmp_files/2301.01059v1.pdf.txt @@ -0,0 +1,1236 @@ +arXiv:2301.01059v1 [math.RT] 3 Jan 2023 +A REFINED MULTIPLICATION FORMULA FOR CLUSTER +CHARACTERS +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +Abstract. We obtain a multiplication formula for cluster characters on (sta- +bly) 2-Calabi–Yau (Frobenius or) triangulated categories. This formula gen- +eralizes those known for arbitrary pairs of objects and for Auslander–Reiten +triangles. As an application, we show that for cluster algebras of acyclic types, +specialization of a cluster variable to 1 sends all cluster variables to elements of +a cluster algebra of smaller rank. We also obtain applications to the reduction +of friezes of acyclic type. +Contents +1. +Introduction +1 +2. +Refined multiplication formula: triangulated case +3 +2.1. +Recollections on 2-Calabi–Yau triangulated categories +3 +2.2. +The refined multiplication formula +6 +3. +Refined multiplication formula: Frobenius case +9 +3.1. +Recollections on Frobenius categories +9 +3.2. +The formula +11 +4. +Applications +11 +4.1. +Specialization of cluster variables in cluster algebras +11 +4.2. +Reduction of friezes +13 +4.3. +A formula for Auslander–Reiten triangles +14 +4.4. +Another restricted formula +15 +Acknowledgements +15 +References +16 +1. Introduction +The additive categorification of cluster algebras has been an important tool +in their study almost from their inception (see for instance the survey papers +[Kel09, Rei10, Ami11, Pla18]). Such a categorification is given by a category C +(usually triangulated or exact) and a cluster character sending objects of C to +Laurent polynomials in several variables so that suitable objects of C are sent to +cluster variables in a cluster algebra. The key property that a cluster character +satisfies is a multiplication formula which recovers the exchange relations in a clus- +ter algebra. Such formulas at various levels of generality have been obtained in +[CC06, CK08, CK06, GLS08, Pal08, FK10, DWZ10, Pal12, Pla11b, DG14, Rup15, +GLS18, CIEFR21] and more. +1 + +2 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +The main result of this paper is a multiplication formula generalizing most pre- +viously known ones in the following context. Let C be a small Hom-finite Krull– +Schmidt 2-Calabi–Yau triangulated category over C, together with a basic cluster +tilting object T (definitions are recalled in Section 2.1.1). Let +CCT : Obj(C) → Z[x±1 +1 , . . . , x±1 +n ] +be the corresponding cluster character (Definition 2.6). For any objects L and M +of C, let +βL,M : HomC(L, ΣM) × HomC(M, ΣL) −→ k +be the non-degenerate bifunctorial bilinear form conferring to C its 2-Calabi–Yau +structure. For an object Y , let HomC(L, ΣM)⟨Y ⟩ be the set of those morphisms ε : +L → ΣM such that, if we have a triangle +M −→ Y ′ −→ L +ε−→ ΣM, +the objects Y and Y ′ have the same index (see Definition 2.5) and for each dimension +vector e, the submodule Grassmannians Gre(HomC(T, Y )) and Gre(HomC(T, Y ′)) +have the same Euler characteristic. It is easy to check that this set is invariant under +multiplication by a non-zero scalar. For a subset X of HomC(L, ΣM), let X⟨Y ⟩ be +the intersection of X with HomC(L, ΣM)⟨Y ⟩. Let YL,M be a set of representatives +of equivalence classes for the equivalence relation defined by HomC(L, ΣM)⟨Y ⟩ = +HomC(L, ΣM)⟨Y ′⟩. +Our main result is the following refined multiplication formula. +Theorem (2.10). Let C be a small Hom-finite Krull–Schmidt 2-Calabi–Yau trian- +gulated category over C with constructible cones (see Section 2.1.3) together with a +basic cluster tilting object T . Let L and M be objects of C such that HomC(L, ΣM) +is non-zero. Finally, let V be a non-zero vector subspace of HomC(L, ΣM). Then +χ(PV )CCT (L)CCT (M) = +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ) + +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ), +where R = P HomC(M, ΣL) \ PKer βL,M(V, ?). +If V is the full space, then this formula specializes to the one proved in [Pal12, +Theorem 1.1]. Our main result also has a counterpart for exact categories. +Theorem (3.6). Let E be an Ext-finite 2-Calabi–Yau Frobenius category with a +cluster tilting object T . Assume that the triangulated category E has constructible +cones. Let L and M be two objects of E such that Ext1 +E(L, M) is non-zero, and let +V be a non-zero vector subspace of Ext1 +E(L, M). Then +χ(PV )CCT (L)CCT (M) = +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ) + +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ). +This generalizes a result of [FK10]. We expect that the refined multiplication +formula generalizes to the setting of suitable extriangulated categories such as the +Higgs category of [Wu21], in which the classical multiplication formula can be +proved [KW]. +We apply our main results to the specialization of cluster variables to 1. Let Q +be finite quiver without loops or 2-cycles and let Q′ be the quiver obtained from Q +by removing a vertex i. Let σ be the specialization of xi at 1. In the case where Q +is mutation-equivalent to an acyclic quiver, it was proved in [ADS14] that the + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +3 +image of the cluster algebra AQ by σ is contained in AQ′ ⊗Z Q. Using our refined +multiplication formula, we can improve on this result. +Corollary (4.4). Assume that Q is mutation-equivalent to an acyclic quiver. Then +the image of the cluster algebra AQ by σ is AQ′. +More generally, we have the following results. +Corollary (4.3). Assume that the quiver Q admits a non-degenerate Jacobi-finite +potential. +If the upper cluster algebra UQ′ is equal to the cluster algebra AQ′, +then σ(AQ) = AQ′. +Corollary (4.5). Assume that the quiver Q admits a non-degenerate Jacobi-finite +potential. If the upper cluster algebra UQ′ is spanned by the cluster characters of +objects of the associated generalized cluster category C, then σ(UQ) = UQ′. +Note that in the above results the variable that gets specialized to 1 is not frozen. +Our formula also finds applications in the reduction of friezes. A frieze is ring +morphism f : AQ → Z that sends all cluster variables to positive integers. Friezes +originated in work of Conway and Coxeter [Cox71, CC73a], but have been vastly +generalized using cluster algebras, see for instance [BM09, ARS10, BFG+21, MG19] +and the survey paper [MG15]. Our result on friezes is the following. +Corollary (4.6). Let Q be an acyclic quiver without loops or 2-cycles, let Q′ be +the quiver obtained by removing the vertex i in Q, and let σ : AQ → AQ′ be the +specialization of xi to 1. Let f ′ : AQ′ → Z be a frieze. Then there exists a unique +frieze f : AQ → Z such that f ′ ◦ σ = f. +The non-trivial part of the above result is the existence. +Finally, in Section 4.3, we give a new proof of a multiplication formula for +Auslander–Reiten triangles first obtained in [DG14], and in Section 4.4, we ob- +tain a formula reminiscent of the one stated in [DX]. +2. Refined multiplication formula: triangulated case +2.1. Recollections on 2-Calabi–Yau triangulated categories. The setting in +which the multiplication formula holds is that of Hom-finite, Krull-Schmidt, tri- +angulated, 2-Calabi–Yau categories with a cluster tilting object and constructible +cones. The aim of this section is to recall the main definitions and properties of +this setting. +2.1.1. 2-Calabi–Yau categories. Let C be a small Hom-finite triangulated category +over a field k, with suspension functor Σ. +Definition 2.1. The category C is 2-Calabi–Yau if, for any objects L and M of C, +it is equipped with a bilinear form +βL,M : HomC(L, ΣM) × HomC(M, ΣL) −→ k +which is non-degenerate and bifunctorial. Here, bifunctorial means that if L, M, +N and P are objects of C, and if ε ∈ HomC(M, ΣN), η ∈ HomC(N, ΣL), δ ∈ +HomC(P, ΣM), f ∈ HomC(L, M) and g ∈ HomC(N, P), then +βL,N(ε ◦ f, η) += +βM,N(ε, Σf ◦ η) +and +βM,P (Σg ◦ ε, δ) += +βM,N(ε, δ ◦ g). + +4 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +Equivalently, C is 2-Calabi–Yau if it is equipped with an isomorphism of bifunc- +tors +HomC(L, ΣM) −→ D HomC(M, ΣL), +where D = Homk(?, k) is the usual duality for vector spaces. +2.1.2. Cluster-tilting objects and associated cluster characters. Let C be a Hom- +finite 2-Calabi–Yau triangulated category. +Definition 2.2 ([BMR+06]). An object T of C is a cluster-tilting object if the +following hold: +(1) T is rigid, that is, the space HomC(T, ΣT ) vanishes; +(2) for any object X of C, if HomC(T, ΣX) vanishes, then X lies in add T (that +is, X is a direct factor of a direct sum of copies of T ). +We will usually assume that T is basic, and write T = T1 ⊕ . . . ⊕ Tn, where the +Ti’s are pairwise non-isomorphic indecomposable objects. +Examples 2.3. +(1) The cluster categories of [BMR+06] are triangulated Hom- +finite 2-Calabi–Yau categories with a cluster-tilting object. +(2) The generalized cluster categories of [Ami09] also have these properties. +(3) The stable categories of all the Frobenius categories of Example 3.3 also +have these properties. +Cluster-tilting objects are essential in the categorification of cluster algebras via +triangulated categories. This is done via cluster characters, whose definition we +recall in Definition 2.6. +Proposition 2.4 ([KR07]). Let T be a basic cluster-tilting object of C. +(1) The functor F = HomC(T, Σ?) induces an equivalence of categories +C/(T ) +F−→ mod EndC(T ), +where (T ) is the ideal of all morphisms factoring through an object of add T . +(2) Any object X of C sits in a triangle +T X +1 → T X +0 → X → ΣT X +1 , +where T X +1 +and T X +0 +lie in add T . +Definition 2.5 ([DK08]). Let T be a basic cluster-tilting object of C. The index +of an object X of C with respect to T is the element of the Grothendieck group +K0(add T ) defined by +indT X = [T X +0 ] − [T X +1 ], +where T X +0 +and T X +1 +are as in Proposition 2.4(2). +Note that, while the triangle in Proposition 2.4(2) is not unique, the index of X +does not depend on the one we choose [Pal08, Lemma 2.1]. Moreover, it was shown +in [Pal08] (in the proof of Lemma 1.3) that for any object X of C, the value of +indT ΣX + indT X only depends on the dimension vector e of FX. We will denote +this value by ι(e). Note that ι extends to a linear map defined on all of Zn. +Let us now assume that the field k is the field C of complex numbers. + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +5 +Definition 2.6 ([CC06][Pal08]). Let T be a basic cluster-tilting object of C. The +cluster character associated with T is the map +CCT : Obj(C) −→ Q(x1, . . . , xn) +defined by +CCT (M) = xindT M � +e∈Nn +χ +� +Gre +� +FM +�� +x−ι(e), +where +• n is the number of indecomposable direct factors of T in a decomposition +T = �n +i=1 Ti; +• xa = xa1 +1 · · · xan +n , for any a = �n +i=1 ai[Ti] ∈ K0(add T ); +• χ is the Euler characteristic for topological spaces; +• FM = HomC(T, ΣM) is considered as a right module over EndC(T ); +• for any module R, Gre(R) is the submodule Grassmannian [CC06, Section +2.3], a projective variety whose points parametrize the submodules of R of +dimension vector e; +• ι(e) is as defined below Definition 2.5. +2.1.3. Constructible cones. The coefficients in the multiplication formula are Euler +characteristics of subsets of certain algebraic varieties. For the formula to be well- +defined, we must ensure that the Euler characteristics of these subsets are well- +defined integers. In [Pal12], this is done by proving that the subsets in question +are constructible. In order to do so, we need to assume that the category C has +constructible cones. Although we will not recall the definition of a category with +consctructible cones (and simply refer to [Pal12, Section 1.3]), we will list the +properties of such categories that we will need. +Let C be a Hom-finite triangulated category with a basic cluster-tilting object +T . Fix two objects L and M. For any object Y of C, let HomC(L, ΣM)⟨Y ⟩ be the +subset of HomC(L, ΣM) of all morphisms ε such that if +M → Y ′ → L +ε→ ΣM +is a triangle, then +• indT Y ′ = indT Y , and +• for all dimension vectors e, we have χ(Gre(FY )) = χ(Gre(FY ′)). +For any subset V of HomC(L, ΣM), let V⟨Y ⟩ be the intersection of V with HomC(L, ΣM)⟨Y ⟩. +Note that the condition +HomC(L, ΣM)⟨Y ⟩ = HomC(L, ΣM)⟨Y ′⟩ +induces an equivalence relation on the set of objects of C. Let YL,M be a set of +representatives for this equivalence relation. +Proposition 2.7 (Proposition 2.8 of [Pal12]). If C has constructible cones, then +HomC(L, ΣM) = +� +Y ∈YL,M +HomC(L, ΣM)⟨Y ⟩ +is a partition of HomC(L, ΣM) into a finite number of constructible subsets. + +6 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +Corollary 2.8. If C has constructible cones, and if V is a constructible subset of +HomC(L, ΣM), then +V = +� +Y ∈YL,M +V⟨Y ⟩ +is a decomposition of V into a finite number of pairwise disjoint constructible sub- +sets. +Example 2.9. All the triangulated categories mentioned in this paper have con- +structible cones, thanks to these two facts proved in [Pal12, Sections 2.4-2.5]: stable +categories of Hom-finite Frobenius categories and the generalized cluster categories +of [Ami09] have constructible cones. +2.2. The refined multiplication formula. This section is devoted to the proof +of the refined multiplication formula. The proof follows the lines of [Pal12] and +relies heavily on results obtained there. +For any vector space E and for any subset U which is stable by scalar multiplica- +tion, we denote by PU the subset of the projective space PE consisting of elements +[u] with u ∈ U, where [u] denotes the class of u in PE. +Theorem 2.10. Let C be a Hom-finite Krull–Schmidt 2-Calabi–Yau triangulated +category over C with constructible cones and admitting a basic cluster-tilting object +T . Let L and M be two objects of C, and let V be a non-zero vector subspace of +HomC(L, ΣM). Then the following equality holds: +χ(PV )CCT (L)CCT (M) = +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ) + +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ), +where R = P HomC(M, ΣL) \ PKer βL,M(V, ?). +Remark 2.11. If V is the whole space HomC(L, ΣM), then the formula recovers +that of Y. Palu [Pal12, Theorem 1.1]. +We assume for the rest of this section that C has constructible cones. +The first step into proving the formula is by replacing CCT (L) and CCT (M) by +their definitions in the left-hand side of the formula. Doing this, we get +χ(PV )CCT (L)CCT (M) += χ(PV ) +� +xindT L � +e +χ +� +Gre(FL) +� +x−ι(e)�� +xindT M � +f +χ +� +Grf(FM) +� +x−ι(f)� += xindT (L⊕M) � +e,f +χ +� +PV × Gre(FL) × Grf(FM) +� +x−ι(e+f). +We will refine this sum by replacing PV × Gre(FL) × Grf(FM) by another con- +structible set with the same Euler characteristic. Let us construct this set. +Define W V +L,M to be the subset of PV × � +d,g +�n +i=1 Grgi(Cdi) consisting of pairs +([ε], E) where E is a subrepresentation of FY , where Y is the middle term of a +triangle M +i +� Y +p +� L +ε � ΣM . + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +7 +Furthermore, define +W V +L,M(e, f, g) = {([ε], E) ∈ W V +L,M | dim E = g, dim Fp(E) = e, dim Fi−1(E) = f}; +W V +L,M(e, f) = {([ε], E) ∈ W V +L,M | dim Fp(E) = e, dim Fi−1(E) = f}; +W V,Y +L,M(e, f, g) = {([ε], E) ∈ W V +L,M | ε ∈ PV⟨Y ⟩, dim E = g, dim Fp(E) = e, dim Fi−1(E) = f}; +W V,Y +L,M(e, f) = {([ε], E) ∈ W V +L,M | ε ∈ PV⟨Y ⟩, dim Fp(E) = e, dim Fi−1(E) = f}. +Then W V +L,M and all the sets defined above are finite disjoint unions of subsets of +the form W V,Y +L,M(e, f, g). Moreover, since we assumed that C has constructible cones, +then the results of Y. Palu give us the following. +Lemma 2.12 (Lemma 3.1 of [Pal12]). The sets W V,Y +L,M(e, f, g), W V,Y +L,M(e, f), W V +L,M(e, f), +W V +L,M(e, f, g) and W V +L,M are constructible. +Proof. In [Pal12, Lemma 3.1], it is shown that certain sets W Y +LM(e, f, g) are +constructible. Our sets W V,Y +L,M(e, f, g) are the intersection of these W Y +LM(e, f, g) with +PV × � +d,g +�n +i=1 Grgi(Cdi); thus they are constructible. Since all the other sets are +finite unions of sets of the form W V,Y +L,M(e, f, g), they must also be constructible. +□ +Now, consider the constructible map +ΨL,M(e, f) : W V +L,M(e, f) +−→ +PV × Gre(FL) × Grf(FM) +([ε], E) +�−→ +([ε], Fp(E), Fi−1(E)). +Let LV +1 (e, f) be the image of this map; let LV +2 (e, f) be the complement of the image. +Then χ(PV × Gre(FL) × Grf(FM)) = χ(LV +1 (e, f)) + χ(LV +2 (e, f)). +Therefore our equation becomes +(⋆) +χ(PV )CCT (L)CCT (M) += +xindT (L⊕M) � +e,f +χ +� +LV +1 (e, f) +� +x−ι(e+f) ++xindT (L⊕M) � +e,f +χ +� +LV +2 (e, f) +� +x−ι(e+f). +We will now study the two terms of the right-hand side of (⋆). +The first term of the RHS of (⋆). +Lemma 2.13. We have an equality +xindT (L⊕M) � +e,f +χ +� +LV +1 (e, f) +� +x−ι(e+f) = +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ). +Proof. +It is proved in [CC06] (see also Section 3 of [Pal12]) that the fibers +of ΨL,M(e, f) are affine spaces. As a consequence, we have that χ(LV +1 (e, f)) = +χ(W V +L,M(e, f)). Thus +xindT (L⊕M) � +e,f +χ +� +LV +1 (e, f) +� +x−ι(e+f) += +xindT (L⊕M) � +e,f +χ +� +W V +L,M(e, f) +� +x−ι(e+f) += +� +e,f,g,⟨Y ⟩ +χ +� +W V,Y +L,M(e, f, g) +� +x−ι(e+f)+indT (L⊕M). + +8 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +Now, by [Pal08, Lemma 5.1], if ([ε], E) lies in W V,Y +L,M(e, f, g), then it implies that +indT (L ⊕ M) − ι(e + f) = indT (Y ) − ι(g). +Moreover, for a fixed g, consider the map +� +e,f +W V,Y +L,M(e, f, g) −→ PV⟨Y ⟩ +sending a pair ([ε], E) to [ε]. This map is obviously surjective if the left-hand side is +non-empty. Moreover, the preimage of any [ε′] is isomorphic to {[ε′]} × Grg(FY ′), +where Y ′ sits in a triangle M → Y ′ → L +ε′ +−→ ΣM. By definition of PV⟨Y ⟩, the Euler +characteristic of all the fibers is the same and is equal to χ(Grg(FY )). Thus +χ( +� +e,f +W V,Y +L,M(e, f, g)) = χ(Grg(FY ))χ(PV⟨Y ⟩). +So the sum becomes +... += +� +e,f,g,Y +χ +� +W V,Y +L,M(e, f, g) +� +x−ι(g)+indT (Y ) += +� +g,Y +χ +� � +e,f +W V,Y +L,M(e, f, g) +� +x−ι(g)+indT (Y ) += +� +g,Y +χ(Grg(FY ))χ(PV⟨Y ⟩)x−ι(g)+indT (Y ) += +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ). +This finishes the proof of the lemma. +□ +The second term of the RHS of (⋆). +Lemma 2.14. We have an equality +xindT (L⊕M) � +e,f +χ +� +LV +2 (e, f) +� +x−ι(e+f) = +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ). +Proof. +Recall that +R = {[η] ∈ P HomC(M, ΣL) | ∃ε ∈ V with βL,M(ε, η) ̸= 0}. +Define W P HomC(L,ΣM),Y +M,L +(f, e, g) as before Lemma 2.12 and let W R,Y +M,L (f, e, g) be +the constructible subset of all pairs ([η], E) with η ∈ R. +For fixed e, f and +g, let CR,Y +L,M(e, f, g) be the subset of LV +2 (e, f) × W R,Y +M,L (f, e, g) consisting of pairs +� +([ε], R, S), ([η], E) +� +such that βL,M(ε, η) ̸= 0, Fi−1(E) = S and Fp(E) = R. +Finally, let CR +L,M(e, f) = � +g,Y ∈YM,L CR,Y +L,M(e, f, g). +Consider the two projections +CR +L,M(e, f) +p1 +−→ +LV +2 (e, f) +CR,Y +L,M(e, f, g) +p2 +−→ +W R,Y +M,L (f, e, g). +By [Pal12, Proposition 3.3], p1 and p2 are surjective. +Moreover, by [Pal12, +Proposition 3.4], the fibers of p1 are extensions of affine spaces, and those of p2 are +affine spaces. + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +9 +Therefore χ(CR +L,M(e, f)) = χ(LV +2 (e, f)) and χ(CR,Y +L,M(e, f, g)) = χ(W R,Y +M,L (f, e, g)). +Thus the left-hand side in the statement is equal to +... += +xindT (L⊕M) � +e,f +χ(LV +2 (e, f))x−ι(e+f) += +� +e,f +χ(CR +L,M(e, f))xindT (L⊕M)−ι(e+f) += +� +e,f,g,Y +χ(CR,Y +L,M(e, f, g))xindT (L⊕M)−ι(e+f) += +� +e,f,g,Y +χ(W R,Y +M,L (f, e, g))xindT (L⊕M)−ι(e+f). +Again, by [Pal08, Lemma 5.1], we have that if ([ε], E) lies in W R,Y +M,L (f, e, g), then +indT (L ⊕ M) − ι(e + f) = indT (Y ) − ι(g). Moreover, the map +� +e,f +W R,Y +M,L (f, e, g) −→ R⟨Y ⟩ +sending ([ε], E) to [ε] is surjective (if the left-hand side is non-empty), and its fibers +have the form {[ε′]} × Grg(FY ′), where Y ′ sits in a triangle L → Y ′ → M +ε′ +−→ ΣL. +Thus +χ( +� +e,f +W R,Y +M,L (f, e, g)) = χ(R⟨Y ⟩)χ(Grg(FY )). +Therefore the above sequence of equalities continues: +. . . += +� +e,f,g,Y +χ(W R,Y +M,L (f, e, g))xindT Y −ι(g) += +� +g,Y +χ( +� +e,f +W R,Y +M,L (f, e, g))xindT Y −ι(g) += +� +g,Y +χ(R⟨Y ⟩)χ(Grg(FY ))xindT Y −ι(g) += +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ). +This finishes the proof. +□ +Theorem 2.10 then follows directly from Lemma 2.13 and Lemma 2.14. +3. Refined multiplication formula: Frobenius case +We will now follow the ideas of [FK10] (see also [Pal12, Section 4]). The main +difference will be that our Frobenius categories can be Hom-infinite; we will only +assume that they are Ext-finite. We will also assume that they are Krull–Schmidt, +and that their stable categories have constructible cones. Since the proofs are very +similar to the ones in the triangulated case, in this section we only provide a detailed +outline for the Frobenius case. +3.1. Recollections on Frobenius categories. + +10 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +3.1.1. 2-Calabi–Yau Frobenius categories. A Frobenius category is an exact category +E in the sense of Quillen with enough projectives and enough injectives, in which +projectives and injectives coincide. It is Ext-finite if for any objects X and Y of E, +the space Ext1 +E(X, Y ) is finite-dimensional. +It was proved in [Hap87, Theorem 9.4] that if E is a Frobenius category, then its +stable category E is triangulated (E is the quotient of E by the ideal of all morphisms +factoring through a projective-injective object). Note that if E is Ext-finite, then +E is Hom-finite. +Definition 3.1 (Section 2.7 of [FK10]). An Ext-finite Frobenius category is 2- +Calabi–Yau if its stable category is 2-Calabi–Yau as a triangulated category. +3.1.2. Cluster-tilting objects. Let E be an Ext-finite Krull–Schmidt 2-Calabi–Yau +Frobenius category. +Definition 3.2 (Section 2.7 of [FK10]). An object T of E is a cluster-tilting object +if +(1) T is rigid, that is, the space Ext1 +E(T, T ) vanishes; +(2) for any object X of E, if Ext1 +E(T, X) vanishes, then X ∈ add T ; and +(3) each object X of E admits a right add T -approximation T X → X and a left +add T -approximation X → TX (in other words, the functors HomE(X, ?)|add T +and HomE(?, X)|(add T )op are finitely generated). +Note that if T is a cluster-tilting object, then every indecomposable projective- +injective object is isomorphic to a direct summand of T . +Examples 3.3. +(1) The module category of a preprojective algebra of Dynkin +type is a Hom-finite stably 2-Calabi–Yau Frobenius category with a cluster +tilting object. +It was used in [GLS06] in the categorification of cluster +algebras. +Its stable category has constructible cones by [Pal12, Section +2.4]. +(2) More generally, subcategories Cw of modules over preprojective algebras +were were used in [BIRS09, GLS12a] to category cluster algebras. These +categories have the same properties as those of the previous example. +(3) Let 0 < k < n be integers, and put ˆR = C[[x, y]]/(xk − yn−k). The group +G = ⟨ζ⟩ of n-th roots of unity acts on ˆR by ζ.x = ζx and ζ.y = ζ−1y. +Then the category E = CMG( ˆR) of G-equivariant Cohen–Macaulay ˆR- +modules is a (not necessarily Hom-finite) Ext-finite Frobenius category with +a cluster tilting object. Its stable category has constructible cones, since +it is equivalent to categories from the previous example. This was used +in [JKS16] to give a categorification of the cluster algebra structure of the +homogeneous coordinate ring C[Gk,n] of the Grassmannian Gk,n. +Let T be a basic cluster-tilting object of E. Write T = T1 ⊕ . . . ⊕ Tn, where each +Ti is indecomposable, and let C = EndE(T ). Since T is cluster-tilting, we have a +functor +F = HomE(T, ?) : E −→ f.g.mod C, +where f.g.mod C is the category of finitely generated right C-modules. +For any +finitely generated C-modules L and N such that N is finite-dimensional, define +⟨L, N⟩τ = dim HomC(L, N) − dim Ext1 +C(L, N), + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +11 +⟨L, N⟩3 = +3 +� +i=0 +(−1)i dim Exti +C(L, N). +Note that these expressions are well-defined integers, since Exti +C(L, N) is finite- +dimensional because E is Ext-finite, and since HomC(L, N) is finite-dimensional +because L is finitely generated and N is finite-dimensional. +Finally, let C = EndE(T ). As in [KR07, Section 4] and [FK10, Section 3], we +view C-modules as C-modules with no composition factors isomorphic to the simple +modules corresponding to the projective-injective direct summands of T . +Proposition 3.4 (Proposition 3.2 of [FK10]). If L and N are finite-dimensional +C-modules of the same dimension vector, then for any finite-dimensional C-module +Y , we have that +⟨L, Y ⟩3 = ⟨N, Y ⟩3. +In view of this proposition, if e is a dimension vector, we can write ⟨e, Y ⟩3 for +the value of ⟨L, Y ⟩3 for any C-module L of dimension vector e. +Definition 3.5 ([FK10]). The cluster character associated with T is the map +CCT : Obj(E) −→ Q(x1, . . . , xn) +defined by +CCT (M) = +n +� +i=1 +x⟨F M,Si⟩τ +i +� +e +χ +� +Gre(Ext1 +E(T, M)) +� n +� +i=1 +x⟨e,Si⟩3 +i +. +3.2. The formula. +Theorem 3.6. Let E be a Hom-finite 2-Calabi–Yau Frobenius category with a clus- +ter tilting object T . Assume that the triangulated category E has constructible cones. +Let L and M be two objects of E, and let V be a vector subspace of Ext1 +E(L, M). +Then +χ(PV )CCT (L)CCT (M) = +� +Y ∈YL,M +χ(PV⟨Y ⟩)CCT (Y ) + +� +Y ∈YM,L +χ(R⟨Y ⟩)CCT (Y ). +The proof follows the lines of that of [Pal12, Theorem 4.1]; it is similar to that +of Theorem 2.10, but uses [FK10, Lemma 3.4] instead of [Pal08, Lemma 5.1]. +4. Applications +4.1. Specialization of cluster variables in cluster algebras. Let C be a Hom- +finite 2-Calabi–Yau triangulated category with a basic cluster tilting object T = +�n +i=1 Ti. Following [CILFS15], let the Caldero-Chapoton algebra AC be the subring +of the ring Z[x±1 +1 , . . . , x±1 +n ] generated by the set of all CCT (X), as X spans all +objects of C. +Motivated by the reduction of friezes (see Section 4.2) and the study of mor- +phisms of rooted cluster algebras of [ADS14], we wish to study the algebra obtained +from AC by specializing xn to 1. To fix notation, let +σ : Z[x±1 +1 , . . . , x±1 +n ] → Z[x±1 +1 , . . . , x±1 +n−1] +be the morphism sending each of x1, . . . , xn−1 to itself and sending xn to 1. + +12 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +The main result of this section is stated in terms of Calabi–Yau reduction: it +was proved in [IY08] that the category +C′ = +� +Σ−1Tn +�⊥ / (Tn) +is a Hom-finite 2-Calabi–Yau triangulated category with a basic cluster tilting ob- +ject T ′, where T ′ is the image of T under the projection functor. +Theorem 4.1. Let C be a Hom-finite 2-Calabi–Yau triangulated category with con- +structible cones and a basic cluster tilting object T = �n +i=1 Ti. Then σ(AC) = AC′. +Remark 4.2. The proof of [ADS14, Theorem 6.13] shows that σ(AC) ⊆ AC′ ⊗Z Q. +It uses the multiplication formula of Palu [Pal12]. Our proof of Theorem 4.1 follows +the same lines using our refined multiplication formula (Theorem 2.10) instead. We +nonetheless include the complete argument below. +Proof. (of Theorem 4.1). It suffices to prove that σ (CCT (X)) ∈ AC′ for all ob- +jects X of C. The proof is by induction on the dimension of the space HomC(Tn, ΣX). +Assume first that dim HomC(Tn, ΣX) = 0. Then X ∈ (Σ−1Tn)⊥. Denote by π the +projection +π : +� +Σ−1Tn +�⊥ → C′. +Then σ (CCT (X)) = CCT ′(πX) ∈ AC′. Note that this implies that AC′ ⊆ σ(AC), +since all objects of C′ have the form π(X) with X an object of C. +Assume now that dim HomC(Tn, ΣX) = d > 0. Choose any non-split triangle +X −→ E −→ Tn +ξ−→ ΣX +and let V be the span on ξ in HomC(Tn, ΣX). Applying Theorem 2.10, we get that +CCT (X)CCT (Tn) = CCT (E) + +� +Y ∈YX,Tn +χ +� +R⟨Y ⟩ +� +CCT (Y ). +Since CCT (Tn) = xn, applying the specialization σ to the left-hand side yields σ (CCT (X)). +Since all χ +� +R⟨Y ⟩ +� +are integers, it thus suffices to prove that CCT (E) and all CCT (Y ) +on the right-hand side are in AC′. We do this by showing that the dimensions +of HomC(Tn, ΣE) and HomC(Tn, ΣY ) are strictly smaller than d and by applying +induction. +To see this, first apply the functor HomC(Tn, ?) to the triangle defined by ξ. We +obtain an exact sequence +(Tn, Tn) +ξ∗ +−→ (Tn, ΣX) +f−→ (Tn, ΣE) −→ (Tn, ΣTn), +where we write (U, V ) instead of HomC(U, V ) to save space. Since Tn is rigid, (Tn, ΣTn) +vanishes, so f is surjective; thus, +(Tn, ΣE) ∼= (Tn, ΣX)/ξ∗ +� +(Tn, Tn) +� +. +Lastly, ξ∗ +� +(Tn, Tn) +� +is non-zero, since it contains ξ∗(idTn) = ξ. Therefore, +dim(Tn, ΣE) < dim(Tn, ΣX) = d, +and by induction, σ (CCT (E)) ∈ AC′. +Now let Y ∈ YTn,X. By definition, there exists a non-split triangle +Tn −→ Y −→ X +δ−→ ΣTn. + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +13 +Applying the functor HomC(?, T ) and repeating the above argument, we get that +dim(Y, ΣTn) < dim(X, ΣTn) = d. +By the 2-Calabi–Yau property, dim(Y, ΣTn) = dim(Tn, ΣY ). Thus, by induction, +we also have that σ (CCT (Y )) ∈ AC′. This finishes the proof. +□ +Theorem 4.1 has an interesting application to cluster algebras. +Corollary 4.3. Let Q be a quiver without loops or 2-cycles, i be a vertex of Q +and Q′ be the quiver obtained from Q by removing the vertex i. Assume that there +exists a non-degenerate potential W such that the generalized cluster category CQ,W +is Hom-finite. +If the cluster algebra AQ′ is equal to its upper cluster algebra UQ′ (see [BFZ05] +for details), then the specialization σ sending xi to 1 satisfies σ(AQ) = AQ′. +Proof. Let C = CQ,W and C′ = CQ′,W ′, where W ′ is the potential obtained by +removing the terms of W involving the vertex i. We know from [Pal08] that AQ ⊂ +AC, and from [Pla11a, Corollary 4.14] that AC ⊂ UQ. +The same is true if we +replace Q with Q′; thus, by our assumption, AQ′ = AC′ = UQ′. Applying σ, we get +that AQ′ ⊂ σ(AQ) ⊂ σ(AC), and this last set is AC′ by Theorem 4.1. This finishes +the proof. +□ +Corollary 4.4. If Q is mutation-equivalent to an acyclic quiver, then we have +that σ(AQ) = AQ′. +Corollary 4.5. Assume that the quiver Q admits a non-degenerate Jacobi-finite +potential, and let C be its generalized cluster category. +If the upper cluster al- +gebra UQ′ is spanned the cluster characters of some objects in C, then we have +that σ(UQ) = UQ′. +Proof. Notice that we have AC′ ⊂ UQ′ and AC ⊂ UQ by the universal Laurent +property of cluster characters. Since UQ′ is spanned by some cluster characters, we +have UQ′ ⊂ AC′. Consequently, Theorem 4.1 implies UQ′ = AC′ = σ(AC) ⊂ σ(UQ). +Notice that every cluster for Q′ (see [BFZ05]) is the image of some cluster for Q +under σ. So we have σ(UQ) ⊂ UQ′. The desired claim follows. +□ +Many upper cluster algebras are known to possess a generic basis, see [GLS12b, +Pla13, Qin19]. They satisfy the assumption in Corollary 4.5. +4.2. Reduction of friezes. Let Q be a quiver without loops or 2-cycles. A frieze +is a morphism of rings f : AQ → Z sending every cluster variable of AQ to a +positive integer. This definition generalizes the originial one of Conway and Cox- +eter [CC73b], and has been an area of active interest in recent years. +In [BFG+21, Section 5], an operation of reduction on friezes is considered. The +purpose of this section is to show that this reduction operation can be “reversed” +by adding a 1 in a frieze. +Corollary 4.6. Let Q be an acyclic quiver without loops or 2-cycles, and let Q′ be +the quiver obtained by removing the vertex i in Q, and let σ : AQ → AQ′ be the +specialization of xi to 1 (this is well-defined thanks to Corollary 4.4). Let f ′ : AQ′ → +Z be a frieze. Then there exists a unique frieze f : AQ → Z such that f ′ ◦ σ = f. + +14 +BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN +Proof. If f exists, then it is unique, since it is determined by its action on the initial +cluster variables of AQ. Let us prove that such an f exists for any frieze f ′. By +Corollary 4.4, f = f ′ ◦σ is a well-defined morphism of rings from AQ to Z. We only +need to check that all cluster variables of AQ are sent to positive values by f; this +follows from the positivity theorem [LS15]: any cluster variable of AQ is a Laurent +polynomial with nonnegative coefficients in the initial cluster variables, and these +are sent to positive values by f. +□ +Remark 4.7. Corollary 4.6 can be deduced for friezes where Q is of type An +from the results of [CC73b], and was shown to be true in types An, Dn and E6 in +[BFG+21, Section 5], where it was also observed to be true for all known friezes of +types E7 and E8 by a direct check. The total number of possible friezes in these +types is still unknown and was conjectured in [FP16]. +4.3. A formula for Auslander–Reiten triangles. In this section, we will show +how Theorem 2.10 allows for a new proof of the following formula of S. Dominguez +and C. Geiss when C has constructible cones. +Theorem 4.8 (Theorem 1 of [DG14]). Let C be a Hom-finite 2-Calabi–Yau category +with constructible cones and a cluster tilting object T . Let Z be an indecomposable +object of C, and assume that it sits in an Auslander–Reiten triangle +ΣZ +α−→ Y +β−→ Z +ε−→ Σ2Z. +Then +CCT (Z)CCT (ΣZ) = CCT (Y ) + 1. +We will give a proof of this theorem under the additionnal assumption that C +has constructible cones. +Let V be the one-dimensional subspace of HomC(Z, Σ2Z) generated by ε. Ap- +plying Theorem 2.10, we get +CCT (Z)CCT (ΣZ) = CCT (Y ) + +� +E∈YΣZ,Z +χ(R⟨E⟩)CCT (E). +Here R = {[η] ∈ P HomC(ΣZ, ΣZ) | βZ,ΣZ(ε, η) ̸= 0}. Let us show that [η] lies in +R if and only if η is an isomorphism. +Since Z is indecomposable, HomC(ΣZ, ΣZ) is a local ring. Hence η is an isomor- +phism if and only if it does not lie in the radical of HomC(ΣZ, ΣZ). Thus we need +to show that η lies in the radical of HomC(ΣZ, ΣZ) if and only if βZ,ΣZ(ε, η) = 0. +Assume that η is in the radical (assume η ̸= 0; the case η = 0 is trivial). Then it +is not an isomorphism, and since Z is indecomposable, it is not a retraction. Then, +by definition of an Auslander–Reiten triangle, we must have that there exists a +morphism f : Z → Y such that Σ−1η = βf. But then +βZ,ΣZ(ε, η) += +βZ,ΣZ(ε, ΣβΣf) += +βY,ΣZ(εβ, Σf) += +βY,ΣZ(0, Σf) += +0. +Assume next that η is an isomorphism. Then η and +rad HomC(ΣZ, ΣZ) gen- +erate HomC(ΣZ, ΣZ) as a vector space. +If βZ,ΣZ(ε, η) were to vanish, it would + +A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS +15 +thus vanish for any η′ in HomC(ΣZ, ΣZ), contradicting the fact that βZ,ΣZ is non- +degenerate. Thus βZ,ΣZ(ε, η) ̸= 0. +This proves that R is the set of [η], with η an isomorphism. But then R = +R⟨0⟩ (since the middle term of a triangle associated with an isomorphism is 0). +Moreover, R = P HomC(ΣZ, ΣZ) \ P rad HomC(ΣZ, ΣZ) is an affine space, since +rad HomC(ΣZ, ΣZ) is a hyperplane in HomC(ΣZ, ΣZ). Thus χ(R) = 1. +Therefore +CCT (Z)CCT (ΣZ) += +CCT (Y ) + +� +E∈YZ,ΣZ +χ(R⟨E⟩)CCT (E) += +CCT (Y ) + χ(R⟨0⟩)CCT (0) += +CCT (Y ) + 1. +This finishes the proof. +4.4. Another restricted formula. Theorem 2.10 allows us to obtain (always +assuming contructibility of cones) the following formula, reminiscent of the one +stated in [DX]. +For two objects L and M of C, let (T )(L, M) be the space of +morphisms from L to M factoring through an object of add T . +Proposition 4.9. Under the hypotheses of Theorem 2.10, we have that +χ +� +P(T )(L, ΣM) +� +CCT (L)CCT (M) = +� +Y ∈YL,M +χ +� +P(T )(L, ΣM)⟨Y ⟩ +� +CCT (Y ) ++ +� +Y ∈YM,L +χ +� +P HomC(M, ΣL)⟨Y ⟩ \ P(T )(M, ΣL)⟨Y ⟩ +� +CCT (Y ). +Proof. This follows from Theorem 2.10 by taking V = (T )(L, ΣM). To see +this, we only need to prove that Ker βL,M(V, ?) = (T )(M, ΣL). +Notice first that (T )(M, ΣL) is contained in Ker βL,M(V, ?); indeed, if f ∈ V and +g ∈ (T )(M, ΣL), then Σg ◦ f = 0 (since T is rigid), so +βL,M(f, g) = βL,ΣL(Σg ◦ f, idΣL) = 0. +Moreover, dim Ker βL,M(V, ?) = dim HomC(L, ΣM)/V , and this last vector space +is isomorphic to the dual of (T )(M, ΣL) thanks to [Pal08, Lemma 3.3]. +Thus +Ker βL,M(V, ?) and (T )(M, ΣL) have the same (finite) dimension, and so they are +equal. +□ +Acknowledgements +The first and second authors were supported by the French ANR grant CHARMS +(ANR-19-CE40-0017-02). The second author was supported by the Institut Univer- +sitaire de France (IUF). The third author was supported by the National Natural +Science Foundation of China (Grant No. 12271347). The final stages of this project +were completed while the first and second author were participating in a trimester +programme at the Isaac Newton Institute. 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K. : Universit´e Paris de Paris, UFR de Math´ematiques, CNRS, Institut de Math´ematiques +de Jussieu–Paris Rive Gauche, IMJ-PRG, Bˆatiment Sophie Germain, 75205 Paris Cedex +13, France +Email address: bernhard.keller@imj-prg.fr +URL: https://webusers.imj-prg.fr/~bernhard.keller/ +P.-G.P. : +Laboratoire de Math´ematiques de Versailles, UVSQ, CNRS, Universit´e +Paris-Saclay, Institut Universitaire de France (IUF) +Email address: pierre-guy.plamondon@uvsq.fr +URL: https://www.imo.universite-paris-saclay.fr/~plamondon/ +F.Q.: +The School of Mathematical Sciences, Shanghai Jiao Tong University, 800 +Dongchuan RD, Shanghai, 200240 China. +Email address: qin.fan.math@gmail.com +URL: https://sites.google.com/site/qinfanmath/ + diff --git a/KNAzT4oBgHgl3EQfIPul/content/tmp_files/load_file.txt b/KNAzT4oBgHgl3EQfIPul/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8058308e7c2bb920c231e083c89eb1f161620946 --- /dev/null +++ b/KNAzT4oBgHgl3EQfIPul/content/tmp_files/load_file.txt @@ -0,0 +1,890 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf,len=889 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='01059v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='RT] 3 Jan 2023 A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We obtain a multiplication formula for cluster characters on (sta- bly) 2-Calabi–Yau (Frobenius or) triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This formula gen- eralizes those known for arbitrary pairs of objects and for Auslander–Reiten triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' As an application, we show that for cluster algebras of acyclic types, specialization of a cluster variable to 1 sends all cluster variables to elements of a cluster algebra of smaller rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We also obtain applications to the reduction of friezes of acyclic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Refined multiplication formula: triangulated case 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Recollections on 2-Calabi–Yau triangulated categories 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The refined multiplication formula 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Refined multiplication formula: Frobenius case 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Recollections on Frobenius categories 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The formula 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Applications 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Specialization of cluster variables in cluster algebras 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Reduction of friezes 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A formula for Auslander–Reiten triangles 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Another restricted formula 15 Acknowledgements 15 References 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Introduction The additive categorification of cluster algebras has been an important tool in their study almost from their inception (see for instance the survey papers [Kel09, Rei10, Ami11, Pla18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Such a categorification is given by a category C (usually triangulated or exact) and a cluster character sending objects of C to Laurent polynomials in several variables so that suitable objects of C are sent to cluster variables in a cluster algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The key property that a cluster character satisfies is a multiplication formula which recovers the exchange relations in a clus- ter algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Such formulas at various levels of generality have been obtained in [CC06, CK08, CK06, GLS08, Pal08, FK10, DWZ10, Pal12, Pla11b, DG14, Rup15, GLS18, CIEFR21] and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 1 2 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN The main result of this paper is a multiplication formula generalizing most pre- viously known ones in the following context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a small Hom-finite Krull– Schmidt 2-Calabi–Yau triangulated category over C, together with a basic cluster tilting object T (definitions are recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let CCT : Obj(C) → Z[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , x±1 n ] be the corresponding cluster character (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For any objects L and M of C, let βL,M : HomC(L, ΣM) × HomC(M, ΣL) −→ k be the non-degenerate bifunctorial bilinear form conferring to C its 2-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For an object Y , let HomC(L, ΣM)⟨Y ⟩ be the set of those morphisms ε : L → ΣM such that, if we have a triangle M −→ Y ′ −→ L ε−→ ΣM, the objects Y and Y ′ have the same index (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5) and for each dimension vector e, the submodule Grassmannians Gre(HomC(T, Y )) and Gre(HomC(T, Y ′)) have the same Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It is easy to check that this set is invariant under multiplication by a non-zero scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For a subset X of HomC(L, ΣM), let X⟨Y ⟩ be the intersection of X with HomC(L, ΣM)⟨Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let YL,M be a set of representatives of equivalence classes for the equivalence relation defined by HomC(L, ΣM)⟨Y ⟩ = HomC(L, ΣM)⟨Y ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our main result is the following refined multiplication formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a small Hom-finite Krull–Schmidt 2-Calabi–Yau trian- gulated category over C with constructible cones (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3) together with a basic cluster tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let L and M be objects of C such that HomC(L, ΣM) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Finally, let V be a non-zero vector subspace of HomC(L, ΣM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then χ(PV )CCT (L)CCT (M) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ) + � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ), where R = P HomC(M, ΣL) \\ PKer βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If V is the full space, then this formula specializes to the one proved in [Pal12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our main result also has a counterpart for exact categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let E be an Ext-finite 2-Calabi–Yau Frobenius category with a cluster tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that the triangulated category E has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let L and M be two objects of E such that Ext1 E(L, M) is non-zero, and let V be a non-zero vector subspace of Ext1 E(L, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then χ(PV )CCT (L)CCT (M) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ) + � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This generalizes a result of [FK10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We expect that the refined multiplication formula generalizes to the setting of suitable extriangulated categories such as the Higgs category of [Wu21], in which the classical multiplication formula can be proved [KW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We apply our main results to the specialization of cluster variables to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Q be finite quiver without loops or 2-cycles and let Q′ be the quiver obtained from Q by removing a vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let σ be the specialization of xi at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In the case where Q is mutation-equivalent to an acyclic quiver, it was proved in [ADS14] that the A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 3 image of the cluster algebra AQ by σ is contained in AQ′ ⊗Z Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Using our refined multiplication formula, we can improve on this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that Q is mutation-equivalent to an acyclic quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then the image of the cluster algebra AQ by σ is AQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' More generally, we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that the quiver Q admits a non-degenerate Jacobi-finite potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If the upper cluster algebra UQ′ is equal to the cluster algebra AQ′, then σ(AQ) = AQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that the quiver Q admits a non-degenerate Jacobi-finite potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If the upper cluster algebra UQ′ is spanned by the cluster characters of objects of the associated generalized cluster category C, then σ(UQ) = UQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that in the above results the variable that gets specialized to 1 is not frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our formula also finds applications in the reduction of friezes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A frieze is ring morphism f : AQ → Z that sends all cluster variables to positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Friezes originated in work of Conway and Coxeter [Cox71, CC73a], but have been vastly generalized using cluster algebras, see for instance [BM09, ARS10, BFG+21, MG19] and the survey paper [MG15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our result on friezes is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Q be an acyclic quiver without loops or 2-cycles, let Q′ be the quiver obtained by removing the vertex i in Q, and let σ : AQ → AQ′ be the specialization of xi to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let f ′ : AQ′ → Z be a frieze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then there exists a unique frieze f : AQ → Z such that f ′ ◦ σ = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The non-trivial part of the above result is the existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3, we give a new proof of a multiplication formula for Auslander–Reiten triangles first obtained in [DG14], and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4, we ob- tain a formula reminiscent of the one stated in [DX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Refined multiplication formula: triangulated case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Recollections on 2-Calabi–Yau triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The setting in which the multiplication formula holds is that of Hom-finite, Krull-Schmidt, tri- angulated, 2-Calabi–Yau categories with a cluster tilting object and constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The aim of this section is to recall the main definitions and properties of this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2-Calabi–Yau categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a small Hom-finite triangulated category over a field k, with suspension functor Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The category C is 2-Calabi–Yau if, for any objects L and M of C, it is equipped with a bilinear form βL,M : HomC(L, ΣM) × HomC(M, ΣL) −→ k which is non-degenerate and bifunctorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Here, bifunctorial means that if L, M, N and P are objects of C, and if ε ∈ HomC(M, ΣN), η ∈ HomC(N, ΣL), δ ∈ HomC(P, ΣM), f ∈ HomC(L, M) and g ∈ HomC(N, P), then βL,N(ε ◦ f, η) = βM,N(ε, Σf ◦ η) and βM,P (Σg ◦ ε, δ) = βM,N(ε, δ ◦ g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 4 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN Equivalently, C is 2-Calabi–Yau if it is equipped with an isomorphism of bifunc- tors HomC(L, ΣM) −→ D HomC(M, ΣL), where D = Homk(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=', k) is the usual duality for vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Cluster-tilting objects and associated cluster characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom- finite 2-Calabi–Yau triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2 ([BMR+06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' An object T of C is a cluster-tilting object if the following hold: (1) T is rigid, that is, the space HomC(T, ΣT ) vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (2) for any object X of C, if HomC(T, ΣX) vanishes, then X lies in add T (that is, X is a direct factor of a direct sum of copies of T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will usually assume that T is basic, and write T = T1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' ⊕ Tn, where the Ti’s are pairwise non-isomorphic indecomposable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (1) The cluster categories of [BMR+06] are triangulated Hom- finite 2-Calabi–Yau categories with a cluster-tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (2) The generalized cluster categories of [Ami09] also have these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (3) The stable categories of all the Frobenius categories of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3 also have these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Cluster-tilting objects are essential in the categorification of cluster algebras via triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This is done via cluster characters, whose definition we recall in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4 ([KR07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let T be a basic cluster-tilting object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (1) The functor F = HomC(T, Σ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') induces an equivalence of categories C/(T ) F−→ mod EndC(T ), where (T ) is the ideal of all morphisms factoring through an object of add T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (2) Any object X of C sits in a triangle T X 1 → T X 0 → X → ΣT X 1 , where T X 1 and T X 0 lie in add T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5 ([DK08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let T be a basic cluster-tilting object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The index of an object X of C with respect to T is the element of the Grothendieck group K0(add T ) defined by indT X = [T X 0 ] − [T X 1 ], where T X 0 and T X 1 are as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that, while the triangle in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4(2) is not unique, the index of X does not depend on the one we choose [Pal08, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, it was shown in [Pal08] (in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3) that for any object X of C, the value of indT ΣX + indT X only depends on the dimension vector e of FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will denote this value by ι(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that ι extends to a linear map defined on all of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let us now assume that the field k is the field C of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6 ([CC06][Pal08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let T be a basic cluster-tilting object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The cluster character associated with T is the map CCT : Obj(C) −→ Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , xn) defined by CCT (M) = xindT M � e∈Nn χ � Gre � FM �� x−ι(e), where n is the number of indecomposable direct factors of T in a decomposition T = �n i=1 Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' xa = xa1 1 · · · xan n , for any a = �n i=1 ai[Ti] ∈ K0(add T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' χ is the Euler characteristic for topological spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' FM = HomC(T, ΣM) is considered as a right module over EndC(T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' for any module R, Gre(R) is the submodule Grassmannian [CC06, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3], a projective variety whose points parametrize the submodules of R of dimension vector e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' ι(e) is as defined below Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The coefficients in the multiplication formula are Euler characteristics of subsets of certain algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For the formula to be well- defined, we must ensure that the Euler characteristics of these subsets are well- defined integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In [Pal12], this is done by proving that the subsets in question are constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In order to do so, we need to assume that the category C has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Although we will not recall the definition of a category with consctructible cones (and simply refer to [Pal12, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3]), we will list the properties of such categories that we will need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom-finite triangulated category with a basic cluster-tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Fix two objects L and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For any object Y of C, let HomC(L, ΣM)⟨Y ⟩ be the subset of HomC(L, ΣM) of all morphisms ε such that if M → Y ′ → L ε→ ΣM is a triangle, then indT Y ′ = indT Y , and for all dimension vectors e, we have χ(Gre(FY )) = χ(Gre(FY ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For any subset V of HomC(L, ΣM), let V⟨Y ⟩ be the intersection of V with HomC(L, ΣM)⟨Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that the condition HomC(L, ΣM)⟨Y ⟩ = HomC(L, ΣM)⟨Y ′⟩ induces an equivalence relation on the set of objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let YL,M be a set of representatives for this equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='7 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='8 of [Pal12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If C has constructible cones, then HomC(L, ΣM) = � Y ∈YL,M HomC(L, ΣM)⟨Y ⟩ is a partition of HomC(L, ΣM) into a finite number of constructible subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 6 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If C has constructible cones, and if V is a constructible subset of HomC(L, ΣM), then V = � Y ∈YL,M V⟨Y ⟩ is a decomposition of V into a finite number of pairwise disjoint constructible sub- sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' All the triangulated categories mentioned in this paper have con- structible cones, thanks to these two facts proved in [Pal12, Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5]: stable categories of Hom-finite Frobenius categories and the generalized cluster categories of [Ami09] have constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The refined multiplication formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This section is devoted to the proof of the refined multiplication formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The proof follows the lines of [Pal12] and relies heavily on results obtained there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For any vector space E and for any subset U which is stable by scalar multiplica- tion, we denote by PU the subset of the projective space PE consisting of elements [u] with u ∈ U, where [u] denotes the class of u in PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom-finite Krull–Schmidt 2-Calabi–Yau triangulated category over C with constructible cones and admitting a basic cluster-tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let L and M be two objects of C, and let V be a non-zero vector subspace of HomC(L, ΣM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then the following equality holds: χ(PV )CCT (L)CCT (M) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ) + � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ), where R = P HomC(M, ΣL) \\ PKer βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If V is the whole space HomC(L, ΣM), then the formula recovers that of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Palu [Pal12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We assume for the rest of this section that C has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The first step into proving the formula is by replacing CCT (L) and CCT (M) by their definitions in the left-hand side of the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Doing this, we get χ(PV )CCT (L)CCT (M) = χ(PV ) � xindT L � e χ � Gre(FL) � x−ι(e)�� xindT M � f χ � Grf(FM) � x−ι(f)� = xindT (L⊕M) � e,f χ � PV × Gre(FL) × Grf(FM) � x−ι(e+f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will refine this sum by replacing PV × Gre(FL) × Grf(FM) by another con- structible set with the same Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let us construct this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Define W V L,M to be the subset of PV × � d,g �n i=1 Grgi(Cdi) consisting of pairs ([ε], E) where E is a subrepresentation of FY , where Y is the middle term of a triangle M i � Y p � L ε � ΣM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 7 Furthermore, define W V L,M(e, f, g) = {([ε], E) ∈ W V L,M | dim E = g, dim Fp(E) = e, dim Fi−1(E) = f};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' W V L,M(e, f) = {([ε], E) ∈ W V L,M | dim Fp(E) = e, dim Fi−1(E) = f};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' W V,Y L,M(e, f, g) = {([ε], E) ∈ W V L,M | ε ∈ PV⟨Y ⟩, dim E = g, dim Fp(E) = e, dim Fi−1(E) = f};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' W V,Y L,M(e, f) = {([ε], E) ∈ W V L,M | ε ∈ PV⟨Y ⟩, dim Fp(E) = e, dim Fi−1(E) = f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then W V L,M and all the sets defined above are finite disjoint unions of subsets of the form W V,Y L,M(e, f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, since we assumed that C has constructible cones, then the results of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Palu give us the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='12 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1 of [Pal12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The sets W V,Y L,M(e, f, g), W V,Y L,M(e, f), W V L,M(e, f), W V L,M(e, f, g) and W V L,M are constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In [Pal12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1], it is shown that certain sets W Y LM(e, f, g) are constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our sets W V,Y L,M(e, f, g) are the intersection of these W Y LM(e, f, g) with PV × � d,g �n i=1 Grgi(Cdi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' thus they are constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since all the other sets are finite unions of sets of the form W V,Y L,M(e, f, g), they must also be constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Now, consider the constructible map ΨL,M(e, f) : W V L,M(e, f) −→ PV × Gre(FL) × Grf(FM) ([ε], E) �−→ ([ε], Fp(E), Fi−1(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let LV 1 (e, f) be the image of this map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' let LV 2 (e, f) be the complement of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then χ(PV × Gre(FL) × Grf(FM)) = χ(LV 1 (e, f)) + χ(LV 2 (e, f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Therefore our equation becomes (⋆) χ(PV )CCT (L)CCT (M) = xindT (L⊕M) � e,f χ � LV 1 (e, f) � x−ι(e+f) +xindT (L⊕M) � e,f χ � LV 2 (e, f) � x−ι(e+f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will now study the two terms of the right-hand side of (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The first term of the RHS of (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We have an equality xindT (L⊕M) � e,f χ � LV 1 (e, f) � x−ι(e+f) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It is proved in [CC06] (see also Section 3 of [Pal12]) that the fibers of ΨL,M(e, f) are affine spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' As a consequence, we have that χ(LV 1 (e, f)) = χ(W V L,M(e, f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus xindT (L⊕M) � e,f χ � LV 1 (e, f) � x−ι(e+f) = xindT (L⊕M) � e,f χ � W V L,M(e, f) � x−ι(e+f) = � e,f,g,⟨Y ⟩ χ � W V,Y L,M(e, f, g) � x−ι(e+f)+indT (L⊕M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 8 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN Now, by [Pal08, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1], if ([ε], E) lies in W V,Y L,M(e, f, g), then it implies that indT (L ⊕ M) − ι(e + f) = indT (Y ) − ι(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, for a fixed g, consider the map � e,f W V,Y L,M(e, f, g) −→ PV⟨Y ⟩ sending a pair ([ε], E) to [ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This map is obviously surjective if the left-hand side is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, the preimage of any [ε′] is isomorphic to {[ε′]} × Grg(FY ′), where Y ′ sits in a triangle M → Y ′ → L ε′ −→ ΣM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' By definition of PV⟨Y ⟩, the Euler characteristic of all the fibers is the same and is equal to χ(Grg(FY )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus χ( � e,f W V,Y L,M(e, f, g)) = χ(Grg(FY ))χ(PV⟨Y ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' So the sum becomes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' = � e,f,g,Y χ � W V,Y L,M(e, f, g) � x−ι(g)+indT (Y ) = � g,Y χ � � e,f W V,Y L,M(e, f, g) � x−ι(g)+indT (Y ) = � g,Y χ(Grg(FY ))χ(PV⟨Y ⟩)x−ι(g)+indT (Y ) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This finishes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ The second term of the RHS of (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We have an equality xindT (L⊕M) � e,f χ � LV 2 (e, f) � x−ι(e+f) = � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Recall that R = {[η] ∈ P HomC(M, ΣL) | ∃ε ∈ V with βL,M(ε, η) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Define W P HomC(L,ΣM),Y M,L (f, e, g) as before Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='12 and let W R,Y M,L (f, e, g) be the constructible subset of all pairs ([η], E) with η ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For fixed e, f and g, let CR,Y L,M(e, f, g) be the subset of LV 2 (e, f) × W R,Y M,L (f, e, g) consisting of pairs � ([ε], R, S), ([η], E) � such that βL,M(ε, η) ̸= 0, Fi−1(E) = S and Fp(E) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Finally, let CR L,M(e, f) = � g,Y ∈YM,L CR,Y L,M(e, f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Consider the two projections CR L,M(e, f) p1 −→ LV 2 (e, f) CR,Y L,M(e, f, g) p2 −→ W R,Y M,L (f, e, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' By [Pal12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3], p1 and p2 are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, by [Pal12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4], the fibers of p1 are extensions of affine spaces, and those of p2 are affine spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 9 Therefore χ(CR L,M(e, f)) = χ(LV 2 (e, f)) and χ(CR,Y L,M(e, f, g)) = χ(W R,Y M,L (f, e, g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus the left-hand side in the statement is equal to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' = xindT (L⊕M) � e,f χ(LV 2 (e, f))x−ι(e+f) = � e,f χ(CR L,M(e, f))xindT (L⊕M)−ι(e+f) = � e,f,g,Y χ(CR,Y L,M(e, f, g))xindT (L⊕M)−ι(e+f) = � e,f,g,Y χ(W R,Y M,L (f, e, g))xindT (L⊕M)−ι(e+f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Again, by [Pal08, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1], we have that if ([ε], E) lies in W R,Y M,L (f, e, g), then indT (L ⊕ M) − ι(e + f) = indT (Y ) − ι(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, the map � e,f W R,Y M,L (f, e, g) −→ R⟨Y ⟩ sending ([ε], E) to [ε] is surjective (if the left-hand side is non-empty), and its fibers have the form {[ε′]} × Grg(FY ′), where Y ′ sits in a triangle L → Y ′ → M ε′ −→ ΣL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus χ( � e,f W R,Y M,L (f, e, g)) = χ(R⟨Y ⟩)χ(Grg(FY )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Therefore the above sequence of equalities continues: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' = � e,f,g,Y χ(W R,Y M,L (f, e, g))xindT Y −ι(g) = � g,Y χ( � e,f W R,Y M,L (f, e, g))xindT Y −ι(g) = � g,Y χ(R⟨Y ⟩)χ(Grg(FY ))xindT Y −ι(g) = � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10 then follows directly from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='13 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Refined multiplication formula: Frobenius case We will now follow the ideas of [FK10] (see also [Pal12, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The main difference will be that our Frobenius categories can be Hom-infinite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' we will only assume that they are Ext-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will also assume that they are Krull–Schmidt, and that their stable categories have constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since the proofs are very similar to the ones in the triangulated case, in this section we only provide a detailed outline for the Frobenius case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Recollections on Frobenius categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 10 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2-Calabi–Yau Frobenius categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A Frobenius category is an exact category E in the sense of Quillen with enough projectives and enough injectives, in which projectives and injectives coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It is Ext-finite if for any objects X and Y of E, the space Ext1 E(X, Y ) is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It was proved in [Hap87, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4] that if E is a Frobenius category, then its stable category E is triangulated (E is the quotient of E by the ideal of all morphisms factoring through a projective-injective object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that if E is Ext-finite, then E is Hom-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='7 of [FK10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' An Ext-finite Frobenius category is 2- Calabi–Yau if its stable category is 2-Calabi–Yau as a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Cluster-tilting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let E be an Ext-finite Krull–Schmidt 2-Calabi–Yau Frobenius category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='7 of [FK10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' An object T of E is a cluster-tilting object if (1) T is rigid, that is, the space Ext1 E(T, T ) vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (2) for any object X of E, if Ext1 E(T, X) vanishes, then X ∈ add T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' and (3) each object X of E admits a right add T -approximation T X → X and a left add T -approximation X → TX (in other words, the functors HomE(X, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' )|add T and HomE(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=', X)|(add T )op are finitely generated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that if T is a cluster-tilting object, then every indecomposable projective- injective object is isomorphic to a direct summand of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (1) The module category of a preprojective algebra of Dynkin type is a Hom-finite stably 2-Calabi–Yau Frobenius category with a cluster tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It was used in [GLS06] in the categorification of cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Its stable category has constructible cones by [Pal12, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (2) More generally, subcategories Cw of modules over preprojective algebras were were used in [BIRS09, GLS12a] to category cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' These categories have the same properties as those of the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (3) Let 0 < k < n be integers, and put ˆR = C[[x, y]]/(xk − yn−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The group G = ⟨ζ⟩ of n-th roots of unity acts on ˆR by ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='x = ζx and ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='y = ζ−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then the category E = CMG( ˆR) of G-equivariant Cohen–Macaulay ˆR- modules is a (not necessarily Hom-finite) Ext-finite Frobenius category with a cluster tilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Its stable category has constructible cones, since it is equivalent to categories from the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This was used in [JKS16] to give a categorification of the cluster algebra structure of the homogeneous coordinate ring C[Gk,n] of the Grassmannian Gk,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let T be a basic cluster-tilting object of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Write T = T1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' ⊕ Tn, where each Ti is indecomposable, and let C = EndE(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since T is cluster-tilting, we have a functor F = HomE(T, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') : E −→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='mod C, where f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='mod C is the category of finitely generated right C-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For any finitely generated C-modules L and N such that N is finite-dimensional, define ⟨L, N⟩τ = dim HomC(L, N) − dim Ext1 C(L, N), A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 11 ⟨L, N⟩3 = 3 � i=0 (−1)i dim Exti C(L, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that these expressions are well-defined integers, since Exti C(L, N) is finite- dimensional because E is Ext-finite, and since HomC(L, N) is finite-dimensional because L is finitely generated and N is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Finally, let C = EndE(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' As in [KR07, Section 4] and [FK10, Section 3], we view C-modules as C-modules with no composition factors isomorphic to the simple modules corresponding to the projective-injective direct summands of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2 of [FK10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If L and N are finite-dimensional C-modules of the same dimension vector, then for any finite-dimensional C-module Y , we have that ⟨L, Y ⟩3 = ⟨N, Y ⟩3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In view of this proposition, if e is a dimension vector, we can write ⟨e, Y ⟩3 for the value of ⟨L, Y ⟩3 for any C-module L of dimension vector e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5 ([FK10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The cluster character associated with T is the map CCT : Obj(E) −→ Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , xn) defined by CCT (M) = n � i=1 x⟨F M,Si⟩τ i � e χ � Gre(Ext1 E(T, M)) � n � i=1 x⟨e,Si⟩3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let E be a Hom-finite 2-Calabi–Yau Frobenius category with a clus- ter tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that the triangulated category E has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let L and M be two objects of E, and let V be a vector subspace of Ext1 E(L, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then χ(PV )CCT (L)CCT (M) = � Y ∈YL,M χ(PV⟨Y ⟩)CCT (Y ) + � Y ∈YM,L χ(R⟨Y ⟩)CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The proof follows the lines of that of [Pal12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' it is similar to that of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10, but uses [FK10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4] instead of [Pal08, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Applications 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Specialization of cluster variables in cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom- finite 2-Calabi–Yau triangulated category with a basic cluster tilting object T = �n i=1 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Following [CILFS15], let the Caldero-Chapoton algebra AC be the subring of the ring Z[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , x±1 n ] generated by the set of all CCT (X), as X spans all objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Motivated by the reduction of friezes (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2) and the study of mor- phisms of rooted cluster algebras of [ADS14], we wish to study the algebra obtained from AC by specializing xn to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' To fix notation, let σ : Z[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , x±1 n ] → Z[x±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , x±1 n−1] be the morphism sending each of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' , xn−1 to itself and sending xn to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 12 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN The main result of this section is stated in terms of Calabi–Yau reduction: it was proved in [IY08] that the category C′ = � Σ−1Tn �⊥ / (Tn) is a Hom-finite 2-Calabi–Yau triangulated category with a basic cluster tilting ob- ject T ′, where T ′ is the image of T under the projection functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom-finite 2-Calabi–Yau triangulated category with con- structible cones and a basic cluster tilting object T = �n i=1 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then σ(AC) = AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The proof of [ADS14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='13] shows that σ(AC) ⊆ AC′ ⊗Z Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It uses the multiplication formula of Palu [Pal12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1 follows the same lines using our refined multiplication formula (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We nonetheless include the complete argument below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' (of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' It suffices to prove that σ (CCT (X)) ∈ AC′ for all ob- jects X of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The proof is by induction on the dimension of the space HomC(Tn, ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume first that dim HomC(Tn, ΣX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then X ∈ (Σ−1Tn)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Denote by π the projection π : � Σ−1Tn �⊥ → C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then σ (CCT (X)) = CCT ′(πX) ∈ AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Note that this implies that AC′ ⊆ σ(AC), since all objects of C′ have the form π(X) with X an object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume now that dim HomC(Tn, ΣX) = d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Choose any non-split triangle X −→ E −→ Tn ξ−→ ΣX and let V be the span on ξ in HomC(Tn, ΣX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10, we get that CCT (X)CCT (Tn) = CCT (E) + � Y ∈YX,Tn χ � R⟨Y ⟩ � CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since CCT (Tn) = xn, applying the specialization σ to the left-hand side yields σ (CCT (X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since all χ � R⟨Y ⟩ � are integers, it thus suffices to prove that CCT (E) and all CCT (Y ) on the right-hand side are in AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We do this by showing that the dimensions of HomC(Tn, ΣE) and HomC(Tn, ΣY ) are strictly smaller than d and by applying induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' To see this, first apply the functor HomC(Tn, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') to the triangle defined by ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We obtain an exact sequence (Tn, Tn) ξ∗ −→ (Tn, ΣX) f−→ (Tn, ΣE) −→ (Tn, ΣTn), where we write (U, V ) instead of HomC(U, V ) to save space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since Tn is rigid, (Tn, ΣTn) vanishes, so f is surjective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' thus, (Tn, ΣE) ∼= (Tn, ΣX)/ξ∗ � (Tn, Tn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Lastly, ξ∗ � (Tn, Tn) � is non-zero, since it contains ξ∗(idTn) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Therefore, dim(Tn, ΣE) < dim(Tn, ΣX) = d, and by induction, σ (CCT (E)) ∈ AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Now let Y ∈ YTn,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' By definition, there exists a non-split triangle Tn −→ Y −→ X δ−→ ΣTn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 13 Applying the functor HomC(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=', T ) and repeating the above argument, we get that dim(Y, ΣTn) < dim(X, ΣTn) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' By the 2-Calabi–Yau property, dim(Y, ΣTn) = dim(Tn, ΣY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus, by induction, we also have that σ (CCT (Y )) ∈ AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1 has an interesting application to cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Q be a quiver without loops or 2-cycles, i be a vertex of Q and Q′ be the quiver obtained from Q by removing the vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that there exists a non-degenerate potential W such that the generalized cluster category CQ,W is Hom-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If the cluster algebra AQ′ is equal to its upper cluster algebra UQ′ (see [BFZ05] for details), then the specialization σ sending xi to 1 satisfies σ(AQ) = AQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C = CQ,W and C′ = CQ′,W ′, where W ′ is the potential obtained by removing the terms of W involving the vertex i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We know from [Pal08] that AQ ⊂ AC, and from [Pla11a, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='14] that AC ⊂ UQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The same is true if we replace Q with Q′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' thus, by our assumption, AQ′ = AC′ = UQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Applying σ, we get that AQ′ ⊂ σ(AQ) ⊂ σ(AC), and this last set is AC′ by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If Q is mutation-equivalent to an acyclic quiver, then we have that σ(AQ) = AQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that the quiver Q admits a non-degenerate Jacobi-finite potential, and let C be its generalized cluster category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If the upper cluster al- gebra UQ′ is spanned the cluster characters of some objects in C, then we have that σ(UQ) = UQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Notice that we have AC′ ⊂ UQ′ and AC ⊂ UQ by the universal Laurent property of cluster characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since UQ′ is spanned by some cluster characters, we have UQ′ ⊂ AC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Consequently, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='1 implies UQ′ = AC′ = σ(AC) ⊂ σ(UQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Notice that every cluster for Q′ (see [BFZ05]) is the image of some cluster for Q under σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' So we have σ(UQ) ⊂ UQ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The desired claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Many upper cluster algebras are known to possess a generic basis, see [GLS12b, Pla13, Qin19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' They satisfy the assumption in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Reduction of friezes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Q be a quiver without loops or 2-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A frieze is a morphism of rings f : AQ → Z sending every cluster variable of AQ to a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This definition generalizes the originial one of Conway and Cox- eter [CC73b], and has been an area of active interest in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In [BFG+21, Section 5], an operation of reduction on friezes is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The purpose of this section is to show that this reduction operation can be “reversed” by adding a 1 in a frieze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Q be an acyclic quiver without loops or 2-cycles, and let Q′ be the quiver obtained by removing the vertex i in Q, and let σ : AQ → AQ′ be the specialization of xi to 1 (this is well-defined thanks to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let f ′ : AQ′ → Z be a frieze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then there exists a unique frieze f : AQ → Z such that f ′ ◦ σ = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 14 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If f exists, then it is unique, since it is determined by its action on the initial cluster variables of AQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let us prove that such an f exists for any frieze f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4, f = f ′ ◦σ is a well-defined morphism of rings from AQ to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We only need to check that all cluster variables of AQ are sent to positive values by f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' this follows from the positivity theorem [LS15]: any cluster variable of AQ is a Laurent polynomial with nonnegative coefficients in the initial cluster variables, and these are sent to positive values by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='6 can be deduced for friezes where Q is of type An from the results of [CC73b], and was shown to be true in types An, Dn and E6 in [BFG+21, Section 5], where it was also observed to be true for all known friezes of types E7 and E8 by a direct check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The total number of possible friezes in these types is still unknown and was conjectured in [FP16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' A formula for Auslander–Reiten triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' In this section, we will show how Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10 allows for a new proof of the following formula of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Dominguez and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Geiss when C has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='8 (Theorem 1 of [DG14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let C be a Hom-finite 2-Calabi–Yau category with constructible cones and a cluster tilting object T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let Z be an indecomposable object of C, and assume that it sits in an Auslander–Reiten triangle ΣZ α−→ Y β−→ Z ε−→ Σ2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then CCT (Z)CCT (ΣZ) = CCT (Y ) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' We will give a proof of this theorem under the additionnal assumption that C has constructible cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let V be the one-dimensional subspace of HomC(Z, Σ2Z) generated by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Ap- plying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10, we get CCT (Z)CCT (ΣZ) = CCT (Y ) + � E∈YΣZ,Z χ(R⟨E⟩)CCT (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Here R = {[η] ∈ P HomC(ΣZ, ΣZ) | βZ,ΣZ(ε, η) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Let us show that [η] lies in R if and only if η is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Since Z is indecomposable, HomC(ΣZ, ΣZ) is a local ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Hence η is an isomor- phism if and only if it does not lie in the radical of HomC(ΣZ, ΣZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus we need to show that η lies in the radical of HomC(ΣZ, ΣZ) if and only if βZ,ΣZ(ε, η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume that η is in the radical (assume η ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' the case η = 0 is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then it is not an isomorphism, and since Z is indecomposable, it is not a retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then, by definition of an Auslander–Reiten triangle, we must have that there exists a morphism f : Z → Y such that Σ−1η = βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' But then βZ,ΣZ(ε, η) = βZ,ΣZ(ε, ΣβΣf) = βY,ΣZ(εβ, Σf) = βY,ΣZ(0, Σf) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Assume next that η is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Then η and rad HomC(ΣZ, ΣZ) gen- erate HomC(ΣZ, ΣZ) as a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' If βZ,ΣZ(ε, η) were to vanish, it would A REFINED MULTIPLICATION FORMULA FOR CLUSTER CHARACTERS 15 thus vanish for any η′ in HomC(ΣZ, ΣZ), contradicting the fact that βZ,ΣZ is non- degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus βZ,ΣZ(ε, η) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This proves that R is the set of [η], with η an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' But then R = R⟨0⟩ (since the middle term of a triangle associated with an isomorphism is 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, R = P HomC(ΣZ, ΣZ) \\ P rad HomC(ΣZ, ΣZ) is an affine space, since rad HomC(ΣZ, ΣZ) is a hyperplane in HomC(ΣZ, ΣZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus χ(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Therefore CCT (Z)CCT (ΣZ) = CCT (Y ) + � E∈YZ,ΣZ χ(R⟨E⟩)CCT (E) = CCT (Y ) + χ(R⟨0⟩)CCT (0) = CCT (Y ) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Another restricted formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10 allows us to obtain (always assuming contructibility of cones) the following formula, reminiscent of the one stated in [DX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' For two objects L and M of C, let (T )(L, M) be the space of morphisms from L to M factoring through an object of add T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Under the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10, we have that χ � P(T )(L, ΣM) � CCT (L)CCT (M) = � Y ∈YL,M χ � P(T )(L, ΣM)⟨Y ⟩ � CCT (Y ) + � Y ∈YM,L χ � P HomC(M, ΣL)⟨Y ⟩ \\ P(T )(M, ΣL)⟨Y ⟩ � CCT (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='10 by taking V = (T )(L, ΣM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' To see this, we only need to prove that Ker βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') = (T )(M, ΣL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Notice first that (T )(M, ΣL) is contained in Ker βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' indeed, if f ∈ V and g ∈ (T )(M, ΣL), then Σg ◦ f = 0 (since T is rigid), so βL,M(f, g) = βL,ΣL(Σg ◦ f, idΣL) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Moreover, dim Ker βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') = dim HomC(L, ΣM)/V , and this last vector space is isomorphic to the dual of (T )(M, ΣL) thanks to [Pal08, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' Thus Ker βL,M(V, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=') and (T )(M, ΣL) have the same (finite) dimension, and so they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' □ Acknowledgements The first and second authors were supported by the French ANR grant CHARMS (ANR-19-CE40-0017-02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The second author was supported by the Institut Univer- sitaire de France (IUF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The third author was supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 12271347).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The final stages of this project were completed while the first and second author were participating in a trimester programme at the Isaac Newton Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hos- pitality during the programme Cluster algebras and representation theory where work on this paper was undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' This work was supported by EPSRC grant no EP/R014604/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} 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+page_content=' Relative cluster categories and Higgs categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='03707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' 18 BERNHARD KELLER, PIERRE-GUY PLAMONDON, AND FAN QIN B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' : Universit´e Paris de Paris, UFR de Math´ematiques, CNRS, Institut de Math´ematiques de Jussieu–Paris Rive Gauche, IMJ-PRG, Bˆatiment Sophie Germain, 75205 Paris Cedex 13, France Email address: bernhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='keller@imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='fr URL: https://webusers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='fr/~bernhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='keller/ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content=' : Laboratoire de Math´ematiques de Versailles, UVSQ, CNRS, Universit´e Paris-Saclay, Institut Universitaire de France (IUF) Email address: pierre-guy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='plamondon@uvsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='fr URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfIPul/content/2301.01059v1.pdf'} +page_content='imo.' metadata={'source': 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b/KtE4T4oBgHgl3EQfJQzO/content/tmp_files/2301.04920v1.pdf.txt @@ -0,0 +1,1730 @@ +On the Validity of Consensus +PIERRE CIVIT, Sorbonne University, France +SETH GILBERT, NUS Singapore, Singapore +RACHID GUERRAOUI, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland +JOVAN KOMATOVIC, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland +MANUEL VIDIGUEIRA, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland +The Byzantine consensus problem involves 𝑛 processes, out of which 𝑡 < 𝑛 could be faulty and +behave arbitrarily. Three properties characterize consensus: (1) termination, requiring correct +(non-faulty) processes to eventually reach a decision, (2) agreement, preventing them from deciding +different values, and (3) validity, precluding “unreasonable” decisions. But, what is a reasonable +decision? Strong validity, a classical property, stipulates that, if all correct processes propose the +same value, only that value can be decided. Weak validity, another established property, stipulates +that, if all processes are correct and they propose the same value, that value must be decided. The +space of possible validity properties is vast. However, their impact on consensus remains unclear. +This paper addresses the question of which validity properties allow Byzantine consensus to +be solvable with partial synchrony, and at what cost. First, we determine necessary and sufficient +conditions for a validity property to make the consensus problem solvable; we say that such validity +properties are solvable. Notably, we prove that, if 𝑛 ≤ 3𝑡, all solvable validity properties are trivial +(there exists an always-admissible decision). Furthermore, we show that, with any non-trivial +(and solvable) validity property, consensus requires Ω(𝑡2) messages. This extends the seminal +Dolev-Reischuk bound, originally proven for strong validity, to all non-trivial validity properties. +Lastly, we give a general Byzantine consensus algorithm, we call Universal, for any solvable (and +non-trivial) validity property. Importantly, Universal incurs 𝑂(𝑛2) message complexity. Thus, +together with our lower bound, Universal implies a fundamental result in partial synchrony: with +𝑡 ∈ Ω(𝑛), the message complexity of all (non-trivial) consensus variants is Θ(𝑛2). +1 +INTRODUCTION +Consensus [50] is the cornerstone of state machine replication (SMR) [1, 8, 9, 21, 46, 47, 57, 61, 77], +as well as various distributed protocols [13, 37, 39, 40]. Recently, it has received a lot of attention +with the advent of blockchain systems [5, 6, 17, 26, 28, 38, 55]. The consensus problem is posed in a +system of 𝑛 processes, out of which 𝑡 < 𝑛 can be faulty, and the rest are correct. Each correct process +proposes a value, and consensus enables correct processes to decide on a common value. In this +paper, we consider Byzantine [50] consensus, where faulty processes can behave arbitrarily. While +the exact definition of the problem might vary, two properties are always present: (1) termination, +requiring correct processes to eventually decide, and (2) agreement, preventing them from deciding +different values. It is not hard to devise an algorithm that satisfies only these two properties: every +correct process decides the same, predetermined value. However, this algorithm is vacuous. To +preclude such trivial solutions and render consensus meaningful, an additional property is required: +validity, defining which decisions are admissible. +The many faces of validity. The literature contains many flavors of validity [4, 23, 24, 28, 36, 45, +59, 73, 74, 78]. One of the most studied properties is Strong Validity [4, 23, 28, 45], stipulating that, +if all correct processes propose the same value, only that value can be decided. Another common +property is Weak Validity [23, 24, 78], affirming that, if all processes are correct and propose the +same value, that value must be decided. In fact, many other variants of the property have been +1 +arXiv:2301.04920v1 [cs.DC] 12 Jan 2023 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +considered [36, 59, 73, 74]. While validity may appear as an inconspicuous property, its exact +definition has a big impact on consensus algorithms. For example, the seminal Dolev-Reischuk +bound [30] states that any solution to consensus with Strong Validity incurs a quadratic number of +messages; it was recently proven that the bound is tight [23, 52, 62]. In contrast, while there have +been several improvements to the performance of consensus with Weak Validity over the last 40 +years [23, 52, 78], the (tight) lower bound on message complexity remains unknown. (Although +the bound is conjectured to be the same as for Strong Validity, this has yet to be formally proven.) +Many other fundamental questions remain unanswered: +• What does it take for a specific validity property to make consensus solvable? +• What are the (best) upper and lower bounds on the message complexity of consensus with +any specific validity property? +• Is there a hierarchy of validity properties (e.g., a “strongest” validity property)? +To the best of our knowledge, no in-depth study of the validity property has ever been conducted, +despite its importance [2]. +Contributions. We propose a precise mathematical formalism for the analysis of validity properties. +We define a validity property as a mapping from assignments of proposals into admissible decisions. +Although simple, our formalism enables us to determine the exact impact of validity on the +solvability and complexity of consensus in the classical partially synchronous model [32], and +answer the aforementioned open questions. Namely, we provide the following contributions: +• We classify all validity properties into solvable and unsolvable ones. (If a validity property +makes consensus solvable, we say that the property itself is solvable.) Specifically, for 𝑛 ≤ 3𝑡, +we show that only trivial validity properties (for which there exists an always-admissible +decision) are solvable. In the case of 𝑛 > 3𝑡, we define the similarity condition, which we +prove to be necessary and sufficient for a validity property to be solvable. +• We prove that all non-trivial (and solvable) validity properties require Ω(𝑡2) exchanged +messages. This result extends the Dolev-Reischuk bound [30], proven only for Strong Validity, +to all “reasonable” validity properties. +• Finally, we present Universal, a general consensus algorithm for all solvable (and non- +trivial) validity properties. Importantly, assuming a threshold signature scheme, Universal +exchanges 𝑂(𝑛2) messages. Thus, together with our lower bound, Universal implies a +fundamental result in partial synchrony: given 𝑡 ∈ Ω(𝑛), all (non-trivial) consensus variants +have Θ(𝑛2) message complexity. Figure 1 summarizes our findings. +Technical overview. In our formalism, we use the notion of input configuration that denotes an +assignment of proposals to correct processes. For example, +� +(𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣) +� represents an +input configuration by which (1) only processes 𝑃1, 𝑃2, and 𝑃3 are correct, and (2) processes 𝑃1, 𝑃2, +and 𝑃3 propose 𝑣. First, we define a similarity relation between input configurations: two input +configurations are similar if and only if (1) they have (at least) one process in common, and (2) for +every common process, the process’s proposal is identical in both input configurations. For example, +an input configuration 𝑐 = +� +(𝑃1, 0), (𝑃2, 1) +� +is similar to +� +(𝑃1, 0), (𝑃3, 0) +� +, but not to +� +(𝑃1, 0), (𝑃2, 0) +� +. +We observe that all similar input configurations must have an admissible value in common; we +call this canonical similarity. Let us illustrate why a common admissible value must exist. Consider +the aforementioned similar input configurations 𝑐 = +� +(𝑃1, 0), (𝑃2, 1) +� and 𝑐′ = +� +(𝑃1, 0), (𝑃3, 0) +�. If +there is no common admissible value for 𝑐 and 𝑐′, consensus cannot be solved: process 𝑃1 cannot +distinguish (1) an execution in which 𝑃2 is correct, and 𝑃3 is faulty and silent, from (2) an execution +in which 𝑃2 is faulty, but behaves correctly, and 𝑃3 is correct, but slow. Thus, 𝑃1 cannot conclude +whether it needs to decide an admissible value for 𝑐 or for 𝑐′. Canonical similarity is a critical +2 + +On the Validity of Consensus +trivial +non-trivial +solvable +Validity +properties +Fig. 1. Illustration of our results: (1) with 𝑛 ≤ 3𝑡, all solvable validity properties are trivial; (2) the exact set of +solvable validity properties (as determined by our necessary and sufficient conditions); (3) all non-trivial (and +solvable) validity properties require Ω(𝑡2) exchanged messages; (4) for any non-trivial (and solvable) validity +property, there exists a consensus algorithm with 𝑂(𝑛2) message complexity. +intermediate result that we use extensively throughout the paper (even if it does not directly imply +any of our results). +In our proof of triviality with 𝑛 ≤ 3𝑡, we intertwine the classical partitioning argument [51] +with our canonical similarity result. Namely, we show that, for any input configuration, there +exists an execution in which the same value 𝑥 is decided, making 𝑥 an always-admissible value. +For our lower bound, while following the idea of the original proof [3, 30], we rely on canonical +similarity to prove the bound for all solvable and non-trivial validity properties. Finally, we design +Universal by relying on vector consensus [27, 31, 69, 76], a problem in which processes agree on +the proposals of 𝑛 − 𝑡 processes: when a correct process decides a vector vec of 𝑛 − 𝑡 proposals +(from vector consensus), it decides (from Universal) the common admissible value for all input +configurations similar to vec. For example, consider an execution which corresponds to an input +configuration 𝑐. If, in this execution, a correct process decides a vector vec from vector consensus, +it is guaranteed that vec is similar to 𝑐 (the proposals of correct processes are identical in 𝑐 and in +vec). Hence, deciding (from Universal) the common admissible value for all input configurations +similar to vec guarantees that the decided value is admissible according to 𝑐. +Roadmap. We provide an overview of related work in §2. In §3, we specify the system model +(§3.1), define the consensus problem (§3.2), describe our formalism for validity properties (§3.3), +and present canonical similarity (§3.4). We define the necessary conditions for the solvability of +validity properties in §4. In §5, we prove a quadratic lower bound on message complexity for all +non-trivial (and solvable) validity properties (§5.1), and introduce Universal, a general consensus +algorithm for any solvable (and non-trivial) validity property (§5.2). We conclude the paper in §6. +The appendix contains (1) detailed proofs of our results, (2) omitted algorithms, and (3) a proposal +for how to extend our formalism to accommodate for blockchain-specific validity properties. +2 +RELATED WORK +Solvability of consensus. The consensus problem has been thoroughly investigated in a variety of +system settings and failure models. It has been known (for long) that consensus can be solved in +3 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +a synchronous setting, both with crash [19, 56, 70] and arbitrary failures [4, 48, 62, 70, 72]. In an +asynchronous environment, however, consensus cannot be solved deterministically even if a single +process can fail, and it does so only by crashing; this is the seminal FLP impossibility result [35]. +A traditional way of circumventing the FLP impossibility result is randomization [7, 10, 11, 33, +54], where termination of consensus is not ensured deterministically. Another well-established +approach to bypass the FLP impossibility is to strengthen the communication model with partial +synchrony [32]: communication is asynchronous until some unknown time, and then it becomes +synchronous. The last couple of decades have produced many partially synchronous consensus +algorithms [18, 21, 23, 24, 28, 32, 49, 52, 56, 78]. +Another line of research has consisted in weakening the definition of consensus to make it +deterministically solvable under asynchrony. In the condition-based approach [64], the specification +of consensus is relaxed to require termination only if the assignment of proposals satisfies some +predetermined conditions. The efficiency of this elegant approach has been studied further in [66], +then extended to the synchronous setting [65, 79], as well as to the 𝑘-set agreement problem [41, 67]. +Solvability of general decision problems. A distributed decision problem has been defined in [25, +44, 60] as a mapping from input assignments to admissible decisions. Our validity formalism is of +the same nature, and it is inspired by the aforementioned specification of decision problems. +The solvability of decision problems has been thoroughly studied in asynchronous, crash-prone +settings. It was shown in [63] that the FLP impossibility result [35] can be extended to many decision +problems. In [15], the authors defined necessary and sufficient conditions for a decision problem to +be solvable with a single crash failure. The asynchronous solvability of problems in which crash +failures occur at the very beginning of an execution was studied in [75]. Necessary and sufficient +conditions for a decision problem to be solvable in a randomized manner were given in [22]. The +topology-based approach on studying the solvability of decision problems in asynchrony has proven +to be extremely effective, both for crash [42, 43, 71] and arbitrary failures [42, 60]. Our results +follow the same spirit as many of these approaches; however, we study the deterministic solvability +and complexity of all consensus variants in a partially synchronous environment. +Validity of consensus. Various validity properties have been associated with the consensus problem +(beyond the aforementioned Strong Validity and Weak Validity). Correct-Proposal Validity [36, 73] +states that a value decided by a correct process must have been proposed by a correct process. +Median Validity [74] is another validity property proposed in the context of synchronous consensus, +requiring the decision to be close to the median of the proposals of correct processes. Interval +Validity [59], on the other hand, requires the decision to be close to the 𝑘-th smallest proposal +of correct processes. The advent of blockchain technologies has resurged the concept of External +Validity [18, 20, 78]. This property requires the decided value to satisfy a predetermined predicate, +typically asserting whether the decided value follows the rules of a blockchain system (e.g., no +double-spending). (For pedagogical purposes, we considered a simple formalism to express basic +validity properties and derive our results. To express External Validity, which is out of the scope of +the paper, we propose an extension of our formalism in Appendix D.) +In interactive consistency [12, 34, 58], correct processes agree on the proposals of all correct +processes. Given that the problem is impossible in a non-synchronous setting, a weaker variant +has been considered: vector consensus [27, 31, 69, 76]. Here, processes need to agree on a vector of +proposals which does not necessarily include the proposals of all correct processes. Interactive +consistency and vector consensus can be seen as specific consensus problems with a validity +property requiring that, if a decided vector contains a proposal 𝑣 of a correct process, that correct +process has indeed proposed value 𝑣. The design of Universal, our general consensus algorithm +4 + +On the Validity of Consensus +for any solvable (and non-trivial) validity property, demonstrates that any non-trivial flavor of +consensus, which is solvable in partial synchrony, can be solved using vector consensus (see §5.2). +3 +PRELIMINARIES +In this section, we present the computational model (§3.1), recall the consensus problem (§3.2), +formally define validity properties (§3.3), and introduce canonical similarity (§3.4). +3.1 +Computational Model +Processes. We consider a system Π = {𝑃1, 𝑃2, ..., 𝑃𝑛} of 𝑛 processes. At most 𝑡 (0 < 𝑡 < 𝑛) processes +can be faulty: these processes can exhibit arbitrary behavior. A non-faulty process is said to be +correct. Processes communicate by exchanging messages over an authenticated point-to-point +network. The communication network is reliable: if a correct process sends a message to a correct +process, the message is eventually received. +Executions. Given an algorithm A, execs(A) denotes the set of all executions of A. Furthermore, +Corr A(E) denotes the set of correct processes in an execution E ∈ execs(A). We say that an +execution E ∈ execs(A) is canonical if and only if no faulty process takes any computational step +in E; note that faulty processes do not send any message in a canonical execution. +Partial synchrony. We consider the standard partially synchronous model [32]. For every execu- +tion of the system, there exists a Global Stabilization Time (GST) and a positive duration 𝛿 such +that message delays are bounded by 𝛿 after GST. GST is not known to processes, whereas 𝛿 is. We +assume that all correct processes start executing their local algorithm before or at GST. +Cryptographic primitives. In one variant of the Universal algorithm, we assume a (𝑘,𝑛)-threshold +signature scheme [53] (see §5.2). In fact, this variant relies on a closed-box consensus algorithm +which internally utilizes threshold signatures. In a threshold signature scheme, each process holds +a distinct private key, and there exists a single public key. Each process 𝑃𝑖 can use its private key to +produce a (partial) signature of a message 𝑚; we denote by ⟨𝑚⟩𝜎𝑖 a message (partially) signed by +the process 𝑃𝑖. Moreover, a signature can be verified by other processes. Finally, a set of signatures +for a message 𝑚 from 𝑘 (the threshold) distinct processes can be combined into a single threshold +signature for 𝑚, which proves that 𝑘 processes have signed 𝑚. +Message complexity. Let A be any algorithm and let E ∈ execs(A) be any execution of A. The +message complexity of E is the number of messages sent by correct processes during [GST, ∞]. +The message complexity of A is defined as +max +E∈execs(A) +� +message complexity of E +� +. +3.2 +Consensus +We denote by V𝐼 the set of values processes can propose, and by V𝑂 the set of values processes +can decide. The consensus1 problem exposes the following interface: +• request propose(𝑣 ∈ V𝐼): a process proposes a value 𝑣. +• indication decide(𝑣 ′ ∈ V𝑂): a process decides a value 𝑣 ′. +A correct process proposes and decides at most once. Consensus requires the following properties: +• Termination: Every correct process eventually decides. +• Agreement: No two correct processes decide different values. +1Throughout the entire paper, we use “consensus” and “Byzantine consensus” interchangeably. +5 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +If the consensus problem was completely defined by Termination and Agreement, a trivial solution +would exist: processes decide on a default value. Therefore, the specification of consensus addition- +ally includes a validity property, which connects the proposals of correct processes to admissible +decisions, precluding the aforementioned trivial solutions. +3.3 +Validity +In a nutshell, our specification of a validity property includes a set of assignments of proposals to +correct processes, and, for each such assignment, a corresponding set of admissible decisions. +We start by defining a process-proposal pair as a pair (𝑃, 𝑣), where (1) 𝑃 ∈ Π is a process, and +(2) 𝑣 ∈ V𝐼 is a proposal. Given a process-proposal pair pp = (𝑃, 𝑣), proposal(pp) = 𝑣 denotes the +proposal associated with pp. +An input configuration is a tuple +� +pp1, pp2, ..., pp𝑥 +� +of 𝑥 process-proposal pairs, where (1) 𝑛 − 𝑡 ≤ +𝑥 ≤ 𝑛, and (2) every process-proposal pair is associated with a distinct process. Intuitively, an +input configuration represents an assignment of proposals to correct processes. For example, an +input configuration +� +(𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣), (𝑃4, 𝑣), (𝑃5, 𝑣) +� +describes an execution in which (1) only +processes 𝑃1, 𝑃2, 𝑃3, 𝑃4 and 𝑃5 are correct, and (2) all of them propose the same value 𝑣. +We denote by I the set of all input configurations. Furthermore, for every 𝑥 ∈ [𝑛 − 𝑡,𝑛], I𝑥 ⊂ I +denotes the set of input configurations with exactly 𝑥 process-proposal pairs. For every input +configuration 𝑐 ∈ I, we denote by 𝑐[𝑖] the process-proposal pair associated with process 𝑃𝑖; if such +a process-proposal pair does not exist, 𝑐[𝑖] = ⊥. Finally, 𝜋(𝑐) = {𝑃𝑖 ∈ Π | 𝑐[𝑖] ≠ ⊥} denotes the set +of all processes included in 𝑐. +Given (1) an execution E of an algorithm A, where A exposes the propose(·)/decide(·) interface, +and (2) an input configuration 𝑐 ∈ I, we say that E corresponds to 𝑐 if and only if (1) 𝜋(𝑐) = +Corr A(E), and (2) for every process 𝑃𝑖 ∈ Corr A(E), 𝑃𝑖’s proposal in E is proposal(𝑐[𝑖]). We +denote by corresponding(E) = 𝑐 the input configuration to which E corresponds. +Finally, we define a validity property val as a function val : I → 2V𝑂 such that, for every +input configuration 𝑐 ∈ I, val(c) ≠ ∅. An algorithm A, where A exposes the propose(·)/decide(·) +interface, satisfies a validity property val if and only if, in every execution E ∈ execs(A), no correct +process decides a value 𝑣 ′ ∉ val�corresponding(E)�. That is, an algorithm satisfies a validity +property if and only if correct processes decide only admissible values. +Weak Validity & Strong Validity in our formalism. To illustrate our formalism, we describe how it +can be used to express these two properties. For both, V𝐼 = V𝑂. Weak Validity can be expressed as: +val(𝑐) = +� +{𝑣}, +if (𝜋(𝑐) = Π) ∧ (∀𝑃𝑖 ∈ 𝜋(𝑐) : proposal(𝑐[𝑖]) = 𝑣) +V𝑂, +otherwise +whereas Strong Validity can be expressed as: +val(𝑐) = +� +{𝑣}, +if ∀𝑃𝑖 ∈ 𝜋(𝑐) : proposal(𝑐[𝑖]) = 𝑣 +V𝑂, +otherwise +Consensus algorithms. An algorithm A solves consensus with a validity property val if and only +if the following holds: +• A exposes the propose(·)/decide(·) interface, and +• A satisfies Termination, Agreement and the validity property val. +Lastly, we formally define the notion of a solvable validity property. +Definition 1 (Solvable validity property). We say that a validity property val is solvable if and +only if there exists an algorithm which solves consensus with val. +6 + +On the Validity of Consensus +3.4 +Canonical Similarity +In this subsection, we introduce canonical similarity, a crucial intermediate result. In order to do so, +we first define an important relation between input configurations, that of similarity. +Similarity. We define the similarity relation (“∼”) between input configurations: +∀𝑐1,𝑐2 ∈ I : 𝑐1 ∼ 𝑐2 ⇐⇒ (𝜋(𝑐1) ∩ 𝜋(𝑐2) ≠ ∅) ∧ (∀𝑃𝑗 ∈ 𝜋(𝑐1) ∩ 𝜋(𝑐2) : 𝑐1[𝑗] = 𝑐2[𝑗]). +In other words, 𝑐1 is similar to 𝑐2 if and only if (1) 𝑐1 and 𝑐2 have at least one process in common, +and (2) for every common process, the process’s proposal is identical in both input configurations. +For example, 𝑐 = +� +(𝑃1, 0), (𝑃2, 1), (𝑃3, 0) +� is similar to +� +(𝑃1, 0), (𝑃3, 0) +�, whereas 𝑐 is not similar to +� +(𝑃1, 0), (𝑃2, 0), (𝑃3, 0) +�. Note that the similarity relation is symmetric (for every pair 𝑐1,𝑐2 ∈ I, +𝑐1 ∼ 𝑐2 ⇔ 𝑐2 ∼ 𝑐1) and reflexive (for every 𝑐 ∈ I, 𝑐 ∼ 𝑐). +For every input configuration 𝑐 ∈ I, we define its similarity set, denoted by sim(𝑐): +sim(𝑐) = {𝑐′ ∈ I | 𝑐′ ∼ 𝑐}. +The canonical similarity result. Let A be an algorithm which solves consensus with some validity +property val. Our canonical similarity result states that A, in any canonical execution which +corresponds to some input configuration 𝑐, can only decide a value which is admissible for all input +configurations similar to 𝑐. Informally, the reason is that correct processes cannot distinguish silent +faulty processes from slow correct ones. +Lemma 1 (Canonical similarity). Let val be any solvable validity property and let A be any +algorithm which solves the consensus problem with val. Let E ∈ execs(A) be any canonical +execution and let corresponding(E) = 𝑐, for some input configuration 𝑐 ∈ I. If a value 𝑣 ′ ∈ V𝑂 is +decided by a correct process in E, then 𝑣 ′ ∈ +� +𝑐′∈sim(𝑐) +val(𝑐′). +Proof. We prove the lemma by contradiction. Suppose that 𝑣 ′ ∉ +� +𝑐′∈sim(𝑐) +val(𝑐′). Hence, there +exists an input configuration 𝑐′ ∈ sim(𝑐) such that 𝑣 ′ ∉ val(𝑐′). Let E𝑃 denote any infinite +continuation of E such that corresponding(E𝑃) = 𝑐. Let 𝑃 be any process such that 𝑃 ∈ 𝜋(𝑐′)∩𝜋(𝑐); +such a process exists as 𝑐′ ∼ 𝑐. As A satisfies Termination and Agreement, E𝑃 is an infinite execution, +and 𝑃 is correct in E𝑃, 𝑃 decides 𝑣 ′ in E𝑃. +We construct another execution E′ ∈ execs(A) such that corresponding(E′) = 𝑐′: +(1) E′ is identical to E𝑃 until process 𝑃 decides 𝑣 ′. +(2) All processes in Π \ 𝜋(𝑐′) are faulty in E′ (they behave correctly until 𝑃 has decided), and all +processes in 𝜋(𝑐′) are correct. +(3) After 𝑃 has decided, processes in 𝜋(𝑐′) \ 𝜋(𝑐) “wake up” with the proposals specified in 𝑐′. +(4) GST is set to after all processes in 𝜋(𝑐′) have taken a computational step. +For every process 𝑃𝑖 ∈ 𝜋(𝑐′)∩𝜋(𝑐), the proposal of 𝑃𝑖 in E′ is proposal(𝑐′[𝑖]); recall that𝑐′[𝑖] = 𝑐[𝑖] +as 𝑐′ ∼ 𝑐. Moreover, for every process 𝑃𝑗 ∈ 𝜋(𝑐′) \ 𝜋(𝑐), the proposal of 𝑃𝑗 in E′ is proposal(𝑐′[𝑗]) +(due to the step 3 of the construction). Hence, corresponding(E′) = 𝑐′. Furthermore, process 𝑃, +which is correct in E′, decides a value 𝑣 ′ ∉ val(𝑐′) (due to the step 1 of the construction). Thus, we +reach a contradiction with the fact that A satisfies val, which proves the lemma. +□ +4 +NECESSARY SOLVABILITY CONDITIONS +We give in this section necessary conditions for the solvability of validity properties. We start by +focusing on the case of 𝑛 ≤ 3𝑡: we prove that, if 𝑛 ≤ 3𝑡, all solvable validity properties are trivial +(§4.1). Then, we consider the case of 𝑛 > 3𝑡: we formally define the similarity condition, and prove +its necessity for solvable validity properties (§4.2). +7 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +4.1 +Triviality of Solvable Validity Properties if 𝑛 ≤ 3𝑡 +Some validity properties, such as Weak Validity and Strong Validity, are known to be unsolvable for +𝑛 ≤ 3𝑡 [32]. This seems to imply a split of validity properties depending on the resiliency threshold. +We prove that such a split indeed exists for 𝑛 ≤ 3𝑡, and, importantly, that it applies to all solvable +validity properties. Implicitly, this means that there is no “useful” relaxation of the validity property +that can tolerate 𝑡 > ⌊𝑛/3⌋ failures. Concretely, we prove the following theorem: +Theorem 1. If a validity property is solvable with 𝑛 ≤ 3𝑡, then the validity property is trivial. i.e., +there exists a value 𝑣 ′ ∈ V𝑂 such that 𝑣 ′ ∈ � +𝑐 ∈I +val(𝑐). +Before presenting the proof of the theorem, we introduce the compatibility relation between +input configurations, which we use throughout this subsection. +Compatibility. We define the compatibility relation (“⋄”) between input configurations: +∀𝑐1,𝑐2 ∈ I : 𝑐1 ⋄𝑐2 ⇐⇒ (|𝜋(𝑐1) ∩ 𝜋(𝑐2)| ≤ 𝑡) ∧ (∃𝑃 ∈ 𝜋(𝑐1) \ 𝜋(𝑐2)) ∧ (∃𝑄 ∈ 𝜋(𝑐2) \ 𝜋(𝑐1)). +That is, 𝑐1 is compatible with 𝑐2 if and only if (1) there are at most 𝑡 processes in common, (2) there +exists a process which belongs to 𝑐1 and does not belong to 𝑐2, and (3) there exists a process which +belongs to 𝑐2 and does not belong to 𝑐1. For example, when 𝑛 = 3 and 𝑡 = 1, 𝑐 = +� +(𝑃1, 0), (𝑃2, 0) +� +is +compatible with +� +(𝑃1, 1), (𝑃3, 1) +� +, whereas 𝑐 is not compatible with +� +(𝑃1, 1), (𝑃2, 1), (𝑃3, 1) +� +. Observe +that the compatibility relation is symmetric and irreflexive. +Proof of Theorem 1. Throughout the rest of the subsection, we fix any validity property val which +is solvable with 𝑛 ≤ 3𝑡. Moreover: +• We assume that 𝑛 ≤ 3𝑡. +• We fix any algorithm A which solves consensus with val. +• We fix any input configuration base ∈ I𝑛−𝑡 with exactly 𝑛 − 𝑡 process-proposal pairs. +• We fix any infinite canonical execution Ebase ∈ execs(A) such that corresponding(Ebase) = +base. As A satisfies Termination and val, and Ebase is infinite, some value 𝑣base ∈ val(base) is +decided by a correct process in Ebase. +To prove Theorem 1, it suffices to prove that val is trivial. Our proof leverages our formalism, +specifically the aforementioned compatibility relation and the established canonical similarity +result (§3.4), and combines the formalism with the classical partitioning argument [50]. +We start by showing that only 𝑣base can be decided in any canonical execution which corresponds +to any input configuration compatible with base. If a value different from 𝑣base is decided, the +adversary would be able to cause a disagreement by partitioning processes into two disagreeing +groups. We delegate the formal proof of the following lemma to Appendix A. +Lemma 2. Let 𝑐 ∈ I be any input configuration such that 𝑐 ⋄ base. Let E𝑐 ∈ execs(A) be any +canonical execution such that corresponding(E𝑐) = 𝑐. If a value 𝑣𝑐 ∈ V𝑂 is decided by a correct +process in E𝑐, then 𝑣𝑐 = 𝑣base. +Observe that the proposals of an input configuration compatible with base do not influence the +decision: given an input configuration 𝑐 ∈ I, 𝑐 ⋄ base, only 𝑣base can be decided in any canonical +execution which corresponds to 𝑐, irrespectively of the proposals. +Next, we prove a direct consequence of Lemma 2: for every input configuration 𝑐𝑛 ∈ I𝑛, there +exists an execution E𝑛 such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. Below, we +give a proof sketch of the claim; the formal proof can be found in Appendix A. +Lemma 3. For every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 ∈ execs(A) such +that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. +8 + +On the Validity of Consensus +Proof Sketch. Fix any input configuration 𝑐𝑛 ∈ I𝑛. There exists an input configuration 𝑐𝑛−𝑡 ∈ +I𝑛−𝑡 such that (1) for every process 𝑃 ∈ 𝜋(𝑐𝑛) ∩ 𝜋(𝑐𝑛−𝑡), proposals of 𝑃 in 𝑐𝑛 and 𝑐𝑛−𝑡 are identical, +and (2)𝑐𝑛−𝑡⋄base. Hence, 𝑣base is decided in any infinite canonical execution E𝑛−𝑡 which corresponds +to 𝑐𝑛−𝑡 (by Lemma 2). Thus, by “waking up” processes in 𝜋(𝑐𝑛) \ 𝜋(𝑐𝑛−𝑡) after 𝑣base is decided in +E𝑛−𝑡, we build an execution E𝑛 such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. □ +We are now ready to prove that val is trivial. We do so by showing that, for every input +configuration 𝑐 ∈ I, 𝑣base ∈ val(𝑐). +Lemma 4. Validity property val is trivial. +Proof. We fix any input configuration 𝑐 ∈ I. Let us distinguish two possible scenarios: +• Let 𝑐 ∈ I𝑛. There exists an execution E𝑐 ∈ execs(A) such that (1) corresponding(E𝑐) = 𝑐, +and (2) 𝑣base is decided in E𝑐 (by Lemma 3). As A satisfies val, 𝑣base ∈ val(𝑐). +• Let 𝑐 ∉ I𝑛. We construct an input configuration 𝑐𝑛 ∈ I𝑛 in the following way: +(1) Let 𝑐𝑛 ← 𝑐. +(2) For every process 𝑃 ∉ 𝜋(𝑐), (𝑃, any proposal) is included in 𝑐𝑛. +Due to the construction of 𝑐𝑛, 𝑐𝑛 ∼ 𝑐. By Lemma 3, there exists an execution E𝑛 ∈ execs(A) +such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. Note that E𝑛 is a canonical +execution. Therefore, canonical similarity ensures that 𝑣base ∈ val(𝑐) (Lemma 1). +In both possible cases, 𝑣base ∈ val(𝑐). Thus, the theorem. +□ +Lemma 4 concludes the proof of Theorem 1, as Lemma 4 proves that val, any solvable validity +property with 𝑛 ≤ 3𝑡, is trivial. Figure 2 depicts the proof of Theorem 1. Since this subsection +showed that no useful consensus variant exists when 𝑛 ≤ 3𝑡, the rest of the paper focuses on the +case of 𝑛 > 3𝑡. +Lemma 2 +base +Lemma 3 +Lemma 4 +All input +configurations +0 +0 +0 +0 +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +A +Lemma 4 +Lemma 4 +Fig. 2. Theorem 1: Overview of the proof in the case 𝑛 = 6, 𝑡 = 2, and base = +� +(𝑃1, 0), (𝑃2, 0), (𝑃3, 0), (𝑃4, 0) +� +. +4.2 +Similarity Condition: Necessary Solvability Condition +This subsection defines the similarity condition, and proves its necessity for solvable properties. +Definition 2 (Similarity condition). A validity property val satisfies the similarity condition (C𝑆, +in short) if and only if there exists a computable function Λ : I𝑛−𝑡 → V𝑂 such that: +∀𝑐 ∈ I𝑛−𝑡 : Λ(𝑐) ∈ +� +𝑐′∈sim(𝑐) +val(𝑐′). +C𝑆 states that, for every input configuration 𝑐 ∈ I𝑛−𝑡, there exists a computable function Λ(𝑐) +which retrieves a common admissible decision among all input configurations similar to 𝑐. The +9 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +necessity of C𝑆 follows from the canonical similarity result: in any infinite canonical execution, a +common admissible value must be decided (Lemma 1). +Theorem 2. Any solvable validity property satisfies C𝑆. +Proof. By the means of contradiction, let there exist a validity property val such that (1) val +does not satisfy C𝑆, and (2) val is solvable. Let A be any algorithm which solves the Byzantine +consensus problem with val. As val does not satisfy C𝑆, there does not exist a computable function +Λ : I𝑛−𝑡 → V𝑂 such that, for every input configuration 𝑐 ∈ I𝑛−𝑡, Λ(𝑐) ∈ +� +𝑐′∈sim(𝑐) +val(𝑐′). +Fix any input configuration 𝑐 ∈ I𝑛−𝑡 for which Λ(𝑐) is not defined or not computable. Let +E𝑐 ∈ execs(A) be an infinite canonical execution such that (1) corresponding(E𝑐) = 𝑐, (2) the +system is synchronous from the very beginning (GST = 0), and (3) message delays are exactly +𝛿. In other words, E𝑐 unfolds in a “lock-step” manner. As A satisfies Termination and E𝑐 is an +infinite execution, some value 𝑣𝑐 ∈ V𝑂 is decided by a correct process in E𝑐. By canonical similarity +(Lemma 1), 𝑣𝑐 ∈ +� +𝑐′∈sim(𝑐) +val(𝑐′). Hence, Λ(𝑐) is defined (Λ(𝑐) = 𝑣𝑐) and computable (due to the +construction of E𝑐). Thus, we reach a contradiction with the fact that Λ(𝑐) is not defined or not +computable, which concludes the proof. +□ +Notice that, for proving the necessity of C𝑆 (Theorem 2), we do not rely on the 𝑛 > 3𝑡 assumption. +Hence, C𝑆 is necessary for all solvable validity properties (irrespectively of the resiliency threshold). +However, as proven in §4.1, C𝑆 is not sufficient when 𝑛 ≤ 3𝑡.2 (Observe that any trivial validity +property satisfies C𝑆.) +Similarity condition (example). Consider Correct-Proposal Validity [36, 73], a validity property +which states that any decided value must have been proposed by a correct process. Consensus with +Correct-Proposal Validity is also known as strong consensus [36, 73]. Strong consensus assumes +V𝐼 = V𝑂. It was shown that, in partial synchrony, strong consensus cannot be solved if 𝑛 ≤ +(|V𝐼 | + 1)𝑡 [36]. We now present an alternative proof. Namely, we show that Correct-Proposal +Validity does not satisfy C𝑆 if 𝑛 ≤ (|V𝐼 | + 1)𝑡, which makes it unsolvable by Theorem 2. +Let |V𝐼 | = |V𝑂 | = 𝑚 and 𝑛 = (𝑚 + 1)𝑡. We define an input configuration base ∈ I𝑛−𝑡 such that +(1) |𝜋(base)| = 𝑛 − 𝑡 = 𝑚𝑡, and (2) every value 𝑣 ∈ V𝐼 is the proposal of exactly 𝑡 processes in +base. Therefore, all values are admissible for base. Next, for each value 𝑣 ∈ V𝐼, we design an input +configuration 𝑐∌𝑣 such that (1) 𝑣 is not admissible for 𝑐∌𝑣, and (2) 𝑐∌𝑣 ∼ base: +(1) Let 𝑐∌𝑣 ← base. +(2) We remove from 𝑐∌𝑣 all process-proposal pairs pp with proposal(pp) = 𝑣. +(3) For every process 𝑃 ∉ 𝜋(base), we add (𝑃, 𝑣 ′) to 𝑐∌𝑣, for any value 𝑣 ′ ≠ 𝑣. +Hence, Correct-Proposal Validity does not satisfy C𝑆 when 𝑛 ≤ (𝑚 + 1)𝑡 as, for every 𝑣 ∈ V𝑂, +there exists an input configuration 𝑐∌𝑣 ∼ base for which 𝑣 is not admissible. Thus, Correct-Proposal +Validity is unsolvable if 𝑛 ≤ (𝑚 + 1)𝑡 (by Theorem 2). +5 +LOWER BOUND & GENERAL ALGORITHM +This section is devoted to the cost of solving consensus. Specifically, we first show that any non- +trivial and solvable validity property requires Ω(𝑡2) messages to be exchanged (§5.1). Then, we +present Universal, a general algorithm which, if 𝑛 > 3𝑡, solves consensus with any validity +property which satisfies C𝑆 (§5.2). Thus, Universal proves the sufficiency of C𝑆 when 𝑛 > 3𝑡. +2For example, Weak Validity satisfies C𝑆, but it is unsolvable with 𝑛 ≤ 3𝑡 [32]. +10 + +On the Validity of Consensus +5.1 +Lower Bound on Message Complexity +In this subsection, we prove the following theorem: +Theorem 3. If an algorithm solves consensus with a non-trivial validity property, the message +complexity of the algorithm is Ω(𝑡2). +Theorem 3 extends the seminal Dolev-Reischuk bound [30], proven only for consensus algorithms +with Strong Validity, to all non-trivial variants of consensus. To prove Theorem 3, we intertwine +the idea of the original proof [30] with the canonical similarity result (§3.4). +Proof of Theorem 3. In our proof, we show that any algorithm which solves the Byzantine +consensus problem with a non-trivial validity property has a synchronous execution (GST = 0) in +which correct processes send ≥ ( 𝑡 +2)2 messages. Hence, throughout the entire subsection, we fix a +non-trivial and solvable validity property val. Moreover, we fix A, an algorithm which solves the +Byzantine consensus problem with val. As val is a non-trivial validity property, 𝑛 > 3𝑡 (§4.1). +Next, we define a specific infinite execution Ebase ∈ execs(A) in the following manner: +(1) GST = 0. That is, the system is synchronous throughout the entire execution. +(2) All processes are separated into two groups: (1) group 𝐴, with |𝐴| = 𝑛 − ⌈ 𝑡 +2⌉, and (2) group 𝐵, +with |𝐵| = ⌈ 𝑡 +2⌉. +(3) All processes in the group 𝐴 are correct, whereas all processes in the group 𝐵 are faulty. +(4) We fix any value 𝑣∗ ∈ V𝐼. For every correct process 𝑃𝐴 ∈ 𝐴, the proposal of 𝑃𝐴 in Ebase is 𝑣∗. +(5) For every faulty process 𝑃𝐵 ∈ 𝐵, 𝑃𝐵 behaves correctly in Ebase with its proposal being 𝑣∗, +except that (1) 𝑃𝐵 ignores the first ⌈ 𝑡 +2⌉ messages received from other processes, and (2) 𝑃𝐵 +omits sending messages to other processes in 𝐵. +To prove Theorem 3, it suffices to show that the message complexity of Ebase is ≥ (⌈ 𝑡 +2⌉)2. By +contradiction, let the correct processes (processes in 𝐴) send less than (⌈ 𝑡 +2⌉)2 messages in Ebase. +The first step of our proof shows that, given that correct processes send less than (⌈ 𝑡 +2⌉)2 messages +in Ebase, there must exist a process 𝑄 ∈ 𝐵 which can correctly decide some value 𝑣𝑄 ∈ V𝑂 without +receiving any message from any other process. We prove this claim using the pigeonhole principle. +Lemma 5. There exist a value 𝑣𝑄 ∈ V𝑂 and a process 𝑄 ∈ 𝐵 such that 𝑄 has a correct behavior +𝛽𝑄 in which (1) 𝑄 decides 𝑣𝑄, and (2) 𝑄 does not receive any message from any other process. +Proof. By assumption, correct processes (i.e., processes in the group 𝐴) send less than (⌈ 𝑡 +2⌉)2 +messages in Ebase. Therefore, due to the pigeonhole principle, there exists a process 𝑄 ∈ 𝐵 which +receives less than ⌈ 𝑡 +2⌉ messages (from other processes) in Ebase. Recall that 𝑄 behaves correctly +in Ebase with its proposal being 𝑣∗ ∈ V𝐼, except that (1) 𝑄 ignores the first ⌈ 𝑡 +2⌉ messages received +from other processes, and (2) 𝑄 does not send any messages to other processes in the group 𝐵. We +denote by 𝑆𝑄 the set of processes, not including 𝑄, which send messages to 𝑄 in Ebase; |𝑆𝑄 | < ⌈ 𝑡 +2⌉. +Next, we construct an infinite execution E′ +base. Execution E′ +base is identical to Ebase, except that +we introduce the following modifications: +(1) Processes in (𝐴 ∪ {𝑄}) \𝑆𝑄 are correct; other processes are faulty. That is, we make 𝑄 correct +in E′ +base, and we make all processes in 𝑆𝑄 faulty in E′ +base. +(2) Processes in 𝐵 \ {𝑄} behave exactly as in Ebase. Moreover, processes in 𝑆𝑄 behave exactly as +in Ebase, except that they do not send any message to 𝑄. +Due to the construction of E′ +base, process 𝑄 does not receive any message (from any other process) +in E′ +base. As 𝑄 is correct in E′ +base and A satisfies Termination, 𝑄 decides some value 𝑣𝑄 ∈ V𝑂 in E′ +base. +Thus, 𝑄 indeed has a correct behavior 𝛽𝑄 in which it decides 𝑣𝑄 ∈ V𝑂 without having received +messages from other processes. +□ +11 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +In the second step of our proof, we show that there exists an execution in which (1) 𝑄 is faulty +and silent, and (2) other processes decide some value 𝑣 ≠ 𝑣𝑄. +Lemma 6. There exists an execution E𝑣 such that (1) 𝑄 is faulty and silent in E𝑣, and (2) a value +𝑣 ≠ 𝑣𝑄 is decided by a correct process. +Proof. As val is a non-trivial validity property, there exists an input configuration 𝑐∌𝑣𝑄 ∈ I +such that 𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 ); recall that 𝑣𝑄 is the value that 𝑄 can correctly decide without having +received any message from any other process (Lemma 5). We consider two possible cases: +• Let 𝑄 ∉ 𝜋(𝑐∌𝑣𝑄 ). Thus, E𝑣 is any infinite canonical execution which corresponds to 𝑐∌𝑣𝑄 . As +𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 ), the value decided in E𝑣 must be different from 𝑣𝑄 (as A satisfies val). +• Let 𝑄 ∈ 𝜋(𝑐∌𝑣𝑄 ). We construct an input configuration 𝑐∌𝑄 ∈ I such that 𝑄 ∉ 𝜋(𝑐∌𝑄): +(1) Let 𝑐∌𝑄 ← 𝑐∌𝑣𝑄 . +(2) We remove (𝑄, ·) from 𝑐∌𝑄. That is, we remove 𝑄’s process-proposal pair from 𝑐∌𝑄. +(3) If |𝜋(𝑐∌𝑣𝑄 )| = 𝑛 − 𝑡, we add (𝑍, any value) to 𝑐∌𝑄, where 𝑍 is any process such that +𝑍 ∉ 𝜋(𝑐∌𝑣𝑄 ); note that such a process 𝑍 exists as 𝑡 > 0. +Due to the construction of 𝑐∌𝑄, 𝑐∌𝑄 ∼ 𝑐∌𝑣𝑄 . Indeed, (1) 𝜋(𝑐∌𝑄) ∩ 𝜋(𝑐∌𝑣𝑄 ) ≠ ∅ (as 𝑛 − 𝑡 − 1 > 0 +when 𝑛 > 3𝑡 and 𝑡 > 0), and (2) for every process 𝑃 ∈ 𝜋(𝑐∌𝑄) ∩ 𝜋(𝑐∌𝑣𝑄 ), the proposal of 𝑃 is +identical in 𝑐∌𝑄 and 𝑐∌𝑣𝑄 (by the step 1 of the construction). +In this case, E𝑣 is any infinite canonical execution such that corresponding(E𝑣) = 𝑐∌𝑄. As A +satisfies Termination and E𝑣 is infinite, some value 𝑣 ∈ V𝑂 is decided by a correct process +in E𝑣. As 𝑐∌𝑄 ∼ 𝑐∌𝑣𝑄 , 𝑣 ∈ val(𝑐∌𝑣𝑄 ) (by canonical similarity; Lemma 1). Finally, 𝑣 ≠ 𝑣𝑄 as (1) +𝑣 ∈ val(𝑐∌𝑣𝑄 ), and (2) 𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 ). +The lemma holds as its statement is true in both possible cases. +□ +As we have shown the existence of E𝑣 (Lemma 6), we can “merge” E𝑣 with the valid behavior 𝛽𝑄 +in which 𝑄 decides 𝑣𝑄 without having received any message (Lemma 5). Hence, we can construct an +execution in which A violates Agreement. Thus, correct processes must send at least (⌈ 𝑡 +2⌉)2 ∈ Ω(𝑡2) +messages in Ebase. The formal proof of the following lemma, from which the lower bound on message +complexity (Theorem 3) follows directly, is given in Appendix B. +Lemma 7. The message complexity of Ebase is at least (⌈ 𝑡 +2⌉)2. +5.2 +General Algorithm Universal: Similarity Condition is Sufficient if 𝑛 > 3𝑡 +In this subsection, we prove that C𝑆 is sufficient for a validity property to be solvable when 𝑛 > 3𝑡. +That is, we prove the following theorem: +Theorem 4. Let 𝑛 > 3𝑡, and let val be any validity property which satisfies C𝑆. Then, val is solvable. +Moreover, assuming a threshold signature scheme, there exists an algorithm which solves Byzantine +consensus with val, and has 𝑂(𝑛2) message complexity. +To prove Theorem 4, we present Universal, an algorithm which solves the Byzantine consensus +problem with any validity property satisfying C𝑆, given that 𝑛 > 3𝑡. That is, Universal solves +consensus with any solvable and non-trivial validity property. Notably, assuming a threshold +signature scheme, Universal achieves 𝑂(𝑛2) message complexity, making it optimal (when 𝑡 ∈ +Ω(𝑛2)) according to our lower bound (§5.1). +To construct Universal, we rely on vector consensus [27, 31, 69, 76] (see §5.2.1), a problem which +requires correct processes to agree on the proposals of 𝑛 − 𝑡 processes. Specifically, when a correct +process decides a vector vec of 𝑛 −𝑡 proposals (from vector consensus), it decides (from Universal) +the common admissible value for all input configurations similar to vec, i.e., the process decides +Λ(vec). Note that the idea of solving consensus from vector consensus is not novel [14, 28, 68]. For +12 + +On the Validity of Consensus +some validity properties it is even natural, such as Strong Validity (choose the most common value) +or Weak Validity (choose any value). However, thanks to the necessity of C𝑆 (§4.2), any solvable +consensus variant can reuse this simple algorithmic design. +In this subsection, we first recall vector consensus (§5.2.1). Then, we utilize vector consensus to +construct Universal (§5.2.2). Throughout the entire subsection, 𝑛 > 3𝑡. +5.2.1 +Vector Consensus. In essence, vector consensus allows each correct process to infer the pro- +posals of 𝑛−𝑡 (correct or faulty) processes. Formally, correct processes agree on input configurations +(of vector consensus) with exactly 𝑛 − 𝑡 process-proposal pairs: V𝑂 = I𝑛−𝑡. +Let us formally define Vector Validity, the validity property of vector consensus: +• Vector Validity: Let a correct process decide vector ∈ V𝑂, which contains exactly 𝑛 −𝑡 process- +proposal pairs, such that (1) (𝑃, 𝑣) belongs to vector, for some process 𝑃 ∈ Π and some value +𝑣 ∈ V𝐼, and (2) 𝑃 is a correct process. Then, 𝑃 proposed 𝑣 to vector consensus. +Intuitively, Vector Validity states that, if a correct process “concludes” that a value 𝑣 was proposed +by a correct process 𝑃, then 𝑃’s proposal was indeed 𝑣. +We provide two implementations of vector consensus: (1) a non-authenticated implementation +(without any cryptographic primitives), and (2) an authenticated implementation (with threshold +signatures). Due to the lack of space, we give the pseudocode of the non-authenticated version +in Appendix C.2. The pseudocode of an authenticated version is presented in Algorithm 1. This +version relies on Quad, a Byzantine consensus algorithm recently introduced in [23]; we briefly +discuss Quad below. +Quad. In essence, Quad is a partially-synchronous, “leader-based” Byzantine consensus algo- +rithm, which achieves 𝑂(𝑛2) message complexity. Internally, Quad relies on a threshold signature +scheme. Formally, Quad is concerned with two sets: (1) VQuad, a set of values, and (2) PQuad, a +set of proofs. In Quad, processes propose and decide value-proof pairs. There exists a function +verify : VQuad ×PQuad → {true, false}. Importantly, PQuad is not known a-priori: it is only assumed +that, if a correct process proposes a pair (𝑣 ∈ VQuad, Σ ∈ PQuad), then verify(𝑣, Σ) = true. Quad +guarantees the following: if a correct process decides a pair (𝑣, Σ), then verify(𝑣, Σ) = true. In other +words, correct processes decide only valid value-proof pairs. (See [23] for full details on Quad.) +In our authenticated implementation of vector consensus (Algorithm 1), we rely on a specific +instance of Quad where (1) VQuad = I𝑛−𝑡 (processes propose to Quad input configurations of vector +consensus), and (2) PQuad is a set of 𝑛−𝑡 proposal messages (sent by processes in vector consensus). +Finally, given an input configuration 𝑐 ∈ VQuad and a set of messages Σ ∈ PQuad, verify(𝑐, Σ) = true +if and only if, for every process-proposal pair (𝑃𝑗, 𝑣 𝑗) which belongs to 𝑐, ⟨proposal, 𝑣 𝑗⟩𝜎𝑗 ∈ Σ (i.e., +every process-proposal pair of 𝑐 is accompanied by a properly signed proposal message). +Description of authenticated vector consensus (Algorithm 1). When a correct process 𝑃𝑖 proposes +a value 𝑣 ∈ V𝐼 to vector consensus (line 8), the process broadcasts a signed proposal message +(line 9). Once 𝑃𝑖 receives 𝑛 − 𝑡 proposal messages (line 14), 𝑃𝑖 constructs an input configuration +vector (line 15), and a proof Σ (line 16) from the received proposal messages. Moreover, 𝑃𝑖 proposes +(vector, Σ) to Quad (line 17). Finally, when 𝑃𝑖 decides a pair (vector′, Σ′) from Quad (line 18), 𝑃𝑖 +decides vector′ from vector consensus (line 19). +The message complexity of Algorithm 1 is 𝑂(𝑛2) as (1) processes only broadcast proposal +messages, and (2) the message complexity of Quad is 𝑂(𝑛2). We underline that the communication +complexity of Algorithm 1, the number of bits sent by correct processes, is 𝑂(𝑛3) as the communi- +cation complexity of Quad is 𝑂(𝑛2 · 𝑥) = 𝑂(𝑛3) (see [23]), where 𝑥 is the size of a Quad proposal +(in our case, 𝑥 ∈ Θ(𝑛)). Due to space constraints, we delegate the full proof of the correctness and +complexity of Algorithm 1 to Appendix C.1. +13 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +Algorithm 1 Authenticated Vector Consensus: Pseudocode (for process 𝑃𝑖) +1: Uses: +2: +Best-Effort Broadcast [19], instance beb +⊲ broadcast with no guarantees if the sender is faulty +3: +Quad [23], instance quad +4: upon init: +5: +Integer received_proposals𝑖 ← 0 +⊲ the number of received proposals +6: +Map(Process → V𝐼 ) proposals𝑖 ← empty +⊲ received proposals +7: +Map(Process → Message) messages𝑖 ← empty +⊲ received proposal messages +8: upon propose(𝑣 ∈ V𝐼 ): +9: +invoke beb.broadcast�⟨proposal, 𝑣⟩𝜎𝑖 +� +⊲ broadcast a signed proposal +10: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩𝜎𝑗 from process 𝑃𝑗 and received_proposals𝑖 < 𝑛 − 𝑡: +11: +received_proposals𝑖 ← received_proposals𝑖 + 1 +12: +proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 +13: +messages𝑖 [𝑃𝑗 ] ← 𝑚 +14: +if received_proposals𝑖 = 𝑛 − 𝑡: +⊲ received 𝑛 − 𝑡 proposals; can propose to Quad +15: +Input_Configuration vector ← input configuration constructed from proposals𝑖 +16: +Proof Σ ← set of messages containing all proposal messages from messages𝑖 +17: +invoke quad.propose�(vector, Σ)� +18: upon quad.decide�(Input_Configuration vector′, Proof Σ′)�: +19: +trigger decide(vector′) +5.2.2 +Universal. We construct Universal (Algorithm 2) directly from vector consensus (§5.2.1). +When a correct process 𝑃𝑖 proposes to Universal (line 3), the proposal is forwarded to vector +consensus (line 4). Once 𝑃𝑖 decides an input configuration 𝑐 from vector consensus (line 5), 𝑃𝑖 +decides Λ(𝑐) (line 6). +Note that our implementation of Universal (Algorithm 2) is independent of the actual imple- +mentation of vector consensus. Thus, by employing our authenticated implementation of vector +consensus (Algorithm 1), we obtain a general consensus algorithm with 𝑂(𝑛2) message complexity. +On the other hand, by employing a non-authenticated implementation of vector consensus (see +Appendix C.2), we obtain a non-authenticated version of Universal, which implies that any validity +property which satisfies C𝑆 is solvable even in a non-authenticated setting (if 𝑛 > 3𝑡). +Algorithm 2 Universal: Pseudocode for process 𝑃𝑖 +1: Uses: +2: +Vector Consensus, instance vec_cons +3: upon propose(𝑣 ∈ V𝐼 ): +4: +invoke vec_cons.propose(𝑣) +5: upon vec_cons.decide(Input_Configuration 𝑐): +6: +trigger decide�Λ(𝑐)� +Finally, we prove that Universal (Algorithm 2) is a general Byzantine consensus algorithm. +Theorem 5. Let val be any validity property which satisfies C𝑆, and let 𝑛 > 3𝑡. Universal solves +the Byzantine consensus problem with val. Moreover, if Universal employs Algorithm 1 as its vector +consensus building block, the message complexity of Universal is 𝑂(𝑛2). +Proof. Termination and Agreement of Universal follow from Termination and Agreement of +vector consensus, respectively. Moreover, the message complexity of Universal is identical to the +message complexity of its vector consensus building block. +14 + +On the Validity of Consensus +Finally, we prove that Universal satisfies val. Consider any execution E of Universal such that +corresponding(E) = 𝑐∗, for some input configuration𝑐∗ ∈ I. Let𝑐 ∈ I𝑛−𝑡 be the input configuration +correct processes decide from vector consensus in E (line 5). As vector consensus satisfies Vector +Validity, we have that, for every process 𝑃 ∈ 𝜋(𝑐∗) ∩ 𝜋(𝑐), 𝑃’s proposals in 𝑐∗ and 𝑐 are identical. +Hence, 𝑐 ∼ 𝑐∗. Therefore, Λ(𝑐) ∈ val(𝑐∗) (by the definition of the Λ function). Thus, val is satisfied +by Universal. +□ +As Universal (Algorithm 2) solves the Byzantine consensus problem with any validity property +which satisfies C𝑆 (Theorem 5) if 𝑛 > 3𝑡, C𝑆 is sufficient for solvable validity properties when 𝑛 > 3𝑡. +Lastly, as Universal relies on vector consensus, we conclude that Vector Validity is a strongest +validity property. That is, a solution to any variant of the consensus problem can be obtained from +vector consensus. +A note on the communication complexity of vector consensus. While the version of Universal +which employs Algorithm 1 (as its vector consensus building block) has optimal message com- +plexity, its communication complexity is 𝑂(𝑛3). This presents a linear gap to the lower bound for +communication complexity (also Ω(𝑛2), implied by Theorem 3), and to known optimal solutions for +some validity properties (e.g., Strong Validity, proven to be Θ(𝑛2) [23]). At first glance, this seems +like an issue inherent to vector consensus: the decided vectors are linear in size, suggesting that +the linear gap could be inevitable. However, this is not the case. In Appendix C.3, we give a vector +consensus algorithm with 𝑂(𝑛2 log𝑛) communication complexity, albeit with exponential latency.3 +Is it possible to construct vector consensus with subcubic communication and polynomial latency? +This is an important open question, as positive answers would lead to (practical) performance +improvements of all consensus variants. +6 +CONCLUDING REMARKS +This paper studies the validity property of partially synchronous Byzantine consensus. Namely, we +mathematically formalize validity properties, and give necessary and sufficient conditions for a +validity property to be solvable (i.e., for the existence of an algorithm which solves a consensus +problem defined with that validity property, in addition to Agreement and Termination). Moreover, +we prove a quadratic lower bound on message complexity for all non-trivial (and solvable) validity +properties. Previously, this bound was mainly known for Strong Validity. Lastly, we introduce +Universal, a general algorithm for consensus with any solvable (and non-trivial) validity property; +Universal achieves 𝑂(𝑛2) message complexity, showing that the aforementioned lower bound is +tight (with 𝑡 ∈ Ω(𝑛)). +We conjecture that our necessary and sufficient conditions for consensus solvability can easily +be adapted to a synchronous environment. However, our proof technique for the lower bound +on message complexity cannot be reused as such: this is because silent processes can be conclu- +sively detected in synchrony. Thus, we believe that the lower bound on message complexity for +synchronous consensus is one of the most important open questions. Another interesting question +is whether our lower bound holds for randomized consensus. In [3], it is proven that randomized +consensus with Strong Validity has Ω(𝑡2) expected message complexity. Can this bound be extended +to all non-trivial validity properties? +Finally, we restate the question posed at the end of §5.2. Is it possible to solve vector consensus +with 𝑜(𝑛3) exchanged bits and polynomial latency? Recall that, due to the design of Universal +(§5.2), any (non-trivial) consensus variant can be solved using vector consensus. Therefore, an +3Both our authenticated (Algorithm 1) and our non-authenticated (see Appendix C.2) variants of vector consensus have +linear latency, which implies linear latency of Universal when employing any of these two algorithms. +15 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +upper bound on the complexity of vector consensus is an upper bound on the complexity of any +consensus variant. 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First, we formally +prove that only 𝑣base can be decided in any canonical execution which corresponds to any input +configuration compatible with base. +Lemma 2 (restated). Let 𝑐 ∈ I be any input configuration such that 𝑐 ⋄ base. Let E𝑐 ∈ execs(A) +be any canonical execution such that corresponding(E𝑐) = 𝑐. If a value 𝑣𝑐 ∈ V𝑂 is decided by a +correct process in E𝑐, then 𝑣𝑐 = 𝑣base. +Proof. By contradiction, suppose that some value 𝑣𝑐 ≠ 𝑣base is decided by a correct process in E𝑐. +Since 𝑐 ⋄ base, there exists a correct process 𝑄 ∈ 𝜋(𝑐) \ 𝜋(base). Let E𝑄 +𝑐 be an infinite continuation +of E𝑐 in which 𝑄 decides; note that E𝑄 +𝑐 exists as A satisfies Termination. The following holds for +process 𝑄: (1) 𝑄 decides 𝑣𝑐 in E𝑄 +𝑐 (as A satisfies Agreement), and (2) process 𝑄 is silent in Ebase. Let +𝑡𝑄 denote the time at which 𝑄 decides in E𝑄 +𝑐 . Similarly, there exists a process 𝑃 ∈ 𝜋(base) \ 𝜋(𝑐); +observe that (1) process 𝑃 decides 𝑣base in Ebase, and (2) process 𝑃 is silent in E𝑄 +𝑐 . Let 𝑡𝑃 denote the +time at which 𝑃 decides in Ebase. +We now construct an execution E ∈ execs(A) by “merging” Ebase and E𝑄 +𝑐 : +(1) Processes in 𝜋(𝑐) ∩ 𝜋(base) behave towards processes in 𝜋(base) \ 𝜋(𝑐) as in Ebase, and +towards processes in 𝜋(𝑐) \ 𝜋(base) as in E𝑄 +𝑐 . +(2) Communication between (1) processes in 𝜋(base) \ 𝜋(𝑐) and (2) processes in 𝜋(𝑐) \ 𝜋(base) +is delayed until after max(𝑡𝑃,𝑡𝑄). +(3) We set GST to after max(𝑡𝑃,𝑡𝑄). +The following holds for E: +• Processes in 𝜋(base) ⊖ 𝜋(𝑐) (symmetric difference) are correct in E. +• Only processes in 𝜋(base) ∩ 𝜋(𝑐) are faulty in E. Recall that |𝜋(base) ∩ 𝜋(𝑐)| ≤ 𝑡 as base ⋄𝑐. +• Process 𝑄, which is correct in E, cannot distinguish E from E𝑄 +𝑐 until time max(𝑡𝑃,𝑡𝑄). Hence, +process 𝑄 decides 𝑣𝑐 in E. +• Process 𝑃, which is correct in E, cannot distinguish E from Ebase until time max(𝑡𝑃,𝑡𝑄). +Hence, process 𝑃 decides 𝑣base ≠ 𝑣𝑐 in E. +Therefore, we reach a contradiction with the fact that A satisfies Agreement. Thus, 𝑣𝑐 = 𝑣base. +□ +Next, we formally prove that, for every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 +such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. +Lemma 3 (restated). For every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 ∈ +execs(A) such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛. +Proof. Fix any input configuration 𝑐𝑛 ∈ I𝑛. We construct an input configuration 𝑐𝑛−𝑡 ∈ I𝑛−𝑡: +(1) For every process 𝑃𝑖 ∉ 𝜋(base), we include a process-proposal pair (𝑃𝑖, 𝑣) in 𝑐𝑛−𝑡 such that +𝑣 = proposal(𝑐𝑛[𝑖]). Note that there are 𝑡 such processes as |𝜋(base)| = 𝑛 − 𝑡. +(2) We include 𝑛 − 2𝑡 process-proposal pairs (𝑃𝑖, 𝑣) in 𝑐𝑛−𝑡 such that (1) 𝑃𝑖 ∈ 𝜋(base), and (2) +𝑣 = proposal(𝑐𝑛[𝑖]). That is, we “complete” 𝑐𝑛−𝑡 (constructed in the step 1) with 𝑛 − 2𝑡 +process-proposal pairs such that the process is “borrowed” from base, and its proposal is +“borrowed” from 𝑐𝑛. +Observe that 𝑐𝑛−𝑡 ⋄base as (1) |𝜋(𝑐𝑛−𝑡) ∩ base| ≤ 𝑡 (because 𝑛 − 2𝑡 ≤ 𝑡 when 𝑛 ≤ 3𝑡), (2) there exists +a process 𝑃 ∈ 𝜋(base) \ 𝜋(𝑐𝑛−𝑡) (because, when constructing 𝑐𝑛−𝑡, we excluded 𝑡 > 0 processes +from base; step 1), and (3) there exists a process 𝑄 ∈ 𝜋(𝑐𝑛−𝑡) \ 𝜋(base) (because we included 𝑡 > 0 +processes in cn−t which are not in base; step 1). +Let E𝑛−𝑡 ∈ execs(A) denote any infinite canonical execution such that corresponding(E𝑛−𝑡) = +𝑐𝑛−𝑡. As A satisfies Termination, some value is decided by correct processes in E𝑛−𝑡; due to Lemma 2, +20 + +On the Validity of Consensus +that value is 𝑣base. Finally, we are able to construct an infinite execution E𝑛 ∈ execs(A) such that +(1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛: +(1) All processes are correct in E𝑛. +(2) Until some correct process 𝑃 ∈ 𝜋(𝑐𝑛−𝑡) decides 𝑣base, E𝑛 is identical to E𝑛−𝑡. +(3) Afterwards, every process 𝑄 ∉ 𝜋(𝑐𝑛−𝑡) “wakes up” with the proposal specified in 𝑐𝑛. +(4) GST is set to after all processes have taken a computational step. +Therefore, 𝑣base is indeed decided in E𝑛 and corresponding(E𝑛) = 𝑐𝑛, which concludes the proof. +□ +B +LOWER BOUND ON MESSAGE COMPLEXITY: FORMAL PROOF +In this section, we give a formal proof of Lemma 7. Recall that we have fixed an algorithm A which +solves the Byzantine consensus problem with a non-trivial validity property val. +Lemma 7 (restated). The message complexity of Ebase is at least (⌈ 𝑡 +2⌉)2. +Proof. By Lemma 5, there exists a behavior 𝛽𝑄 of process 𝑄 in which 𝑄 decides a value 𝑣𝑄 +without having received any message (from any other process). Let 𝑡𝑄 denote the time at which 𝑄 +decides 𝑣𝑄 in 𝛽𝑄. Moreover, there exists an execution E𝑣 in which (1) 𝑄 is faulty and silent, and (2) +correct processes decide a value 𝑣 ≠ 𝑣𝑄 (by Lemma 6). Let 𝑡𝑣 denote the time at which a correct +process decides 𝑣 ≠ 𝑣𝑄 in E𝑣. +We now construct an execution E in the following way: +(1) Processes in Corr A(E𝑣) ∪ {𝑄} are correct in E. All other processes are faulty. +(2) All messages from and to 𝑄 are delayed until after max(𝑡𝑄,𝑡𝑣). +(3) Process 𝑄 exhibits the behavior 𝛽𝑄. +(4) Until max(𝑡𝑄,𝑡𝑣), no process in Corr A(E𝑣) can distinguish E from E𝑣. +(5) GST is set to after max(𝑡𝑄,𝑡𝑣). +As no process in Corr A(E𝑣) can distinguish E from E𝑣 until max(𝑡𝑄,𝑡𝑣), 𝑣 ≠ 𝑣𝑄 is decided by a +correct process in E. Moreover, 𝑄 decides 𝑣𝑄 in E (step 3 of the construction). Thus, Agreement is +violated in E, which contradicts the fact that A satisfies Agreement. Hence, the starting assumption +is not correct: in Ebase, correct processes send (at least) (⌈ 𝑡 +2⌉)2 messages. +□ +C +VECTOR CONSENUS: FORMAL PROOFS & OMITTED ALGORITHMS +In Appendix C.1, we prove the correctness and complexity of our authenticated implementation of +vector consensus (Algorithm 1). We dedicate Appendix C.2 to a non-authenticated implementation +of vector consensus. Finally, in Appendix C.3, we give an implementation of vector consensus with +𝑂(𝑛2 log𝑛) communication complexity. Throughout the entire section, we assume that 𝑛 > 3𝑡. +C.1 +Authenticated Implementation (Algorithm 1): Formal Proofs +In this subsection, we prove the correctness and complexity of our authenticated implementation +of vector consensus. We start with the correctness. +Theorem 6. Algorithm 1 is correct. +Proof. Agreement follows directly from the fact that Quad satisfies Agreement. Termination +follows from (1) Termination of Quad, and (2) the fact that, eventually, all correct processes receive +𝑛 − 𝑡 proposal messages (as there are at least 𝑛 − 𝑡 correct processes). +We now prove that Algorithm 1 satisfies Vector Validity. Let a correct process 𝑃 decide vector′ ∈ +I𝑛−𝑡 from vector consensus (line 19). Hence, 𝑃 has decided (vector′, Σ′) from Quad, where (1) Σ′ is +some proof, and (2) verify(vector′, Σ′) = true (due to the specification of Quad). Furthermore, if +there exists a process-proposal pair (𝑃, 𝑣 ∈ V𝐼) in vector′, where 𝑃 is a correct process, a properly +21 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +signed proposal message belongs to Σ′. As correct processes send proposal messages only for +their proposals (line 9), 𝑣 was indeed proposed by 𝑃. Thus, the theorem. +□ +Finally, we prove the complexity of Algorithm 1. +Theorem 7. The message complexity of Algorithm 1 is 𝑂(𝑛2). +Proof. The message complexity of the specific instance of Quad utilized in Algorithm 1 is 𝑂(𝑛2). +Additionally, correct processes exchange 𝑂(𝑛2) proposal messages. Thus, the message complexity +is 𝑂(𝑛2) + 𝑂(𝑛2) = 𝑂(𝑛2). +□ +C.2 +Non-Authenticated Implementation: Pseudocode & Formal Proofs +We now present a non-authenticated implementation (Algorithm 3) of vector consensus. The +design of Algorithm 3 follows the reduction from the binary consensus to the multivalue consensus +(e.g., [28]). Namely, we use the following two building blocks in Algorithm 3: +(1) Byzantine Reliable Broadcast [16, 19]: This primitive allows processes to disseminate infor- +mation in a reliable manner. Formally, the Byzantine reliable broadcast exposes the following +interface: (1) request broadcast(𝑚), and (2) indication deliver(𝑚′). The primitive satisfies +the following properties: +• Validity: If a correct process 𝑃 broadcasts a message 𝑚, 𝑃 eventually delivers 𝑚. +• Consistency: No two correct processes deliver different messages. +• Integrity: Every correct process delivers at most one message. Moreover, if a correct process +delivers a message 𝑚 from a process 𝑃 and 𝑃 is correct, then 𝑃 broadcast 𝑚. +• Totality: If a correct process delivers a message, every correct process delivers a message. +In Algorithm 3, we use a non-authenticated implementation [16] of the Byzantine Reliable +Broadcast primitive. +(2) Binary DBFT [28], a non-authenticated algorithm which solves the Byzantine consensus +problem with Strong Validity. +Let us briefly explain how Algorithm 3 works; we focus on a correct process 𝑃𝑖. First, 𝑃𝑖 reliably +broadcasts its proposal (line 11). Once 𝑃𝑖 delivers a proposal of some process 𝑃𝑗 (line 12), 𝑃𝑖 proposes +1 to the corresponding DBFT instance (line 17). Eventually, 𝑛 − 𝑡 DBFT instances decide 1 (line 18). +Once that happens, 𝑃𝑖 proposes 0 to all DBFT instance to which 𝑃𝑖 has not proposed (line 22). When +all DBFT instances have decided (line 23), 𝑃𝑖 decides an input configuration associated with the +first 𝑛 − 𝑡 processes whose DBFT instances decided 1 (constructed at line 24). +Theorem 8. Algorithm 3 is correct. +Proof. We start by proving Termination of Algorithm 3. Eventually, at least 𝑛 −𝑡 DBFT instances +decide 1 due to the fact that (1) no correct process proposes 0 to any DBFT instance unless 𝑛 − 𝑡 +DBFT instances have decided 1 (line 18), and (2) all correct processes eventually propose 1 to the +DBFT instances which correspond to the correct processes (unless 𝑛 − 𝑡 DBFT instances have +already decided 1). When 𝑛 − 𝑡 DBFT instances decide 1 (line 18), each correct process proposes to +all instances to which it has not yet proposed (line 22). Hence, eventually all DBFT instances decide, +and (at least) 𝑛 − 𝑡 DBFT instances decide 1. Therefore, the rule at line 23 eventually activates at +every correct process, which implies that every correct process eventually decides (line 25). +Next, we prove Vector Validity. If a correct process 𝑃 decides an input configuration with a +process-proposal pair (𝑄, 𝑣), 𝑃 has delivered a proposal message from 𝑄. If 𝑄 is correct, due to +integrity of the reliable broadcast primitive, 𝑄’s proposal was indeed 𝑣. +Finally, Agreement follows from (1) Agreement of DBFT, and (2) consistency of the reliable +broadcast primitive. Therefore, Algorithm 3 is correct. +□ +22 + +On the Validity of Consensus +Algorithm 3 Non-Authenticated Vector Consensus: Pseudocode (for process 𝑃𝑖) +1: Uses: +2: +Non-Authenticated Byzantine Reliable Broadcast [16], instance brb +3: +Binary DBFT [28], instances dbft[1], ..., dbft[𝑛] +⊲ one instance of the binary DBFT protocol per process +4: upon init: +5: +Map(Process → V𝐼 ) proposals𝑖 ← empty +⊲ received proposals +6: +Map(Process → Message) messages𝑖 ← empty +⊲ received proposal messages +7: +Boolean dbft_proposing𝑖 = true +⊲ is 𝑃𝑖 still proposing to the DBFT instances +8: +Map(Process → Boolean) dbft_proposed𝑖 ← {false, for every Process} +9: +Integer dbft_decisions𝑖 ← 0 +⊲ the number of the DBFT instances which have decided +10: upon propose(𝑣 ∈ V𝐼 ): +11: +invoke brb.broadcast�⟨proposal, 𝑣⟩� +⊲ broadcast a proposal +12: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩ from process 𝑃𝑗: +13: +proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 +14: +messages𝑖 [𝑃𝑗 ] ← 𝑚 +15: +if dbft_proposing𝑖 = true: +16: +dbft_proposed𝑖 [𝑃𝑗 ] ← true +17: +invoke dbft[𝑗 ].propose(1) +18: upon 𝑛 − 𝑡 DBFT instances have decided 1 (for the first time): +19: +dbft_proposing𝑖 ← false +20: +for every Process 𝑃𝑗 such that dbft_proposed𝑖 [𝑃𝑗 ] = false: +21: +dbft_proposed𝑖 [𝑃𝑗 ] ← true +22: +invoke dbft[𝑗 ].propose(0) +23: upon all DBFT instances decided, and, for the first 𝑛 − 𝑡 processes 𝑃𝑗 such that dbft[𝑗 ] decided 1, proposals𝑖 [𝑃𝑗 ] ≠ ⊥: +24: +Input_Configuration vector ← input configuration with 𝑛 − 𝑡 process-proposal pairs corresponding to the first +𝑛 − 𝑡 DBFT instances which decided 1 +25: +trigger decide(vector) +The main downside of Algorithm 3 is that its message complexity is 𝑂(𝑛4). Therefore, non- +authenticated version of Universal has 𝑂(𝑛4) message complexity, which is not optimal according +to our lower bound (§5.1). +C.3 +Implementation with 𝑂(𝑛2 log𝑛) Communication: Pseudocode & Formal Proofs +In this subsection, we give an implementation of vector consensus with 𝑂(𝑛2 log𝑛) communication +complexity, which comes within a logarithmic factor of the lower bound (§5.1). This implementation +represents a near-linear communication improvement over Algorithm 1 (§5.2), which achieves +𝑂(𝑛3) communication complexity. We note that the following solution is highly impractical due to +its exponential latency (worst-case 𝑂(𝑛𝑡), requiring idealized cryptographic primitives). However, +our solution does represent a step towards closing the existing gap in the communication complexity +of non-trivial and solvable validity properties. +C.3.1 +Vector Dissemination. First, we formally define the vector dissemination problem, which +plays the crucial role in our vector consensus algorithm with improved communication complexity. +In this problem, each correct process disseminates a vector of 𝑛 − 𝑡 values, and all correct processes +eventually obtain (1) a hash of some disseminated vector, and (2) a storage proof. For every hash +value 𝐻 and every storage proof sp, we define valid_SP(𝐻, sp) ∈ {true, false}. Formally, the vector +dissemination problem exposes the following interface: +• request disseminate(Vector vec): a process disseminates a vector vec. +• indication obtain(Hash_Value 𝐻 ′, Storage_Proof sp′): a process obtains a hash value 𝐻 ′ +and a storage proof sp′. +23 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +The following properties are required: +• Termination: Every correct process eventually obtains a hash value and a storage proof. +• 𝛿-Closeness: Let 𝑡first denote the first time a correct process obtains a hash value and a +storage proof. Then, every correct process obtains a hash value and a storage proof by time +max(GST,𝑡first) + 𝛿. +• Redundancy: Let a (faulty or correct) process obtain a storage proof sp′ such that, for some +hash value 𝐻 ′, valid_SP(𝐻 ′, sp′) = true. Then, (at least) 𝑡 + 1 correct processes have cached a +vector vec′ such that hash(vec′) = 𝐻 ′. +• Integrity: If a correct process obtains a hash value 𝐻 ′ and a storage proof sp′, the following +holds: valid_SP(𝐻 ′, sp′) = true. +Slow broadcast. In order to solve the vector dissemination problem, we present a simple algorithm +(Algorithm 4) which implements slow broadcast. In slow broadcast, each process disseminates its +vector in “one-by-one” fashion, with a “waiting step” between any two sending events. Specifically, +process 𝑃1 broadcasts its vector by (1) sending the vector to 𝑃1 (line 3), and then waiting 𝛿 time +(line 4), (2) sending the vector to 𝑃2 (line 3), and then waiting 𝛿 time (line 4), etc. Process 𝑃2 +broadcasts its vector in the same manner, but it waits 𝛿 · 𝑛 time (line 4). Crucially, if the system +is synchronous, the waiting time of 𝑃2 is (roughly) sufficient for 𝑃1 to completely disseminate its +vector. This holds for any two processes 𝑃𝑖 and 𝑃𝑗 such that 𝑖 < 𝑗. +Algorithm 4 Slow Broadcast: Pseudocode (for process 𝑃𝑖) +1: upon broadcast(Vector vec): +2: +for each Process 𝑃𝑗: +3: +send ⟨slow_broadcast, vec⟩ to 𝑃𝑗 +4: +wait for 𝛿 · 𝑛(𝑖−1) time +5: upon reception of ⟨slow_broadcast, Vector vec′⟩ from process 𝑃𝑗: +6: +trigger deliver(vec′, 𝑃𝑗) +Vector dissemination algorithm. Our solution is given in Algorithm 5. First, we give a con- +crete implementation of the valid_SP(·, ·) function. Given a hash value 𝐻 and a storage proof +sp, valid_SP(𝐻, sp) = true if and only if sp is a valid (𝑛 − 𝑡)-threshold signature of ⟨stored, 𝐻⟩. +Let us explain Algorithm 5 from the perspective of a correct process 𝑃𝑖. When 𝑃𝑖 starts dissem- +inating its vector vec (line 8), 𝑃𝑖 stores its hash (line 9) and slow-broadcasts the vector (line 10). +Once 𝑃𝑖 receives stored messages from 𝑛 − 𝑡 distinct processes (line 17), 𝑃𝑖 combines received +partial signatures into a storage proof (line 18). Then, 𝑃𝑖 broadcasts (using the best-effort broadcast +primitive) the constructed storage proof (line 19). +Whenever 𝑃𝑖 receives a storage proof (line 21), 𝑃𝑖 checks whether the storage proof is valid +(line 22). If it is, 𝑃𝑖 rebroadcasts the storage proof (line 23), obtains a hash value and the storage +proof (line 24), and stops participating (i.e., sending and processing messages) in vector dissemina- +tion (line 25). Observe that, once 𝑃𝑖 stops participating in vector dissemination (line 25), it stops +participating in slow broadcast, as well. +Proof of correctness and complexity. We start by proving redundancy of Algorithm 5. +Lemma 8. Algorithm 5 satisfies redundancy. +Proof. Let a (correct or faulty) process obtain a storage proof sp′ such that valid_SP(𝐻 ′, sp′) = +true, for some hash value 𝐻 ′. Hence, 𝑛 − 𝑡 processes have signed a stored message for 𝐻 ′ (as +valid_SP(𝐻 ′, sp′) = true). Among these 𝑛 − 𝑡 processes, at least 𝑡 + 1 are correct (as 𝑛 > 3𝑡). Before +24 + +On the Validity of Consensus +Algorithm 5 Vector Dissemination: Pseudocode (for process 𝑃𝑖) +1: Uses: +2: +Best-Effort Broadcast [19], instance beb +⊲ broadcast with no guarantees if the sender is faulty +3: +Slow Broadcast, instance slow +⊲ see Algorithm 4 +4: upon init: +5: +Hash_Value 𝐻𝑖 ← ⊥ +⊲ hash of the message 𝑃𝑖 slow-broadcasts +6: +Map(Hash_Value → Vector) vectors𝑖 ← empty +⊲ received vectors +7: +Set(Process) disseminated𝑖 ← empty +⊲ processes who have disseminated a vector +8: upon disseminate(Vector vec): +9: +𝐻𝑖 ← hash(vec) +10: +invoke slow.broadcast(vec) +11: upon slow.deliver(Vector vec′, Process 𝑃𝑗): +12: +if 𝑃𝑗 ∉ disseminated𝑖: +13: +disseminated𝑖 ← disseminated𝑖 ∪ {𝑃𝑗 } +14: +vectors𝑖 [hash(vec′)] ← vec′ +⊲ cache vec′ +15: +send ⟨stored, hash(vec′)⟩𝜎𝑖 to 𝑃𝑗 +⊲ acknowledge the reception by sending a signature to 𝑃𝑗 +16: ⊲ acknowledgements are received +17: upon reception of Message 𝑚𝑗 = ⟨stored, Hash_Value 𝐻′⟩𝜎𝑗 such that 𝐻′ = 𝐻𝑖 from 𝑛 − 𝑡 distinct processes: +18: +Storage_Proof sp ← Combine�{𝜎 | 𝜎 is a signature of a received stored message}� +19: +invoke beb.broadcast�⟨storage_proof, 𝐻𝑖, sp⟩� +⊲ disseminate the storage proof +20: ⊲ a storage proof is received +21: upon reception of Message 𝑚 = ⟨storage_proof, Hash_Value 𝐻′, Storage_Proof sp′⟩: +22: +if sp′ is a valid (𝑛 − 𝑡)-threshold signature of ⟨stored, 𝐻′⟩: +⊲ check that the storage proof is valid +23: +invoke beb.broadcast�⟨storage_proof, 𝐻′, sp′⟩� +⊲ rebroadcast the storage proof +24: +trigger obtain(𝐻′, sp′) +25: +stop participating in vector dissemination and slow broadcast +sending (and signing) a stored message for 𝐻 ′, all these correct processes have cached a vector +vec′ (line 14), where hash(vec′) = 𝐻 ′. Thus, the lemma. +□ +Next, we prove 𝛿-closeness. +Lemma 9. Algorithm 5 satisfies 𝛿-closeness. +Proof. Let 𝑃first be a correct process which obtains a hash value and a storage proof at time +𝑡first. Before the aforementioned attainment, 𝑃first rebroadcasts the hash value and the storage +proof (line 23). Hence, every correct process receives a hash value and storage proof by time +max(GST,𝑡first) + 𝛿. +□ +The following lemma proves that, if a correct process 𝑃𝑖 starts the dissemination of its vector at +time 𝑡𝑖, then every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + +𝛿 · 𝑛𝑖 + 3𝛿. We emphasize that the max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿 time is not tight; we choose it due to +the simplicity of the presentation. +Lemma 10. If a correct process 𝑃𝑖 starts the dissemination of its vector at time 𝑡𝑖, every correct +process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿. +Proof. We separate the proof into two cases: +• There exists a correct process which obtains a hash value and a storage proof by time +max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿. In this case, the statement of the lemma holds as every correct +process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿 due to the +“rebroadcasting step” (line 23). +25 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +• There does not exist a correct process which obtains a hash value and a storage proof by time +𝑇 = max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿. Hence, no process stops participating in vector dissemination +by time 𝑇, i.e., no process executes line 25 by time 𝑇. Every correct process receives a +slow_broadcast message from process 𝑃𝑖 by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 𝛿. +Thus, by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿, 𝑃𝑖 receives 𝑛 − 𝑡 partial signatures (line 17). Finally, +by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿, every correct process receives a storage_proof message +from 𝑃𝑖 (line 21), and obtains a hash value and a storage proof (line 24). In this case, the +statement of the lemma holds. +As the statement of the lemma holds in both cases, the proof is concluded. +□ +The next lemma proves that Algorithm 5 satisfies termination. +Lemma 11. Algorithm 5 satisfies termination. +Proof. Follows directly from Lemma 10. +□ +Next, we prove integrity. +Lemma 12. Algorithm 5 satisfies integrity. +Proof. Follows from the check at line 22. +□ +Therefore, Algorithm 5 solves the vector dissemination problem. +Theorem 9. Algorithm 5 is correct. +Lastly, we prove that the communication complexity of Algorithm 5 is 𝑂(𝑛2). Recall that the +communication complexity denotes the number of bits sent by correct processes at and after GST. +Theorem 10. Let no correct process start the dissemination of its vector after time GST + 𝛿. Then, +the communication complexity of Algorithm 5 is 𝑂(𝑛2). +Proof. Let 𝑖 be the minimum index such that (1) process 𝑃𝑖 is correct, and (2) 𝑃𝑖 sends a +slow_broadcast message at some time ≥ GST. If 𝑖 does not exist, the lemma trivially holds. +Let 𝑡𝑖 denote the time at which 𝑃𝑖 starts the dissemination of its vector (line 8). By assumption, +𝑡𝑖 ≤ GST+𝛿. Every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + +𝛿 · 𝑛𝑖 + 3𝛿 (by Lemma 10). Thus, as 𝑡𝑖 ≤ GST + 𝛿, every correct process obtains a hash value and a +storage proof by time GST + 𝛿 · 𝑛𝑖 + 4𝛿. Moreover, by time GST + 𝛿 · 𝑛𝑖 + 4𝛿, all correct processes +stop sending slow_broadcast messages (due to line 25). +Let 𝑃𝑗 be a correct process such that 𝑗 > 𝑖. Due to the slow broadcast primitive (Algorithm 4), +𝑃𝑗 has a “waiting step” of (at least) 𝛿 · 𝑛𝑖 time. Therefore, during the [GST, GST + 𝛿 · 𝑛𝑖 + 4𝛿] +period, 𝑃𝑗 can send only 𝑂(1) slow_broadcast messages. Thus, at most one correct process +(i.e., 𝑃𝑖) sends more than 𝑂(1) slow_broadcast messages during the [GST, GST + 𝛿 · 𝑛𝑖 + 4𝛿] +period; that process sends at most 𝑛 slow_broadcast messages. As each message is of size +𝑂(𝑛) (since it carries a vector of 𝑛 − 𝑡 values), the communication complexity of Algorithm 5 is +𝑂(𝑛) · 𝑂(1) · 𝑂(𝑛) + 1 · 𝑂(𝑛) · 𝑂(𝑛) = 𝑂(𝑛2). +□ +C.3.2 +Vector Consensus with 𝑂(𝑛2 log𝑛) Communication Complexity. Finally, we are ready to +present our vector consensus algorithm (Algorithm 6) with subcubic communication complexity. +Our algorithm consists of three building blocks: (1) vector dissemination (Appendix C.3.1), (2) Quad +(§5.2.1), and (3) add [29], an algorithm for asynchronous data dissemination. In Algorithm 6, we +rely on a specific instance of Quad in which (1) each proposal is a hash value, and (2) given a hash +value 𝐻 and a (Quad) proof Σ,4 verify(𝐻, Σ) = true if and only if valid_SP(𝐻, Σ) = true (recall the +vector dissemination problem; Appendix C.3.1). Below, we briefly explain add. +4Do not confuse Quad proofs with storage proofs of the vector dissemination problem (Appendix C.3.1). +26 + +On the Validity of Consensus +add. This algorithm solves the data dissemination [29] problem defined in the following way. Let +𝑀 be a data blob which is an input of (at least) 𝑡 +1 correct processes; other correct processes have ⊥ +as their input. The data dissemination problem requires every correct process to eventually output +𝑀, and no other message. The key feature of add is that it solves the problem with 𝑂(𝑛2 log𝑛) +communication complexity. (For full details on add, refer to [29].) +Description of vector consensus. We give the description of Algorithm 6 from the perspective +of a correct process 𝑃𝑖. When 𝑃𝑖 proposes its value (line 10), it disseminates the value (using +the best-effort broadcast primitive) to all processes (line 11). Once 𝑃𝑖 receives proposals of 𝑛 − 𝑡 +distinct processes (line 16), it constructs an input configuration (line 17), and starts disseminating +it (line 18).5 +When 𝑃𝑖 obtains a hash value 𝐻 and a storage proof sp (line 19), 𝑃𝑖 proposes (𝐻, sp) to Quad +(line 21). Observe that verify(𝐻, sp) = true (due to the integrity property of vector dissemination). +Once 𝑃𝑖 decides from Quad (line 22), it starts add (line 24). Specifically, 𝑃𝑖 checks whether it +has cached an input configuration whose hash value is 𝐻 ′ (line 23). If so, 𝑃𝑖 inputs the input +configuration to add; otherwise, 𝑃𝑖 inputs ⊥. Once 𝑃𝑖 outputs an input configuration from add +(line 25), it decides it (line 26). +Algorithm 6 𝑂(𝑛2 log𝑛) Vector Consensus: Pseudocode (for process 𝑃𝑖) +1: Uses: +2: +Best-Effort Broadcast [19], instance beb +⊲ broadcast with no guarantees if the sender is faulty +3: +Vector Dissemination, instance disseminator +⊲ see Algorithm 5 +4: +Quad [23], instance quad +5: +add [29], instance add +6: upon init: +7: +Integer received_proposals𝑖 ← 0 +⊲ the number of received proposals +8: +Map(Process → V𝐼 ) proposals𝑖 ← empty +⊲ received proposals +9: +Map(Process → Message) messages𝑖 ← empty +⊲ received proposal messages +10: upon propose(𝑣 ∈ V𝐼 ): +11: +invoke beb.broadcast�⟨proposal, 𝑣⟩𝜎𝑖 +� +⊲ broadcast a signed proposal +12: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩𝜎𝑗 from process 𝑃𝑗 and received_proposals𝑖 < 𝑛 − 𝑡: +13: +received_proposals𝑖 ← received_proposals𝑖 + 1 +14: +proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 +15: +messages𝑖 [𝑃𝑗 ] ← 𝑚 +16: +if received_proposals𝑖 = 𝑛 − 𝑡: +⊲ received 𝑛 − 𝑡 proposals; can start disseminating +17: +Input_Configuration vector ← input configuration constructed from proposals𝑖 +18: +invoke disseminator.disseminate(vector) +19: upon disseminator.obtain�(Hash_Value H, Storage_Proof sp)�: +20: +if have not yet proposed to Quad: +21: +invoke quad.propose�(H, sp)� +22: upon quad.decide�(Hash_Value H′, Storage_Proof sp′)�: +23: +Input_Configuration vector′ ← a cached vector whose hash value is 𝐻′ +⊲ can be ⊥ +24: +invoke add.input(vector′) +25: upon add.output�Input_Configuration vector′′�: +26: +trigger decide(vector′′) +5Recall that this input configuration is actually a vector of 𝑛 − 𝑡 values. +27 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +Proof of correctness and complexity. We start by proving that Algorithm 6 is correct. +Theorem 11. Algorithm 6 is correct. +Proof. Let us prove that Algorithm 6 satisfies all properties of vector consensus. +• Agreement: Due to Agreement of Quad, no two correct processes decide different pairs from +Quad (line 22). Hence, for all correct processes which input a non-⊥ input configuration to +add (line 24), they input the same input configuration. Thus, due to the specification of add, +all correct processes output the same input configuration from add (line 25). Agreement is +satisfied. +• Termination: Every correct process broadcasts its proposal (line 11). Thus, every correct even- +tually receives 𝑛 −𝑡 proposals (line 16), and starts the dissemination of an input configuration +(line 18). Due to the termination property of vector dissemination (Lemma 11), every correct +process eventually obtains a hash value and a storage proof (line 19). Thus, every correct +process eventually proposes to Quad (line 21). Due to Termination of Quad, every correct +process eventually decides from Quad (line 22). As the pair decided from Quad includes a +storage proof (due to the specification of Quad), at least 𝑡 +1 correct processes have cached an +input configuration whose hash value is decided from Quad (by the redundancy property of +vector dissemination). Thus, due to the specification of add, every correct process eventually +outputs an input configuration from add (line 25). Termination is satisfied. +• Vector Validity: Let vec′ be an input configuration of 𝑛 − 𝑡 proposals decided by a correct +process (line 26). Hence, a storage proof sp′ such that valid_SP(hash(vec′), sp′) = true is +obtained (due to the specification of add). Therefore, vec′ is cached by (at least) 𝑡 + 1 correct +processes (due to the redundancy property of vector dissemination). Before a correct process +caches a vector (Algorithm 5), it verifies that it is associated with corresponding proposal +messages; we omit this check for brevity. As correct processes only send proposal messages +for their proposals (line 11), Vector Validity is satisfied. +The theorem holds. +□ +Lastly, we show the communication complexity of Algorithm 6. +Theorem 12. The communication complexity of Algorithm 6 is 𝑂(𝑛2 log𝑛). +Proof. The communication complexity of a single best-effort broadcast instance is 𝑂(𝑛). Every +correct process starts the dissemination of its vector by time GST + 𝛿 (as every correct process +receives 𝑛 −𝑡 proposals by this time). Thus, the communication complexity of vector dissemination +is 𝑂(𝑛2) (by Theorem 10). The communication complexity of Quad is 𝑂(𝑛2) (see [23]). Moreover, +the communication complexity of add is 𝑂(𝑛2 log𝑛) (see [29]). As Algorithm 6 is a composition of +the aforementioned building blocks, its communication complexity is 𝑛 · 𝑂(𝑛) + 𝑂(𝑛2) + 𝑂(𝑛2) + +𝑂(𝑛2 log𝑛) = 𝑂(𝑛2 log𝑛). +□ +D +EXTENDED FORMALISM +In this section, we give intuition behind an extension of our formalism which is suitable for the +analysis of blockchain-specific validity properties, such as External Validity [18, 20, 78]. External +Validity stipulates that any decided value must satisfy a predetermined logical predicate. However, +the “difficulty” of this property is that the logical predicate (usually) verifies a cryptographic proof, +which processes might not know “in advance” (see Appendix D.1). +In a nutshell, we make our original formalism more expressive by (1) making the input (V𝐼) and +output (V𝑂) spaces “unknown” to the processes, and (2) taking into account “proposals” of faulty +processes. In the rest of the paper: +28 + +On the Validity of Consensus +• We refer to the formalism introduced in the main body of the paper as the “original formalism”. +• We refer to the formalism we introduce below as the “extended formalism”. +We start by giving an intuition behind our extended formalism (Appendix D.1). Then, we introduce +some preliminaries (Appendix D.2). Finally, we define our extended formalism (Appendix D.3). +D.1 +Intuition +In the original formalism, processes “know” the entire input space V𝐼 and the entire output space +V𝑂. That is, processes are able to “produce” any value which belongs to V𝐼 or V𝑂. However, this +assumption limits the expressiveness of our formalism as it is impossible to describe a Byzantine +consensus problem in which input or output spaces are not “known”. Let us give an example. +Imagine a committee-based blockchain which establishes two roles: +• Clients are the users of the blockchain. They issue signed transactions to the blockchain. +• Servers are the operating nodes of the blockchain. Servers receive signed transactions issued +by the clients, and solve the Byzantine consensus problem to agree on the exact order +transactions are processed. +As servers propose transactions signed by the clients and they do not have access to the private keys +of the clients, servers do not “know” the input space V𝐼 nor the output space V𝑂 of the Byzantine +consensus problem. Hence, our original formalism cannot describe the Byzantine consensus problem +in the core of the aforementioned blockchain. +Extended vs. original formalism. As highlighted above, the main difference between the two +formalisms is that the extended one allows us to specify the “knowledge level” of the input and +output spaces. In the extended formalism, a process is able to “learn” output values by observing +input values. That is, we define a discovery function that defines which output values are learned +given observed input values. In the committee-based blockchain example, once a server observes +signed (by the issuing clients) transactions tx1 and tx2, it learns the following output values: (1) +tx1, (2) tx2, (3) tx1||tx2, and (4) tx2||tx1.6 +The second difference between the original and the extended formalism is that the extended +formalism takes into account “proposals” of faulty processes. Indeed, the original formalism does +not enable us to define which values are admissible given the adversary’s knowledge of the input +space. Think of the aforementioned example with a blockchain system. If no process (correct or +faulty) obtains a transaction tx, tx cannot be decided. However, if only a faulty process obtains a +transaction tx, tx could be an admissible decision. This scenario can be described by the extended +formalism, while it cannot by the original one. +D.2 +Preliminaries +We denote by V𝐼 the input space of Byzantine consensus, i.e., processes propose values contained +in V𝐼. Similarly, V𝑂 denotes the output space, i.e., processes decide values which belong to V𝑂. +Membership functions. We define two membership functions: +• valid_input : {0, 1}∗ → {true, false}: Intuitively, the valid_input(·) function specifies whether +a bit-sequence belongs to the input space V𝐼. +• valid_output : {0, 1}∗ → {true, false}: Intuitively, the valid_output(·) function specifies +whether a bit-sequence belongs to the output space V𝑂. +We assume that each process has access to these two functions. That is, each process can verify +whether an arbitrary sequence of bits belongs to the input (V𝐼) or output (V𝑂) space. In the case of +6We denote by “||” the concatenation operation. +29 + +Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira +a committee-based blockchain (Appendix D.1), the membership functions are simply signature- +verification functions. +Discovery function. We define a function discover: 2V𝐼 → 2V𝑂 . Given a set of proposals 𝑉𝐼 ⊆ V𝐼, +discover(𝑉𝐼) ⊆ V𝑂 specifies the set of decisions which are “discoverable” by 𝑉𝐼. We assume +that each process has access to the discover(·) function. Moreover, for any two sets 𝑉 1 +𝐼 ,𝑉 2 +𝐼 with +𝑉 1 +𝐼 ⊆ 𝑉 2 +𝐼 , discover(𝑉 1 +𝐼 ) ⊆ discover(𝑉 2 +𝐼 ); in other words, “knowledge” of the output space can only +be improved upon learning more input values. +Let us take a look at the committee-based blockchain example again (Appendix D.1). If a server +obtains a proposal tx, it learns tx as a potential decision. We model this “deduction” concept using +the discover(·) function: discover�{tx}� = {tx}. +Adversary pool. Given an execution E, P(E) ⊆ V𝐼 defines the adversary pool in E. Informally, the +adversary pool represents the input values the adversary “knows”. In the example of a committee- +based blockchain (Appendix D.1), the adversary pool is a set of signed transactions which the +adversary “learns” from the clients. +We underline that the adversary pool is an abstract concept. Specifically, the adversary pool rep- +resents the “starting knowledge” the adversary has. However, the notion of the “starting knowledge” +must be precisely defined once all particularities of the exact considered system are taken into +account. Due to the sophisticated details as the aforementioned one, we believe that the formalism +suitable for blockchain-specific validity properties deserves a standalone paper. +D.3 +Validity +We start by restating the definition of process-proposal pairs. A process-proposal pair is a pair (𝑃, 𝑣), +where (1) 𝑃 ∈ Π is a process, and (2) 𝑣 ∈ V𝐼 is a proposal. Given a process-proposal pair pp = (𝑃, 𝑣), +proposal(pp) = 𝑣 denotes the proposal associated with pp. +An input configuration is a tuple +� +pp1, pp2, ..., pp𝑥, 𝜌 +� +of 𝑥 process-proposal pairs and a set 𝜌 ⊆ V𝐼, +where (1) 𝑛 −𝑡 ≤ 𝑥 ≤ 𝑛, (2) every process-proposal pair is associated with a distinct process, and (3) +if 𝑥 = 𝑛, 𝜌 = ∅. Intuitively, an input configuration represents an assignment of proposals to correct +processes, as well as a “part” of the input space known to the adversary. For example, an input +configuration +� +(𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣), {𝑣, 𝑣 ′, 𝑣 ′′} +� +describes an execution in which (1) only processes +𝑃1, 𝑃2, and 𝑃3 are correct, (2) processes 𝑃1, 𝑃2, and 𝑃3 propose the same value 𝑣, and (3) faulty +processes know only 𝑣, 𝑣 ′, and 𝑣 ′′. +We denote by I the set of all input configurations. For every input configuration 𝑐 ∈ I, we +denote by 𝑐[𝑖] the process-proposal pair associated with process 𝑃𝑖; if such a process-proposal pair +does not exist, 𝑐[𝑖] = ⊥. Moreover, we define by pool(𝑐) the set of input values associated with 𝑐 +(the “𝜌” field of 𝑐). Next, 𝜋(𝑐) = {𝑃𝑖 ∈ Π | 𝑐[𝑖] ≠ ⊥} denotes the set of all processes included in 𝑐. +Finally, correct_proposals(𝑐) = {𝑣 ∈ V𝐼 | ∃𝑖 ∈ [1,𝑛] : 𝑐[𝑖] ≠ ⊥ ∧ proposal(𝑐[𝑖]) = 𝑣} denotes the +set of all proposals of correct processes (as specified by 𝑐). +Given (1) an execution E ∈ execs(A), where A is an algorithm which exposes the propose(·)/decide(·) +interface, and (2) an input configuration 𝑐 ∈ I, we say that E corresponds to 𝑐 if and only if (1) +𝜋(𝑐) = Corr A(E), (2) for every process 𝑃𝑖 ∈ Corr A(E), 𝑃𝑖’s proposal in E is proposal(𝑐[𝑖]), and +(3) P(E) = pool(𝑐). We denote by corresponding(E) = 𝑐 the input configuration to which E +corresponds. +A validity property val is a function val : I → 2V𝑂 such that, for every input configuration +𝑐 ∈ I, val(c) ≠ ∅. Algorithm A, where A exposes the propose(·)/decide(·) interface, satisfies a +validity property val if and only if, in every execution E ∈ execs(A), no correct process decides a +value 𝑣 ′ ∉ val�corresponding(E)�. That is, an algorithm satisfies a validity property if and only if +correct processes decide only admissible values. +30 + +On the Validity of Consensus +Assumptions on executions. Lastly, we introduce two assumptions that conclude our proposal for +the extended formalism. +Assumption 1. For every execution E of any algorithm A which solves the Byzantine consensus +problem with some validity property, if a correct process 𝑃 decides a value 𝑣 ′ ∈ V𝑂 in E, then +𝑣 ′ ∈ discover�correct_proposals(𝑐) ∪ pool(𝑐)�, where corresponding(E) = 𝑐. +Assumption 1 states that correct processes can only decide values which are “discoverable” using +all the proposals of correct processes and the knowledge of the adversary. For example, if every +correct process proposes the same value 𝑣 ∈ V𝐼 and the adversary pool contains only 𝑣 ′ ∈ V𝐼, then +a correct process can only decide a value from discover({𝑣, 𝑣 ′}). +Next, we introduce an assumption concerned only with the canonical executions (executions in +which faulty processes do not take any computational step). +Assumption 2. For every canonical execution E of any algorithm A which solves the Byzantine +consensus problem with some validity property, if a correct process 𝑃 decides a value 𝑣 ′ ∈ V𝑂 in +E, then 𝑣 ′ ∈ discover�correct_proposals(𝑐)�, where corresponding(E) = 𝑐. +Intuitively, Assumption 2 states that, if faulty processes are silent, correct processes can only +decide values which can be discovered using their own proposals. In other words, correct processes +cannot use “hidden” proposals (possessed by the “silent” adversary) to discover a decision. +Finally, we underline that these two assumptions do not completely prevent “unreasonable” +executions. For example, given these two assumptions, a (correct or faulty) process is still able to +send a message with a value which cannot be discovered using the proposals of correct processes +and the adversary pool. Hence, an assumption that prevents such an execution should be introduced. +Thus, due to the complexity we envision for the extended formalism, we leave it out of this paper. +In the future, we will focus on this interesting and important problem. +31 + diff --git a/KtE4T4oBgHgl3EQfJQzO/content/tmp_files/load_file.txt b/KtE4T4oBgHgl3EQfJQzO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f5c6c2fd85ed28a7442a85b9448ef709227198c --- /dev/null +++ b/KtE4T4oBgHgl3EQfJQzO/content/tmp_files/load_file.txt @@ -0,0 +1,1562 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf,len=1561 +page_content='On the Validity of Consensus PIERRE CIVIT, Sorbonne University, France SETH GILBERT, NUS Singapore, Singapore RACHID GUERRAOUI, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland JOVAN KOMATOVIC, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland MANUEL VIDIGUEIRA, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland The Byzantine consensus problem involves 𝑛 processes, out of which 𝑡 < 𝑛 could be faulty and behave arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Three properties characterize consensus: (1) termination, requiring correct (non-faulty) processes to eventually reach a decision, (2) agreement, preventing them from deciding different values, and (3) validity, precluding “unreasonable” decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' But, what is a reasonable decision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Strong validity, a classical property, stipulates that, if all correct processes propose the same value, only that value can be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Weak validity, another established property, stipulates that, if all processes are correct and they propose the same value, that value must be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The space of possible validity properties is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, their impact on consensus remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This paper addresses the question of which validity properties allow Byzantine consensus to be solvable with partial synchrony, and at what cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, we determine necessary and sufficient conditions for a validity property to make the consensus problem solvable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we say that such validity properties are solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Notably, we prove that, if 𝑛 ≤ 3𝑡, all solvable validity properties are trivial (there exists an always-admissible decision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Furthermore, we show that, with any non-trivial (and solvable) validity property, consensus requires Ω(𝑡2) messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This extends the seminal Dolev-Reischuk bound, originally proven for strong validity, to all non-trivial validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, we give a general Byzantine consensus algorithm, we call Universal, for any solvable (and non-trivial) validity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Importantly, Universal incurs 𝑂(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, together with our lower bound, Universal implies a fundamental result in partial synchrony: with 𝑡 ∈ Ω(𝑛), the message complexity of all (non-trivial) consensus variants is Θ(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 1 INTRODUCTION Consensus [50] is the cornerstone of state machine replication (SMR) [1, 8, 9, 21, 46, 47, 57, 61, 77], as well as various distributed protocols [13, 37, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recently, it has received a lot of attention with the advent of blockchain systems [5, 6, 17, 26, 28, 38, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The consensus problem is posed in a system of 𝑛 processes, out of which 𝑡 < 𝑛 can be faulty, and the rest are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Each correct process proposes a value, and consensus enables correct processes to decide on a common value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this paper, we consider Byzantine [50] consensus, where faulty processes can behave arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' While the exact definition of the problem might vary, two properties are always present: (1) termination, requiring correct processes to eventually decide, and (2) agreement, preventing them from deciding different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' It is not hard to devise an algorithm that satisfies only these two properties: every correct process decides the same, predetermined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, this algorithm is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To preclude such trivial solutions and render consensus meaningful, an additional property is required: validity, defining which decisions are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The many faces of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The literature contains many flavors of validity [4, 23, 24, 28, 36, 45, 59, 73, 74, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' One of the most studied properties is Strong Validity [4, 23, 28, 45], stipulating that, if all correct processes propose the same value, only that value can be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Another common property is Weak Validity [23, 24, 78], affirming that, if all processes are correct and propose the same value, that value must be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In fact, many other variants of the property have been 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='04920v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='DC] 12 Jan 2023 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira considered [36, 59, 73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' While validity may appear as an inconspicuous property, its exact definition has a big impact on consensus algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, the seminal Dolev-Reischuk bound [30] states that any solution to consensus with Strong Validity incurs a quadratic number of messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' it was recently proven that the bound is tight [23, 52, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In contrast, while there have been several improvements to the performance of consensus with Weak Validity over the last 40 years [23, 52, 78], the (tight) lower bound on message complexity remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (Although the bound is conjectured to be the same as for Strong Validity, this has yet to be formally proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') Many other fundamental questions remain unanswered: What does it take for a specific validity property to make consensus solvable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' What are the (best) upper and lower bounds on the message complexity of consensus with any specific validity property?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Is there a hierarchy of validity properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', a “strongest” validity property)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To the best of our knowledge, no in-depth study of the validity property has ever been conducted, despite its importance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We propose a precise mathematical formalism for the analysis of validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define a validity property as a mapping from assignments of proposals into admissible decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Although simple, our formalism enables us to determine the exact impact of validity on the solvability and complexity of consensus in the classical partially synchronous model [32], and answer the aforementioned open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Namely, we provide the following contributions: We classify all validity properties into solvable and unsolvable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (If a validity property makes consensus solvable, we say that the property itself is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') Specifically, for 𝑛 ≤ 3𝑡, we show that only trivial validity properties (for which there exists an always-admissible decision) are solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the case of 𝑛 > 3𝑡, we define the similarity condition, which we prove to be necessary and sufficient for a validity property to be solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We prove that all non-trivial (and solvable) validity properties require Ω(𝑡2) exchanged messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This result extends the Dolev-Reischuk bound [30], proven only for Strong Validity, to all “reasonable” validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we present Universal, a general consensus algorithm for all solvable (and non- trivial) validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Importantly, assuming a threshold signature scheme, Universal exchanges 𝑂(𝑛2) messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, together with our lower bound, Universal implies a fundamental result in partial synchrony: given 𝑡 ∈ Ω(𝑛), all (non-trivial) consensus variants have Θ(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Figure 1 summarizes our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Technical overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In our formalism, we use the notion of input configuration that denotes an assignment of proposals to correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, � (𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣) � represents an input configuration by which (1) only processes 𝑃1, 𝑃2, and 𝑃3 are correct, and (2) processes 𝑃1, 𝑃2, and 𝑃3 propose 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, we define a similarity relation between input configurations: two input configurations are similar if and only if (1) they have (at least) one process in common, and (2) for every common process, the process’s proposal is identical in both input configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, an input configuration 𝑐 = � (𝑃1, 0), (𝑃2, 1) � is similar to � (𝑃1, 0), (𝑃3, 0) � , but not to � (𝑃1, 0), (𝑃2, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We observe that all similar input configurations must have an admissible value in common;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we call this canonical similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us illustrate why a common admissible value must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consider the aforementioned similar input configurations 𝑐 = � (𝑃1, 0), (𝑃2, 1) � and 𝑐′ = � (𝑃1, 0), (𝑃3, 0) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If there is no common admissible value for 𝑐 and 𝑐′, consensus cannot be solved: process 𝑃1 cannot distinguish (1) an execution in which 𝑃2 is correct, and 𝑃3 is faulty and silent, from (2) an execution in which 𝑃2 is faulty, but behaves correctly, and 𝑃3 is correct, but slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, 𝑃1 cannot conclude whether it needs to decide an admissible value for 𝑐 or for 𝑐′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Canonical similarity is a critical 2 On the Validity of Consensus trivial non-trivial solvable Validity properties Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Illustration of our results: (1) with 𝑛 ≤ 3𝑡, all solvable validity properties are trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) the exact set of solvable validity properties (as determined by our necessary and sufficient conditions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) all non-trivial (and solvable) validity properties require Ω(𝑡2) exchanged messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (4) for any non-trivial (and solvable) validity property, there exists a consensus algorithm with 𝑂(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' intermediate result that we use extensively throughout the paper (even if it does not directly imply any of our results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In our proof of triviality with 𝑛 ≤ 3𝑡, we intertwine the classical partitioning argument [51] with our canonical similarity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Namely, we show that, for any input configuration, there exists an execution in which the same value 𝑥 is decided, making 𝑥 an always-admissible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For our lower bound, while following the idea of the original proof [3, 30], we rely on canonical similarity to prove the bound for all solvable and non-trivial validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we design Universal by relying on vector consensus [27, 31, 69, 76], a problem in which processes agree on the proposals of 𝑛 − 𝑡 processes: when a correct process decides a vector vec of 𝑛 − 𝑡 proposals (from vector consensus), it decides (from Universal) the common admissible value for all input configurations similar to vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, consider an execution which corresponds to an input configuration 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If, in this execution, a correct process decides a vector vec from vector consensus, it is guaranteed that vec is similar to 𝑐 (the proposals of correct processes are identical in 𝑐 and in vec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, deciding (from Universal) the common admissible value for all input configurations similar to vec guarantees that the decided value is admissible according to 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We provide an overview of related work in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In §3, we specify the system model (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), define the consensus problem (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), describe our formalism for validity properties (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3), and present canonical similarity (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define the necessary conditions for the solvability of validity properties in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In §5, we prove a quadratic lower bound on message complexity for all non-trivial (and solvable) validity properties (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), and introduce Universal, a general consensus algorithm for any solvable (and non-trivial) validity property (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We conclude the paper in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The appendix contains (1) detailed proofs of our results, (2) omitted algorithms, and (3) a proposal for how to extend our formalism to accommodate for blockchain-specific validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 2 RELATED WORK Solvability of consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The consensus problem has been thoroughly investigated in a variety of system settings and failure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' It has been known (for long) that consensus can be solved in 3 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira a synchronous setting, both with crash [19, 56, 70] and arbitrary failures [4, 48, 62, 70, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In an asynchronous environment, however, consensus cannot be solved deterministically even if a single process can fail, and it does so only by crashing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' this is the seminal FLP impossibility result [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A traditional way of circumventing the FLP impossibility result is randomization [7, 10, 11, 33, 54], where termination of consensus is not ensured deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Another well-established approach to bypass the FLP impossibility is to strengthen the communication model with partial synchrony [32]: communication is asynchronous until some unknown time, and then it becomes synchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The last couple of decades have produced many partially synchronous consensus algorithms [18, 21, 23, 24, 28, 32, 49, 52, 56, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Another line of research has consisted in weakening the definition of consensus to make it deterministically solvable under asynchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the condition-based approach [64], the specification of consensus is relaxed to require termination only if the assignment of proposals satisfies some predetermined conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The efficiency of this elegant approach has been studied further in [66], then extended to the synchronous setting [65, 79], as well as to the 𝑘-set agreement problem [41, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Solvability of general decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A distributed decision problem has been defined in [25, 44, 60] as a mapping from input assignments to admissible decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our validity formalism is of the same nature, and it is inspired by the aforementioned specification of decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The solvability of decision problems has been thoroughly studied in asynchronous, crash-prone settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' It was shown in [63] that the FLP impossibility result [35] can be extended to many decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In [15], the authors defined necessary and sufficient conditions for a decision problem to be solvable with a single crash failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The asynchronous solvability of problems in which crash failures occur at the very beginning of an execution was studied in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Necessary and sufficient conditions for a decision problem to be solvable in a randomized manner were given in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The topology-based approach on studying the solvability of decision problems in asynchrony has proven to be extremely effective, both for crash [42, 43, 71] and arbitrary failures [42, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our results follow the same spirit as many of these approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' however, we study the deterministic solvability and complexity of all consensus variants in a partially synchronous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Validity of consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Various validity properties have been associated with the consensus problem (beyond the aforementioned Strong Validity and Weak Validity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Correct-Proposal Validity [36, 73] states that a value decided by a correct process must have been proposed by a correct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Median Validity [74] is another validity property proposed in the context of synchronous consensus, requiring the decision to be close to the median of the proposals of correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Interval Validity [59], on the other hand, requires the decision to be close to the 𝑘-th smallest proposal of correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The advent of blockchain technologies has resurged the concept of External Validity [18, 20, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This property requires the decided value to satisfy a predetermined predicate, typically asserting whether the decided value follows the rules of a blockchain system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', no double-spending).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (For pedagogical purposes, we considered a simple formalism to express basic validity properties and derive our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To express External Validity, which is out of the scope of the paper, we propose an extension of our formalism in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') In interactive consistency [12, 34, 58], correct processes agree on the proposals of all correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given that the problem is impossible in a non-synchronous setting, a weaker variant has been considered: vector consensus [27, 31, 69, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Here, processes need to agree on a vector of proposals which does not necessarily include the proposals of all correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Interactive consistency and vector consensus can be seen as specific consensus problems with a validity property requiring that, if a decided vector contains a proposal 𝑣 of a correct process, that correct process has indeed proposed value 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The design of Universal, our general consensus algorithm 4 On the Validity of Consensus for any solvable (and non-trivial) validity property, demonstrates that any non-trivial flavor of consensus, which is solvable in partial synchrony, can be solved using vector consensus (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 3 PRELIMINARIES In this section, we present the computational model (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), recall the consensus problem (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), formally define validity properties (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3), and introduce canonical similarity (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Computational Model Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We consider a system Π = {𝑃1, 𝑃2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', 𝑃𝑛} of 𝑛 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' At most 𝑡 (0 < 𝑡 < 𝑛) processes can be faulty: these processes can exhibit arbitrary behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A non-faulty process is said to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Processes communicate by exchanging messages over an authenticated point-to-point network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The communication network is reliable: if a correct process sends a message to a correct process, the message is eventually received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given an algorithm A, execs(A) denotes the set of all executions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Furthermore, Corr A(E) denotes the set of correct processes in an execution E ∈ execs(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We say that an execution E ∈ execs(A) is canonical if and only if no faulty process takes any computational step in E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' note that faulty processes do not send any message in a canonical execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Partial synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We consider the standard partially synchronous model [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every execu- tion of the system, there exists a Global Stabilization Time (GST) and a positive duration 𝛿 such that message delays are bounded by 𝛿 after GST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' GST is not known to processes, whereas 𝛿 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We assume that all correct processes start executing their local algorithm before or at GST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Cryptographic primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In one variant of the Universal algorithm, we assume a (𝑘,𝑛)-threshold signature scheme [53] (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In fact, this variant relies on a closed-box consensus algorithm which internally utilizes threshold signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In a threshold signature scheme, each process holds a distinct private key, and there exists a single public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Each process 𝑃𝑖 can use its private key to produce a (partial) signature of a message 𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we denote by ⟨𝑚⟩𝜎𝑖 a message (partially) signed by the process 𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, a signature can be verified by other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, a set of signatures for a message 𝑚 from 𝑘 (the threshold) distinct processes can be combined into a single threshold signature for 𝑚, which proves that 𝑘 processes have signed 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let A be any algorithm and let E ∈ execs(A) be any execution of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of E is the number of messages sent by correct processes during [GST, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of A is defined as max E∈execs(A) � message complexity of E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Consensus We denote by V𝐼 the set of values processes can propose, and by V𝑂 the set of values processes can decide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The consensus1 problem exposes the following interface: request propose(𝑣 ∈ V𝐼): a process proposes a value 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' indication decide(𝑣 ′ ∈ V𝑂): a process decides a value 𝑣 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A correct process proposes and decides at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consensus requires the following properties: Termination: Every correct process eventually decides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Agreement: No two correct processes decide different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 1Throughout the entire paper, we use “consensus” and “Byzantine consensus” interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 5 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira If the consensus problem was completely defined by Termination and Agreement, a trivial solution would exist: processes decide on a default value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, the specification of consensus addition- ally includes a validity property, which connects the proposals of correct processes to admissible decisions, precluding the aforementioned trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3 Validity In a nutshell, our specification of a validity property includes a set of assignments of proposals to correct processes, and, for each such assignment, a corresponding set of admissible decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by defining a process-proposal pair as a pair (𝑃, 𝑣), where (1) 𝑃 ∈ Π is a process, and (2) 𝑣 ∈ V𝐼 is a proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given a process-proposal pair pp = (𝑃, 𝑣), proposal(pp) = 𝑣 denotes the proposal associated with pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' An input configuration is a tuple � pp1, pp2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', pp𝑥 � of 𝑥 process-proposal pairs, where (1) 𝑛 − 𝑡 ≤ 𝑥 ≤ 𝑛, and (2) every process-proposal pair is associated with a distinct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Intuitively, an input configuration represents an assignment of proposals to correct processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, an input configuration � (𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣), (𝑃4, 𝑣), (𝑃5, 𝑣) � describes an execution in which (1) only processes 𝑃1, 𝑃2, 𝑃3, 𝑃4 and 𝑃5 are correct, and (2) all of them propose the same value 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We denote by I the set of all input configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Furthermore, for every 𝑥 ∈ [𝑛 − 𝑡,𝑛], I𝑥 ⊂ I denotes the set of input configurations with exactly 𝑥 process-proposal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every input configuration 𝑐 ∈ I, we denote by 𝑐[𝑖] the process-proposal pair associated with process 𝑃𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' if such a process-proposal pair does not exist, 𝑐[𝑖] = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, 𝜋(𝑐) = {𝑃𝑖 ∈ Π | 𝑐[𝑖] ≠ ⊥} denotes the set of all processes included in 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given (1) an execution E of an algorithm A, where A exposes the propose(·)/decide(·) interface, and (2) an input configuration 𝑐 ∈ I, we say that E corresponds to 𝑐 if and only if (1) 𝜋(𝑐) = Corr A(E), and (2) for every process 𝑃𝑖 ∈ Corr A(E), 𝑃𝑖’s proposal in E is proposal(𝑐[𝑖]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We denote by corresponding(E) = 𝑐 the input configuration to which E corresponds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we define a validity property val as a function val : I → 2V𝑂 such that, for every input configuration 𝑐 ∈ I, val(c) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' An algorithm A, where A exposes the propose(·)/decide(·) interface, satisfies a validity property val if and only if, in every execution E ∈ execs(A), no correct process decides a value 𝑣 ′ ∉ val�corresponding(E)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, an algorithm satisfies a validity property if and only if correct processes decide only admissible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Weak Validity & Strong Validity in our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To illustrate our formalism, we describe how it can be used to express these two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For both, V𝐼 = V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Weak Validity can be expressed as: val(𝑐) = � {𝑣}, if (𝜋(𝑐) = Π) ∧ (∀𝑃𝑖 ∈ 𝜋(𝑐) : proposal(𝑐[𝑖]) = 𝑣) V𝑂, otherwise whereas Strong Validity can be expressed as: val(𝑐) = � {𝑣}, if ∀𝑃𝑖 ∈ 𝜋(𝑐) : proposal(𝑐[𝑖]) = 𝑣 V𝑂, otherwise Consensus algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' An algorithm A solves consensus with a validity property val if and only if the following holds: A exposes the propose(·)/decide(·) interface, and A satisfies Termination, Agreement and the validity property val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, we formally define the notion of a solvable validity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Definition 1 (Solvable validity property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We say that a validity property val is solvable if and only if there exists an algorithm which solves consensus with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 6 On the Validity of Consensus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='4 Canonical Similarity In this subsection, we introduce canonical similarity, a crucial intermediate result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In order to do so, we first define an important relation between input configurations, that of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define the similarity relation (“∼”) between input configurations: ∀𝑐1,𝑐2 ∈ I : 𝑐1 ∼ 𝑐2 ⇐⇒ (𝜋(𝑐1) ∩ 𝜋(𝑐2) ≠ ∅) ∧ (∀𝑃𝑗 ∈ 𝜋(𝑐1) ∩ 𝜋(𝑐2) : 𝑐1[𝑗] = 𝑐2[𝑗]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In other words, 𝑐1 is similar to 𝑐2 if and only if (1) 𝑐1 and 𝑐2 have at least one process in common, and (2) for every common process, the process’s proposal is identical in both input configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, 𝑐 = � (𝑃1, 0), (𝑃2, 1), (𝑃3, 0) � is similar to � (𝑃1, 0), (𝑃3, 0) �, whereas 𝑐 is not similar to � (𝑃1, 0), (𝑃2, 0), (𝑃3, 0) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Note that the similarity relation is symmetric (for every pair 𝑐1,𝑐2 ∈ I, 𝑐1 ∼ 𝑐2 ⇔ 𝑐2 ∼ 𝑐1) and reflexive (for every 𝑐 ∈ I, 𝑐 ∼ 𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every input configuration 𝑐 ∈ I, we define its similarity set, denoted by sim(𝑐): sim(𝑐) = {𝑐′ ∈ I | 𝑐′ ∼ 𝑐}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The canonical similarity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let A be an algorithm which solves consensus with some validity property val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our canonical similarity result states that A, in any canonical execution which corresponds to some input configuration 𝑐, can only decide a value which is admissible for all input configurations similar to 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Informally, the reason is that correct processes cannot distinguish silent faulty processes from slow correct ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 1 (Canonical similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let val be any solvable validity property and let A be any algorithm which solves the consensus problem with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E ∈ execs(A) be any canonical execution and let corresponding(E) = 𝑐, for some input configuration 𝑐 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a value 𝑣 ′ ∈ V𝑂 is decided by a correct process in E, then 𝑣 ′ ∈ � 𝑐′∈sim(𝑐) val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We prove the lemma by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Suppose that 𝑣 ′ ∉ � 𝑐′∈sim(𝑐) val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, there exists an input configuration 𝑐′ ∈ sim(𝑐) such that 𝑣 ′ ∉ val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑃 denote any infinite continuation of E such that corresponding(E𝑃) = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑃 be any process such that 𝑃 ∈ 𝜋(𝑐′)∩𝜋(𝑐);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' such a process exists as 𝑐′ ∼ 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies Termination and Agreement, E𝑃 is an infinite execution, and 𝑃 is correct in E𝑃, 𝑃 decides 𝑣 ′ in E𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We construct another execution E′ ∈ execs(A) such that corresponding(E′) = 𝑐′: (1) E′ is identical to E𝑃 until process 𝑃 decides 𝑣 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) All processes in Π \\ 𝜋(𝑐′) are faulty in E′ (they behave correctly until 𝑃 has decided), and all processes in 𝜋(𝑐′) are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) After 𝑃 has decided, processes in 𝜋(𝑐′) \\ 𝜋(𝑐) “wake up” with the proposals specified in 𝑐′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (4) GST is set to after all processes in 𝜋(𝑐′) have taken a computational step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every process 𝑃𝑖 ∈ 𝜋(𝑐′)∩𝜋(𝑐), the proposal of 𝑃𝑖 in E′ is proposal(𝑐′[𝑖]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' recall that𝑐′[𝑖] = 𝑐[𝑖] as 𝑐′ ∼ 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, for every process 𝑃𝑗 ∈ 𝜋(𝑐′) \\ 𝜋(𝑐), the proposal of 𝑃𝑗 in E′ is proposal(𝑐′[𝑗]) (due to the step 3 of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, corresponding(E′) = 𝑐′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Furthermore, process 𝑃, which is correct in E′, decides a value 𝑣 ′ ∉ val(𝑐′) (due to the step 1 of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, we reach a contradiction with the fact that A satisfies val, which proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ 4 NECESSARY SOLVABILITY CONDITIONS We give in this section necessary conditions for the solvability of validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by focusing on the case of 𝑛 ≤ 3𝑡: we prove that, if 𝑛 ≤ 3𝑡, all solvable validity properties are trivial (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, we consider the case of 𝑛 > 3𝑡: we formally define the similarity condition, and prove its necessity for solvable validity properties (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 7 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Triviality of Solvable Validity Properties if 𝑛 ≤ 3𝑡 Some validity properties, such as Weak Validity and Strong Validity, are known to be unsolvable for 𝑛 ≤ 3𝑡 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This seems to imply a split of validity properties depending on the resiliency threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We prove that such a split indeed exists for 𝑛 ≤ 3𝑡, and, importantly, that it applies to all solvable validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Implicitly, this means that there is no “useful” relaxation of the validity property that can tolerate 𝑡 > ⌊𝑛/3⌋ failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Concretely, we prove the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a validity property is solvable with 𝑛 ≤ 3𝑡, then the validity property is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', there exists a value 𝑣 ′ ∈ V𝑂 such that 𝑣 ′ ∈ � 𝑐 ∈I val(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Before presenting the proof of the theorem, we introduce the compatibility relation between input configurations, which we use throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define the compatibility relation (“⋄”) between input configurations: ∀𝑐1,𝑐2 ∈ I : 𝑐1 ⋄𝑐2 ⇐⇒ (|𝜋(𝑐1) ∩ 𝜋(𝑐2)| ≤ 𝑡) ∧ (∃𝑃 ∈ 𝜋(𝑐1) \\ 𝜋(𝑐2)) ∧ (∃𝑄 ∈ 𝜋(𝑐2) \\ 𝜋(𝑐1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, 𝑐1 is compatible with 𝑐2 if and only if (1) there are at most 𝑡 processes in common, (2) there exists a process which belongs to 𝑐1 and does not belong to 𝑐2, and (3) there exists a process which belongs to 𝑐2 and does not belong to 𝑐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, when 𝑛 = 3 and 𝑡 = 1, 𝑐 = � (𝑃1, 0), (𝑃2, 0) � is compatible with � (𝑃1, 1), (𝑃3, 1) � , whereas 𝑐 is not compatible with � (𝑃1, 1), (𝑃2, 1), (𝑃3, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Observe that the compatibility relation is symmetric and irreflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Throughout the rest of the subsection, we fix any validity property val which is solvable with 𝑛 ≤ 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover: We assume that 𝑛 ≤ 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We fix any algorithm A which solves consensus with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We fix any input configuration base ∈ I𝑛−𝑡 with exactly 𝑛 − 𝑡 process-proposal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We fix any infinite canonical execution Ebase ∈ execs(A) such that corresponding(Ebase) = base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies Termination and val, and Ebase is infinite, some value 𝑣base ∈ val(base) is decided by a correct process in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To prove Theorem 1, it suffices to prove that val is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our proof leverages our formalism, specifically the aforementioned compatibility relation and the established canonical similarity result (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='4), and combines the formalism with the classical partitioning argument [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by showing that only 𝑣base can be decided in any canonical execution which corresponds to any input configuration compatible with base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a value different from 𝑣base is decided, the adversary would be able to cause a disagreement by partitioning processes into two disagreeing groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We delegate the formal proof of the following lemma to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑐 ∈ I be any input configuration such that 𝑐 ⋄ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑐 ∈ execs(A) be any canonical execution such that corresponding(E𝑐) = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a value 𝑣𝑐 ∈ V𝑂 is decided by a correct process in E𝑐, then 𝑣𝑐 = 𝑣base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Observe that the proposals of an input configuration compatible with base do not influence the decision: given an input configuration 𝑐 ∈ I, 𝑐 ⋄ base, only 𝑣base can be decided in any canonical execution which corresponds to 𝑐, irrespectively of the proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, we prove a direct consequence of Lemma 2: for every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Below, we give a proof sketch of the claim;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' the formal proof can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 ∈ execs(A) such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 8 On the Validity of Consensus Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Fix any input configuration 𝑐𝑛 ∈ I𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' There exists an input configuration 𝑐𝑛−𝑡 ∈ I𝑛−𝑡 such that (1) for every process 𝑃 ∈ 𝜋(𝑐𝑛) ∩ 𝜋(𝑐𝑛−𝑡), proposals of 𝑃 in 𝑐𝑛 and 𝑐𝑛−𝑡 are identical, and (2)𝑐𝑛−𝑡⋄base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, 𝑣base is decided in any infinite canonical execution E𝑛−𝑡 which corresponds to 𝑐𝑛−𝑡 (by Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, by “waking up” processes in 𝜋(𝑐𝑛) \\ 𝜋(𝑐𝑛−𝑡) after 𝑣base is decided in E𝑛−𝑡, we build an execution E𝑛 such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ We are now ready to prove that val is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We do so by showing that, for every input configuration 𝑐 ∈ I, 𝑣base ∈ val(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Validity property val is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We fix any input configuration 𝑐 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us distinguish two possible scenarios: Let 𝑐 ∈ I𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' There exists an execution E𝑐 ∈ execs(A) such that (1) corresponding(E𝑐) = 𝑐, and (2) 𝑣base is decided in E𝑐 (by Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies val, 𝑣base ∈ val(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑐 ∉ I𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We construct an input configuration 𝑐𝑛 ∈ I𝑛 in the following way: (1) Let 𝑐𝑛 ← 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) For every process 𝑃 ∉ 𝜋(𝑐), (𝑃, any proposal) is included in 𝑐𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the construction of 𝑐𝑛, 𝑐𝑛 ∼ 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By Lemma 3, there exists an execution E𝑛 ∈ execs(A) such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Note that E𝑛 is a canonical execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, canonical similarity ensures that 𝑣base ∈ val(𝑐) (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In both possible cases, 𝑣base ∈ val(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Lemma 4 concludes the proof of Theorem 1, as Lemma 4 proves that val, any solvable validity property with 𝑛 ≤ 3𝑡, is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Figure 2 depicts the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Since this subsection showed that no useful consensus variant exists when 𝑛 ≤ 3𝑡, the rest of the paper focuses on the case of 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 2 base Lemma 3 Lemma 4 All input configurations 0 0 0 0 A A A A A A A A A A A A A A A A A A A A A A A A Lemma 4 Lemma 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 1: Overview of the proof in the case 𝑛 = 6, 𝑡 = 2, and base = � (𝑃1, 0), (𝑃2, 0), (𝑃3, 0), (𝑃4, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Similarity Condition: Necessary Solvability Condition This subsection defines the similarity condition, and proves its necessity for solvable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Definition 2 (Similarity condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A validity property val satisfies the similarity condition (C𝑆, in short) if and only if there exists a computable function Λ : I𝑛−𝑡 → V𝑂 such that: ∀𝑐 ∈ I𝑛−𝑡 : Λ(𝑐) ∈ � 𝑐′∈sim(𝑐) val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' C𝑆 states that, for every input configuration 𝑐 ∈ I𝑛−𝑡, there exists a computable function Λ(𝑐) which retrieves a common admissible decision among all input configurations similar to 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The 9 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira necessity of C𝑆 follows from the canonical similarity result: in any infinite canonical execution, a common admissible value must be decided (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Any solvable validity property satisfies C𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By the means of contradiction, let there exist a validity property val such that (1) val does not satisfy C𝑆, and (2) val is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let A be any algorithm which solves the Byzantine consensus problem with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As val does not satisfy C𝑆, there does not exist a computable function Λ : I𝑛−𝑡 → V𝑂 such that, for every input configuration 𝑐 ∈ I𝑛−𝑡, Λ(𝑐) ∈ � 𝑐′∈sim(𝑐) val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Fix any input configuration 𝑐 ∈ I𝑛−𝑡 for which Λ(𝑐) is not defined or not computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑐 ∈ execs(A) be an infinite canonical execution such that (1) corresponding(E𝑐) = 𝑐, (2) the system is synchronous from the very beginning (GST = 0), and (3) message delays are exactly 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In other words, E𝑐 unfolds in a “lock-step” manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies Termination and E𝑐 is an infinite execution, some value 𝑣𝑐 ∈ V𝑂 is decided by a correct process in E𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By canonical similarity (Lemma 1), 𝑣𝑐 ∈ � 𝑐′∈sim(𝑐) val(𝑐′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, Λ(𝑐) is defined (Λ(𝑐) = 𝑣𝑐) and computable (due to the construction of E𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, we reach a contradiction with the fact that Λ(𝑐) is not defined or not computable, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Notice that, for proving the necessity of C𝑆 (Theorem 2), we do not rely on the 𝑛 > 3𝑡 assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, C𝑆 is necessary for all solvable validity properties (irrespectively of the resiliency threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, as proven in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1, C𝑆 is not sufficient when 𝑛 ≤ 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 (Observe that any trivial validity property satisfies C𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') Similarity condition (example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consider Correct-Proposal Validity [36, 73], a validity property which states that any decided value must have been proposed by a correct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consensus with Correct-Proposal Validity is also known as strong consensus [36, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Strong consensus assumes V𝐼 = V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' It was shown that, in partial synchrony, strong consensus cannot be solved if 𝑛 ≤ (|V𝐼 | + 1)𝑡 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We now present an alternative proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Namely, we show that Correct-Proposal Validity does not satisfy C𝑆 if 𝑛 ≤ (|V𝐼 | + 1)𝑡, which makes it unsolvable by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let |V𝐼 | = |V𝑂 | = 𝑚 and 𝑛 = (𝑚 + 1)𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define an input configuration base ∈ I𝑛−𝑡 such that (1) |𝜋(base)| = 𝑛 − 𝑡 = 𝑚𝑡, and (2) every value 𝑣 ∈ V𝐼 is the proposal of exactly 𝑡 processes in base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, all values are admissible for base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, for each value 𝑣 ∈ V𝐼, we design an input configuration 𝑐∌𝑣 such that (1) 𝑣 is not admissible for 𝑐∌𝑣, and (2) 𝑐∌𝑣 ∼ base: (1) Let 𝑐∌𝑣 ← base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) We remove from 𝑐∌𝑣 all process-proposal pairs pp with proposal(pp) = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) For every process 𝑃 ∉ 𝜋(base), we add (𝑃, 𝑣 ′) to 𝑐∌𝑣, for any value 𝑣 ′ ≠ 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, Correct-Proposal Validity does not satisfy C𝑆 when 𝑛 ≤ (𝑚 + 1)𝑡 as, for every 𝑣 ∈ V𝑂, there exists an input configuration 𝑐∌𝑣 ∼ base for which 𝑣 is not admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, Correct-Proposal Validity is unsolvable if 𝑛 ≤ (𝑚 + 1)𝑡 (by Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 5 LOWER BOUND & GENERAL ALGORITHM This section is devoted to the cost of solving consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Specifically, we first show that any non- trivial and solvable validity property requires Ω(𝑡2) messages to be exchanged (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, we present Universal, a general algorithm which, if 𝑛 > 3𝑡, solves consensus with any validity property which satisfies C𝑆 (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, Universal proves the sufficiency of C𝑆 when 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 2For example, Weak Validity satisfies C𝑆, but it is unsolvable with 𝑛 ≤ 3𝑡 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 10 On the Validity of Consensus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Lower Bound on Message Complexity In this subsection, we prove the following theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If an algorithm solves consensus with a non-trivial validity property, the message complexity of the algorithm is Ω(𝑡2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 3 extends the seminal Dolev-Reischuk bound [30], proven only for consensus algorithms with Strong Validity, to all non-trivial variants of consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To prove Theorem 3, we intertwine the idea of the original proof [30] with the canonical similarity result (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In our proof, we show that any algorithm which solves the Byzantine consensus problem with a non-trivial validity property has a synchronous execution (GST = 0) in which correct processes send ≥ ( 𝑡 2)2 messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, throughout the entire subsection, we fix a non-trivial and solvable validity property val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, we fix A, an algorithm which solves the Byzantine consensus problem with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As val is a non-trivial validity property, 𝑛 > 3𝑡 (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, we define a specific infinite execution Ebase ∈ execs(A) in the following manner: (1) GST = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, the system is synchronous throughout the entire execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) All processes are separated into two groups: (1) group 𝐴, with |𝐴| = 𝑛 − ⌈ 𝑡 2⌉, and (2) group 𝐵, with |𝐵| = ⌈ 𝑡 2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) All processes in the group 𝐴 are correct, whereas all processes in the group 𝐵 are faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (4) We fix any value 𝑣∗ ∈ V𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every correct process 𝑃𝐴 ∈ 𝐴, the proposal of 𝑃𝐴 in Ebase is 𝑣∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (5) For every faulty process 𝑃𝐵 ∈ 𝐵, 𝑃𝐵 behaves correctly in Ebase with its proposal being 𝑣∗, except that (1) 𝑃𝐵 ignores the first ⌈ 𝑡 2⌉ messages received from other processes, and (2) 𝑃𝐵 omits sending messages to other processes in 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To prove Theorem 3, it suffices to show that the message complexity of Ebase is ≥ (⌈ 𝑡 2⌉)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By contradiction, let the correct processes (processes in 𝐴) send less than (⌈ 𝑡 2⌉)2 messages in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The first step of our proof shows that, given that correct processes send less than (⌈ 𝑡 2⌉)2 messages in Ebase, there must exist a process 𝑄 ∈ 𝐵 which can correctly decide some value 𝑣𝑄 ∈ V𝑂 without receiving any message from any other process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We prove this claim using the pigeonhole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' There exist a value 𝑣𝑄 ∈ V𝑂 and a process 𝑄 ∈ 𝐵 such that 𝑄 has a correct behavior 𝛽𝑄 in which (1) 𝑄 decides 𝑣𝑄, and (2) 𝑄 does not receive any message from any other process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By assumption, correct processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', processes in the group 𝐴) send less than (⌈ 𝑡 2⌉)2 messages in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, due to the pigeonhole principle, there exists a process 𝑄 ∈ 𝐵 which receives less than ⌈ 𝑡 2⌉ messages (from other processes) in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recall that 𝑄 behaves correctly in Ebase with its proposal being 𝑣∗ ∈ V𝐼, except that (1) 𝑄 ignores the first ⌈ 𝑡 2⌉ messages received from other processes, and (2) 𝑄 does not send any messages to other processes in the group 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We denote by 𝑆𝑄 the set of processes, not including 𝑄, which send messages to 𝑄 in Ebase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' |𝑆𝑄 | < ⌈ 𝑡 2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, we construct an infinite execution E′ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Execution E′ base is identical to Ebase, except that we introduce the following modifications: (1) Processes in (𝐴 ∪ {𝑄}) \\𝑆𝑄 are correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' other processes are faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, we make 𝑄 correct in E′ base, and we make all processes in 𝑆𝑄 faulty in E′ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) Processes in 𝐵 \\ {𝑄} behave exactly as in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, processes in 𝑆𝑄 behave exactly as in Ebase, except that they do not send any message to 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the construction of E′ base, process 𝑄 does not receive any message (from any other process) in E′ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As 𝑄 is correct in E′ base and A satisfies Termination, 𝑄 decides some value 𝑣𝑄 ∈ V𝑂 in E′ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, 𝑄 indeed has a correct behavior 𝛽𝑄 in which it decides 𝑣𝑄 ∈ V𝑂 without having received messages from other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ 11 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira In the second step of our proof, we show that there exists an execution in which (1) 𝑄 is faulty and silent, and (2) other processes decide some value 𝑣 ≠ 𝑣𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' There exists an execution E𝑣 such that (1) 𝑄 is faulty and silent in E𝑣, and (2) a value 𝑣 ≠ 𝑣𝑄 is decided by a correct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As val is a non-trivial validity property, there exists an input configuration 𝑐∌𝑣𝑄 ∈ I such that 𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' recall that 𝑣𝑄 is the value that 𝑄 can correctly decide without having received any message from any other process (Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We consider two possible cases: Let 𝑄 ∉ 𝜋(𝑐∌𝑣𝑄 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, E𝑣 is any infinite canonical execution which corresponds to 𝑐∌𝑣𝑄 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As 𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 ), the value decided in E𝑣 must be different from 𝑣𝑄 (as A satisfies val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑄 ∈ 𝜋(𝑐∌𝑣𝑄 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We construct an input configuration 𝑐∌𝑄 ∈ I such that 𝑄 ∉ 𝜋(𝑐∌𝑄): (1) Let 𝑐∌𝑄 ← 𝑐∌𝑣𝑄 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) We remove (𝑄, ·) from 𝑐∌𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, we remove 𝑄’s process-proposal pair from 𝑐∌𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) If |𝜋(𝑐∌𝑣𝑄 )| = 𝑛 − 𝑡, we add (𝑍, any value) to 𝑐∌𝑄, where 𝑍 is any process such that 𝑍 ∉ 𝜋(𝑐∌𝑣𝑄 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' note that such a process 𝑍 exists as 𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the construction of 𝑐∌𝑄, 𝑐∌𝑄 ∼ 𝑐∌𝑣𝑄 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Indeed, (1) 𝜋(𝑐∌𝑄) ∩ 𝜋(𝑐∌𝑣𝑄 ) ≠ ∅ (as 𝑛 − 𝑡 − 1 > 0 when 𝑛 > 3𝑡 and 𝑡 > 0), and (2) for every process 𝑃 ∈ 𝜋(𝑐∌𝑄) ∩ 𝜋(𝑐∌𝑣𝑄 ), the proposal of 𝑃 is identical in 𝑐∌𝑄 and 𝑐∌𝑣𝑄 (by the step 1 of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this case, E𝑣 is any infinite canonical execution such that corresponding(E𝑣) = 𝑐∌𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies Termination and E𝑣 is infinite, some value 𝑣 ∈ V𝑂 is decided by a correct process in E𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As 𝑐∌𝑄 ∼ 𝑐∌𝑣𝑄 , 𝑣 ∈ val(𝑐∌𝑣𝑄 ) (by canonical similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, 𝑣 ≠ 𝑣𝑄 as (1) 𝑣 ∈ val(𝑐∌𝑣𝑄 ), and (2) 𝑣𝑄 ∉ val(𝑐∌𝑣𝑄 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The lemma holds as its statement is true in both possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ As we have shown the existence of E𝑣 (Lemma 6), we can “merge” E𝑣 with the valid behavior 𝛽𝑄 in which 𝑄 decides 𝑣𝑄 without having received any message (Lemma 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, we can construct an execution in which A violates Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, correct processes must send at least (⌈ 𝑡 2⌉)2 ∈ Ω(𝑡2) messages in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The formal proof of the following lemma, from which the lower bound on message complexity (Theorem 3) follows directly, is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of Ebase is at least (⌈ 𝑡 2⌉)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 General Algorithm Universal: Similarity Condition is Sufficient if 𝑛 > 3𝑡 In this subsection, we prove that C𝑆 is sufficient for a validity property to be solvable when 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, we prove the following theorem: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑛 > 3𝑡, and let val be any validity property which satisfies C𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, val is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, assuming a threshold signature scheme, there exists an algorithm which solves Byzantine consensus with val, and has 𝑂(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To prove Theorem 4, we present Universal, an algorithm which solves the Byzantine consensus problem with any validity property satisfying C𝑆, given that 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, Universal solves consensus with any solvable and non-trivial validity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Notably, assuming a threshold signature scheme, Universal achieves 𝑂(𝑛2) message complexity, making it optimal (when 𝑡 ∈ Ω(𝑛2)) according to our lower bound (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' To construct Universal, we rely on vector consensus [27, 31, 69, 76] (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), a problem which requires correct processes to agree on the proposals of 𝑛 − 𝑡 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Specifically, when a correct process decides a vector vec of 𝑛 −𝑡 proposals (from vector consensus), it decides (from Universal) the common admissible value for all input configurations similar to vec, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', the process decides Λ(vec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Note that the idea of solving consensus from vector consensus is not novel [14, 28, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For 12 On the Validity of Consensus some validity properties it is even natural, such as Strong Validity (choose the most common value) or Weak Validity (choose any value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, thanks to the necessity of C𝑆 (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), any solvable consensus variant can reuse this simple algorithmic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this subsection, we first recall vector consensus (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, we utilize vector consensus to construct Universal (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Throughout the entire subsection, 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Vector Consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In essence, vector consensus allows each correct process to infer the pro- posals of 𝑛−𝑡 (correct or faulty) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Formally, correct processes agree on input configurations (of vector consensus) with exactly 𝑛 − 𝑡 process-proposal pairs: V𝑂 = I𝑛−𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us formally define Vector Validity, the validity property of vector consensus: Vector Validity: Let a correct process decide vector ∈ V𝑂, which contains exactly 𝑛 −𝑡 process- proposal pairs, such that (1) (𝑃, 𝑣) belongs to vector, for some process 𝑃 ∈ Π and some value 𝑣 ∈ V𝐼, and (2) 𝑃 is a correct process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, 𝑃 proposed 𝑣 to vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Intuitively, Vector Validity states that, if a correct process “concludes” that a value 𝑣 was proposed by a correct process 𝑃, then 𝑃’s proposal was indeed 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We provide two implementations of vector consensus: (1) a non-authenticated implementation (without any cryptographic primitives), and (2) an authenticated implementation (with threshold signatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the lack of space, we give the pseudocode of the non-authenticated version in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The pseudocode of an authenticated version is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This version relies on Quad, a Byzantine consensus algorithm recently introduced in [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we briefly discuss Quad below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In essence, Quad is a partially-synchronous, “leader-based” Byzantine consensus algo- rithm, which achieves 𝑂(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Internally, Quad relies on a threshold signature scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Formally, Quad is concerned with two sets: (1) VQuad, a set of values, and (2) PQuad, a set of proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In Quad, processes propose and decide value-proof pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' There exists a function verify : VQuad ×PQuad → {true, false}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Importantly, PQuad is not known a-priori: it is only assumed that, if a correct process proposes a pair (𝑣 ∈ VQuad, Σ ∈ PQuad), then verify(𝑣, Σ) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Quad guarantees the following: if a correct process decides a pair (𝑣, Σ), then verify(𝑣, Σ) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In other words, correct processes decide only valid value-proof pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (See [23] for full details on Quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') In our authenticated implementation of vector consensus (Algorithm 1), we rely on a specific instance of Quad where (1) VQuad = I𝑛−𝑡 (processes propose to Quad input configurations of vector consensus), and (2) PQuad is a set of 𝑛−𝑡 proposal messages (sent by processes in vector consensus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, given an input configuration 𝑐 ∈ VQuad and a set of messages Σ ∈ PQuad, verify(𝑐, Σ) = true if and only if, for every process-proposal pair (𝑃𝑗, 𝑣 𝑗) which belongs to 𝑐, ⟨proposal, 𝑣 𝑗⟩𝜎𝑗 ∈ Σ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', every process-proposal pair of 𝑐 is accompanied by a properly signed proposal message).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Description of authenticated vector consensus (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When a correct process 𝑃𝑖 proposes a value 𝑣 ∈ V𝐼 to vector consensus (line 8), the process broadcasts a signed proposal message (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 receives 𝑛 − 𝑡 proposal messages (line 14), 𝑃𝑖 constructs an input configuration vector (line 15), and a proof Σ (line 16) from the received proposal messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, 𝑃𝑖 proposes (vector, Σ) to Quad (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, when 𝑃𝑖 decides a pair (vector′, Σ′) from Quad (line 18), 𝑃𝑖 decides vector′ from vector consensus (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of Algorithm 1 is 𝑂(𝑛2) as (1) processes only broadcast proposal messages, and (2) the message complexity of Quad is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We underline that the communication complexity of Algorithm 1, the number of bits sent by correct processes, is 𝑂(𝑛3) as the communi- cation complexity of Quad is 𝑂(𝑛2 · 𝑥) = 𝑂(𝑛3) (see [23]), where 𝑥 is the size of a Quad proposal (in our case, 𝑥 ∈ Θ(𝑛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to space constraints, we delegate the full proof of the correctness and complexity of Algorithm 1 to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 13 Pierre Civit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Seth Gilbert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Rachid Guerraoui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Jovan Komatovic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' and Manuel Vidigueira Algorithm 1 Authenticated Vector Consensus: Pseudocode (for process 𝑃𝑖) 1: Uses: 2: Best-Effort Broadcast [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance beb ⊲ broadcast with no guarantees if the sender is faulty 3: Quad [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance quad 4: upon init: 5: Integer received_proposals𝑖 ← 0 ⊲ the number of received proposals 6: Map(Process → V𝐼 ) proposals𝑖 ← empty ⊲ received proposals 7: Map(Process → Message) messages𝑖 ← empty ⊲ received proposal messages 8: upon propose(𝑣 ∈ V𝐼 ): 9: invoke beb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast�⟨proposal, 𝑣⟩𝜎𝑖 � ⊲ broadcast a signed proposal 10: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩𝜎𝑗 from process 𝑃𝑗 and received_proposals𝑖 < 𝑛 − 𝑡: 11: received_proposals𝑖 ← received_proposals𝑖 + 1 12: proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 13: messages𝑖 [𝑃𝑗 ] ← 𝑚 14: if received_proposals𝑖 = 𝑛 − 𝑡: ⊲ received 𝑛 − 𝑡 proposals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' can propose to Quad 15: Input_Configuration vector ← input configuration constructed from proposals𝑖 16: Proof Σ ← set of messages containing all proposal messages from messages𝑖 17: invoke quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='propose�(vector, Σ)� 18: upon quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='decide�(Input_Configuration vector′, Proof Σ′)�: 19: trigger decide(vector′) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We construct Universal (Algorithm 2) directly from vector consensus (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When a correct process 𝑃𝑖 proposes to Universal (line 3), the proposal is forwarded to vector consensus (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 decides an input configuration 𝑐 from vector consensus (line 5), 𝑃𝑖 decides Λ(𝑐) (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Note that our implementation of Universal (Algorithm 2) is independent of the actual imple- mentation of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, by employing our authenticated implementation of vector consensus (Algorithm 1), we obtain a general consensus algorithm with 𝑂(𝑛2) message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' On the other hand, by employing a non-authenticated implementation of vector consensus (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), we obtain a non-authenticated version of Universal, which implies that any validity property which satisfies C𝑆 is solvable even in a non-authenticated setting (if 𝑛 > 3𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 2 Universal: Pseudocode for process 𝑃𝑖 1: Uses: 2: Vector Consensus, instance vec_cons 3: upon propose(𝑣 ∈ V𝐼 ): 4: invoke vec_cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='propose(𝑣) 5: upon vec_cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='decide(Input_Configuration 𝑐): 6: trigger decide�Λ(𝑐)� Finally, we prove that Universal (Algorithm 2) is a general Byzantine consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let val be any validity property which satisfies C𝑆, and let 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Universal solves the Byzantine consensus problem with val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, if Universal employs Algorithm 1 as its vector consensus building block, the message complexity of Universal is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Termination and Agreement of Universal follow from Termination and Agreement of vector consensus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, the message complexity of Universal is identical to the message complexity of its vector consensus building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 14 On the Validity of Consensus Finally, we prove that Universal satisfies val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consider any execution E of Universal such that corresponding(E) = 𝑐∗, for some input configuration𝑐∗ ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let𝑐 ∈ I𝑛−𝑡 be the input configuration correct processes decide from vector consensus in E (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As vector consensus satisfies Vector Validity, we have that, for every process 𝑃 ∈ 𝜋(𝑐∗) ∩ 𝜋(𝑐), 𝑃’s proposals in 𝑐∗ and 𝑐 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, 𝑐 ∼ 𝑐∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, Λ(𝑐) ∈ val(𝑐∗) (by the definition of the Λ function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, val is satisfied by Universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ As Universal (Algorithm 2) solves the Byzantine consensus problem with any validity property which satisfies C𝑆 (Theorem 5) if 𝑛 > 3𝑡, C𝑆 is sufficient for solvable validity properties when 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, as Universal relies on vector consensus, we conclude that Vector Validity is a strongest validity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, a solution to any variant of the consensus problem can be obtained from vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A note on the communication complexity of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' While the version of Universal which employs Algorithm 1 (as its vector consensus building block) has optimal message com- plexity, its communication complexity is 𝑂(𝑛3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This presents a linear gap to the lower bound for communication complexity (also Ω(𝑛2), implied by Theorem 3), and to known optimal solutions for some validity properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', Strong Validity, proven to be Θ(𝑛2) [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' At first glance, this seems like an issue inherent to vector consensus: the decided vectors are linear in size, suggesting that the linear gap could be inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3, we give a vector consensus algorithm with 𝑂(𝑛2 log𝑛) communication complexity, albeit with exponential latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3 Is it possible to construct vector consensus with subcubic communication and polynomial latency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This is an important open question, as positive answers would lead to (practical) performance improvements of all consensus variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 6 CONCLUDING REMARKS This paper studies the validity property of partially synchronous Byzantine consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Namely, we mathematically formalize validity properties, and give necessary and sufficient conditions for a validity property to be solvable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', for the existence of an algorithm which solves a consensus problem defined with that validity property, in addition to Agreement and Termination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, we prove a quadratic lower bound on message complexity for all non-trivial (and solvable) validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Previously, this bound was mainly known for Strong Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, we introduce Universal, a general algorithm for consensus with any solvable (and non-trivial) validity property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Universal achieves 𝑂(𝑛2) message complexity, showing that the aforementioned lower bound is tight (with 𝑡 ∈ Ω(𝑛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We conjecture that our necessary and sufficient conditions for consensus solvability can easily be adapted to a synchronous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, our proof technique for the lower bound on message complexity cannot be reused as such: this is because silent processes can be conclu- sively detected in synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, we believe that the lower bound on message complexity for synchronous consensus is one of the most important open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Another interesting question is whether our lower bound holds for randomized consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In [3], it is proven that randomized consensus with Strong Validity has Ω(𝑡2) expected message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Can this bound be extended to all non-trivial validity properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we restate the question posed at the end of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Is it possible to solve vector consensus with 𝑜(𝑛3) exchanged bits and polynomial latency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recall that, due to the design of Universal (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), any (non-trivial) consensus variant can be solved using vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, an 3Both our authenticated (Algorithm 1) and our non-authenticated (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2) variants of vector consensus have linear latency, which implies linear latency of Universal when employing any of these two algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 15 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira upper bound on the complexity of vector 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In International Symposium on Distributed Computing (2003), Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 239–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 19 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira A TRIVIALITY OF SOLVABLE VALIDITY PROPERTIES IF 𝑛 ≤ 3𝑡: FORMAL PROOF In this section, we give formal proofs of the intermediate results from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, we formally prove that only 𝑣base can be decided in any canonical execution which corresponds to any input configuration compatible with base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 2 (restated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑐 ∈ I be any input configuration such that 𝑐 ⋄ base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑐 ∈ execs(A) be any canonical execution such that corresponding(E𝑐) = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a value 𝑣𝑐 ∈ V𝑂 is decided by a correct process in E𝑐, then 𝑣𝑐 = 𝑣base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By contradiction, suppose that some value 𝑣𝑐 ≠ 𝑣base is decided by a correct process in E𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Since 𝑐 ⋄ base, there exists a correct process 𝑄 ∈ 𝜋(𝑐) \\ 𝜋(base).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑄 𝑐 be an infinite continuation of E𝑐 in which 𝑄 decides;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' note that E𝑄 𝑐 exists as A satisfies Termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The following holds for process 𝑄: (1) 𝑄 decides 𝑣𝑐 in E𝑄 𝑐 (as A satisfies Agreement), and (2) process 𝑄 is silent in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑡𝑄 denote the time at which 𝑄 decides in E𝑄 𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Similarly, there exists a process 𝑃 ∈ 𝜋(base) \\ 𝜋(𝑐);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' observe that (1) process 𝑃 decides 𝑣base in Ebase, and (2) process 𝑃 is silent in E𝑄 𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑡𝑃 denote the time at which 𝑃 decides in Ebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We now construct an execution E ∈ execs(A) by “merging” Ebase and E𝑄 𝑐 : (1) Processes in 𝜋(𝑐) ∩ 𝜋(base) behave towards processes in 𝜋(base) \\ 𝜋(𝑐) as in Ebase, and towards processes in 𝜋(𝑐) \\ 𝜋(base) as in E𝑄 𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) Communication between (1) processes in 𝜋(base) \\ 𝜋(𝑐) and (2) processes in 𝜋(𝑐) \\ 𝜋(base) is delayed until after max(𝑡𝑃,𝑡𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) We set GST to after max(𝑡𝑃,𝑡𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The following holds for E: Processes in 𝜋(base) ⊖ 𝜋(𝑐) (symmetric difference) are correct in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Only processes in 𝜋(base) ∩ 𝜋(𝑐) are faulty in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recall that |𝜋(base) ∩ 𝜋(𝑐)| ≤ 𝑡 as base ⋄𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Process 𝑄, which is correct in E, cannot distinguish E from E𝑄 𝑐 until time max(𝑡𝑃,𝑡𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, process 𝑄 decides 𝑣𝑐 in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Process 𝑃, which is correct in E, cannot distinguish E from Ebase until time max(𝑡𝑃,𝑡𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, process 𝑃 decides 𝑣base ≠ 𝑣𝑐 in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, we reach a contradiction with the fact that A satisfies Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, 𝑣𝑐 = 𝑣base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Next, we formally prove that, for every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 such that (1) E𝑛 corresponds to 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 3 (restated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every input configuration 𝑐𝑛 ∈ I𝑛, there exists an execution E𝑛 ∈ execs(A) such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Fix any input configuration 𝑐𝑛 ∈ I𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We construct an input configuration 𝑐𝑛−𝑡 ∈ I𝑛−𝑡: (1) For every process 𝑃𝑖 ∉ 𝜋(base), we include a process-proposal pair (𝑃𝑖, 𝑣) in 𝑐𝑛−𝑡 such that 𝑣 = proposal(𝑐𝑛[𝑖]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Note that there are 𝑡 such processes as |𝜋(base)| = 𝑛 − 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) We include 𝑛 − 2𝑡 process-proposal pairs (𝑃𝑖, 𝑣) in 𝑐𝑛−𝑡 such that (1) 𝑃𝑖 ∈ 𝜋(base), and (2) 𝑣 = proposal(𝑐𝑛[𝑖]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, we “complete” 𝑐𝑛−𝑡 (constructed in the step 1) with 𝑛 − 2𝑡 process-proposal pairs such that the process is “borrowed” from base, and its proposal is “borrowed” from 𝑐𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Observe that 𝑐𝑛−𝑡 ⋄base as (1) |𝜋(𝑐𝑛−𝑡) ∩ base| ≤ 𝑡 (because 𝑛 − 2𝑡 ≤ 𝑡 when 𝑛 ≤ 3𝑡), (2) there exists a process 𝑃 ∈ 𝜋(base) \\ 𝜋(𝑐𝑛−𝑡) (because, when constructing 𝑐𝑛−𝑡, we excluded 𝑡 > 0 processes from base;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' step 1), and (3) there exists a process 𝑄 ∈ 𝜋(𝑐𝑛−𝑡) \\ 𝜋(base) (because we included 𝑡 > 0 processes in cn−t which are not in base;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let E𝑛−𝑡 ∈ execs(A) denote any infinite canonical execution such that corresponding(E𝑛−𝑡) = 𝑐𝑛−𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As A satisfies Termination, some value is decided by correct processes in E𝑛−𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' due to Lemma 2, 20 On the Validity of Consensus that value is 𝑣base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we are able to construct an infinite execution E𝑛 ∈ execs(A) such that (1) corresponding(E𝑛) = 𝑐𝑛, and (2) 𝑣base is decided in E𝑛: (1) All processes are correct in E𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) Until some correct process 𝑃 ∈ 𝜋(𝑐𝑛−𝑡) decides 𝑣base, E𝑛 is identical to E𝑛−𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) Afterwards, every process 𝑄 ∉ 𝜋(𝑐𝑛−𝑡) “wakes up” with the proposal specified in 𝑐𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (4) GST is set to after all processes have taken a computational step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, 𝑣base is indeed decided in E𝑛 and corresponding(E𝑛) = 𝑐𝑛, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ B LOWER BOUND ON MESSAGE COMPLEXITY: FORMAL PROOF In this section, we give a formal proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recall that we have fixed an algorithm A which solves the Byzantine consensus problem with a non-trivial validity property val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 7 (restated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of Ebase is at least (⌈ 𝑡 2⌉)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By Lemma 5, there exists a behavior 𝛽𝑄 of process 𝑄 in which 𝑄 decides a value 𝑣𝑄 without having received any message (from any other process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑡𝑄 denote the time at which 𝑄 decides 𝑣𝑄 in 𝛽𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, there exists an execution E𝑣 in which (1) 𝑄 is faulty and silent, and (2) correct processes decide a value 𝑣 ≠ 𝑣𝑄 (by Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑡𝑣 denote the time at which a correct process decides 𝑣 ≠ 𝑣𝑄 in E𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We now construct an execution E in the following way: (1) Processes in Corr A(E𝑣) ∪ {𝑄} are correct in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' All other processes are faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) All messages from and to 𝑄 are delayed until after max(𝑡𝑄,𝑡𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (3) Process 𝑄 exhibits the behavior 𝛽𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (4) Until max(𝑡𝑄,𝑡𝑣), no process in Corr A(E𝑣) can distinguish E from E𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (5) GST is set to after max(𝑡𝑄,𝑡𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As no process in Corr A(E𝑣) can distinguish E from E𝑣 until max(𝑡𝑄,𝑡𝑣), 𝑣 ≠ 𝑣𝑄 is decided by a correct process in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, 𝑄 decides 𝑣𝑄 in E (step 3 of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, Agreement is violated in E, which contradicts the fact that A satisfies Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, the starting assumption is not correct: in Ebase, correct processes send (at least) (⌈ 𝑡 2⌉)2 messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ C VECTOR CONSENUS: FORMAL PROOFS & OMITTED ALGORITHMS In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1, we prove the correctness and complexity of our authenticated implementation of vector consensus (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We dedicate Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 to a non-authenticated implementation of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3, we give an implementation of vector consensus with 𝑂(𝑛2 log𝑛) communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Throughout the entire section, we assume that 𝑛 > 3𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Authenticated Implementation (Algorithm 1): Formal Proofs In this subsection, we prove the correctness and complexity of our authenticated implementation of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start with the correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 1 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Agreement follows directly from the fact that Quad satisfies Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Termination follows from (1) Termination of Quad, and (2) the fact that, eventually, all correct processes receive 𝑛 − 𝑡 proposal messages (as there are at least 𝑛 − 𝑡 correct processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We now prove that Algorithm 1 satisfies Vector Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let a correct process 𝑃 decide vector′ ∈ I𝑛−𝑡 from vector consensus (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, 𝑃 has decided (vector′, Σ′) from Quad, where (1) Σ′ is some proof, and (2) verify(vector′, Σ′) = true (due to the specification of Quad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Furthermore, if there exists a process-proposal pair (𝑃, 𝑣 ∈ V𝐼) in vector′, where 𝑃 is a correct process, a properly 21 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira signed proposal message belongs to Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As correct processes send proposal messages only for their proposals (line 9), 𝑣 was indeed proposed by 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Finally, we prove the complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of Algorithm 1 is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The message complexity of the specific instance of Quad utilized in Algorithm 1 is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Additionally, correct processes exchange 𝑂(𝑛2) proposal messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, the message complexity is 𝑂(𝑛2) + 𝑂(𝑛2) = 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Non-Authenticated Implementation: Pseudocode & Formal Proofs We now present a non-authenticated implementation (Algorithm 3) of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The design of Algorithm 3 follows the reduction from the binary consensus to the multivalue consensus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Namely, we use the following two building blocks in Algorithm 3: (1) Byzantine Reliable Broadcast [16, 19]: This primitive allows processes to disseminate infor- mation in a reliable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Formally, the Byzantine reliable broadcast exposes the following interface: (1) request broadcast(𝑚), and (2) indication deliver(𝑚′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The primitive satisfies the following properties: Validity: If a correct process 𝑃 broadcasts a message 𝑚, 𝑃 eventually delivers 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Consistency: No two correct processes deliver different messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Integrity: Every correct process delivers at most one message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, if a correct process delivers a message 𝑚 from a process 𝑃 and 𝑃 is correct, then 𝑃 broadcast 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Totality: If a correct process delivers a message, every correct process delivers a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In Algorithm 3, we use a non-authenticated implementation [16] of the Byzantine Reliable Broadcast primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (2) Binary DBFT [28], a non-authenticated algorithm which solves the Byzantine consensus problem with Strong Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us briefly explain how Algorithm 3 works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we focus on a correct process 𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, 𝑃𝑖 reliably broadcasts its proposal (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 delivers a proposal of some process 𝑃𝑗 (line 12), 𝑃𝑖 proposes 1 to the corresponding DBFT instance (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Eventually, 𝑛 − 𝑡 DBFT instances decide 1 (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once that happens, 𝑃𝑖 proposes 0 to all DBFT instance to which 𝑃𝑖 has not proposed (line 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When all DBFT instances have decided (line 23), 𝑃𝑖 decides an input configuration associated with the first 𝑛 − 𝑡 processes whose DBFT instances decided 1 (constructed at line 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 3 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by proving Termination of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Eventually, at least 𝑛 −𝑡 DBFT instances decide 1 due to the fact that (1) no correct process proposes 0 to any DBFT instance unless 𝑛 − 𝑡 DBFT instances have decided 1 (line 18), and (2) all correct processes eventually propose 1 to the DBFT instances which correspond to the correct processes (unless 𝑛 − 𝑡 DBFT instances have already decided 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When 𝑛 − 𝑡 DBFT instances decide 1 (line 18), each correct process proposes to all instances to which it has not yet proposed (line 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, eventually all DBFT instances decide, and (at least) 𝑛 − 𝑡 DBFT instances decide 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, the rule at line 23 eventually activates at every correct process, which implies that every correct process eventually decides (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, we prove Vector Validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a correct process 𝑃 decides an input configuration with a process-proposal pair (𝑄, 𝑣), 𝑃 has delivered a proposal message from 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If 𝑄 is correct, due to integrity of the reliable broadcast primitive, 𝑄’s proposal was indeed 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, Agreement follows from (1) Agreement of DBFT, and (2) consistency of the reliable broadcast primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, Algorithm 3 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ 22 On the Validity of Consensus Algorithm 3 Non-Authenticated Vector Consensus: Pseudocode (for process 𝑃𝑖) 1: Uses: 2: Non-Authenticated Byzantine Reliable Broadcast [16], instance brb 3: Binary DBFT [28], instances dbft[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' dbft[𝑛] ⊲ one instance of the binary DBFT protocol per process 4: upon init: 5: Map(Process → V𝐼 ) proposals𝑖 ← empty ⊲ received proposals 6: Map(Process → Message) messages𝑖 ← empty ⊲ received proposal messages 7: Boolean dbft_proposing𝑖 = true ⊲ is 𝑃𝑖 still proposing to the DBFT instances 8: Map(Process → Boolean) dbft_proposed𝑖 ← {false,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' for every Process} 9: Integer dbft_decisions𝑖 ← 0 ⊲ the number of the DBFT instances which have decided 10: upon propose(𝑣 ∈ V𝐼 ): 11: invoke brb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast�⟨proposal, 𝑣⟩� ⊲ broadcast a proposal 12: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩ from process 𝑃𝑗: 13: proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 14: messages𝑖 [𝑃𝑗 ] ← 𝑚 15: if dbft_proposing𝑖 = true: 16: dbft_proposed𝑖 [𝑃𝑗 ] ← true 17: invoke dbft[𝑗 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='propose(1) 18: upon 𝑛 − 𝑡 DBFT instances have decided 1 (for the first time): 19: dbft_proposing𝑖 ← false 20: for every Process 𝑃𝑗 such that dbft_proposed𝑖 [𝑃𝑗 ] = false: 21: dbft_proposed𝑖 [𝑃𝑗 ] ← true 22: invoke dbft[𝑗 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='propose(0) 23: upon all DBFT instances decided, and, for the first 𝑛 − 𝑡 processes 𝑃𝑗 such that dbft[𝑗 ] decided 1, proposals𝑖 [𝑃𝑗 ] ≠ ⊥: 24: Input_Configuration vector ← input configuration with 𝑛 − 𝑡 process-proposal pairs corresponding to the first 𝑛 − 𝑡 DBFT instances which decided 1 25: trigger decide(vector) The main downside of Algorithm 3 is that its message complexity is 𝑂(𝑛4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, non- authenticated version of Universal has 𝑂(𝑛4) message complexity, which is not optimal according to our lower bound (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3 Implementation with 𝑂(𝑛2 log𝑛) Communication: Pseudocode & Formal Proofs In this subsection, we give an implementation of vector consensus with 𝑂(𝑛2 log𝑛) communication complexity, which comes within a logarithmic factor of the lower bound (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This implementation represents a near-linear communication improvement over Algorithm 1 (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2), which achieves 𝑂(𝑛3) communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We note that the following solution is highly impractical due to its exponential latency (worst-case 𝑂(𝑛𝑡), requiring idealized cryptographic primitives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, our solution does represent a step towards closing the existing gap in the communication complexity of non-trivial and solvable validity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Vector Dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, we formally define the vector dissemination problem, which plays the crucial role in our vector consensus algorithm with improved communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this problem, each correct process disseminates a vector of 𝑛 − 𝑡 values, and all correct processes eventually obtain (1) a hash of some disseminated vector, and (2) a storage proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every hash value 𝐻 and every storage proof sp, we define valid_SP(𝐻, sp) ∈ {true, false}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Formally, the vector dissemination problem exposes the following interface: request disseminate(Vector vec): a process disseminates a vector vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' indication obtain(Hash_Value 𝐻 ′, Storage_Proof sp′): a process obtains a hash value 𝐻 ′ and a storage proof sp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 23 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira The following properties are required: Termination: Every correct process eventually obtains a hash value and a storage proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 𝛿-Closeness: Let 𝑡first denote the first time a correct process obtains a hash value and a storage proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, every correct process obtains a hash value and a storage proof by time max(GST,𝑡first) + 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Redundancy: Let a (faulty or correct) process obtain a storage proof sp′ such that, for some hash value 𝐻 ′, valid_SP(𝐻 ′, sp′) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, (at least) 𝑡 + 1 correct processes have cached a vector vec′ such that hash(vec′) = 𝐻 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Integrity: If a correct process obtains a hash value 𝐻 ′ and a storage proof sp′, the following holds: valid_SP(𝐻 ′, sp′) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Slow broadcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In order to solve the vector dissemination problem, we present a simple algorithm (Algorithm 4) which implements slow broadcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In slow broadcast, each process disseminates its vector in “one-by-one” fashion, with a “waiting step” between any two sending events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Specifically, process 𝑃1 broadcasts its vector by (1) sending the vector to 𝑃1 (line 3), and then waiting 𝛿 time (line 4), (2) sending the vector to 𝑃2 (line 3), and then waiting 𝛿 time (line 4), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Process 𝑃2 broadcasts its vector in the same manner, but it waits 𝛿 · 𝑛 time (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Crucially, if the system is synchronous, the waiting time of 𝑃2 is (roughly) sufficient for 𝑃1 to completely disseminate its vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This holds for any two processes 𝑃𝑖 and 𝑃𝑗 such that 𝑖 < 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 4 Slow Broadcast: Pseudocode (for process 𝑃𝑖) 1: upon broadcast(Vector vec): 2: for each Process 𝑃𝑗: 3: send ⟨slow_broadcast, vec⟩ to 𝑃𝑗 4: wait for 𝛿 · 𝑛(𝑖−1) time 5: upon reception of ⟨slow_broadcast, Vector vec′⟩ from process 𝑃𝑗: 6: trigger deliver(vec′, 𝑃𝑗) Vector dissemination algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our solution is given in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' First, we give a con- crete implementation of the valid_SP(·, ·) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given a hash value 𝐻 and a storage proof sp, valid_SP(𝐻, sp) = true if and only if sp is a valid (𝑛 − 𝑡)-threshold signature of ⟨stored, 𝐻⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us explain Algorithm 5 from the perspective of a correct process 𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When 𝑃𝑖 starts dissem- inating its vector vec (line 8), 𝑃𝑖 stores its hash (line 9) and slow-broadcasts the vector (line 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 receives stored messages from 𝑛 − 𝑡 distinct processes (line 17), 𝑃𝑖 combines received partial signatures into a storage proof (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, 𝑃𝑖 broadcasts (using the best-effort broadcast primitive) the constructed storage proof (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Whenever 𝑃𝑖 receives a storage proof (line 21), 𝑃𝑖 checks whether the storage proof is valid (line 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If it is, 𝑃𝑖 rebroadcasts the storage proof (line 23), obtains a hash value and the storage proof (line 24), and stops participating (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', sending and processing messages) in vector dissemina- tion (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Observe that, once 𝑃𝑖 stops participating in vector dissemination (line 25), it stops participating in slow broadcast, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof of correctness and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by proving redundancy of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 5 satisfies redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let a (correct or faulty) process obtain a storage proof sp′ such that valid_SP(𝐻 ′, sp′) = true, for some hash value 𝐻 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, 𝑛 − 𝑡 processes have signed a stored message for 𝐻 ′ (as valid_SP(𝐻 ′, sp′) = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Among these 𝑛 − 𝑡 processes, at least 𝑡 + 1 are correct (as 𝑛 > 3𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Before 24 On the Validity of Consensus Algorithm 5 Vector Dissemination: Pseudocode (for process 𝑃𝑖) 1: Uses: 2: Best-Effort Broadcast [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance beb ⊲ broadcast with no guarantees if the sender is faulty 3: Slow Broadcast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance slow ⊲ see Algorithm 4 4: upon init: 5: Hash_Value 𝐻𝑖 ← ⊥ ⊲ hash of the message 𝑃𝑖 slow-broadcasts 6: Map(Hash_Value → Vector) vectors𝑖 ← empty ⊲ received vectors 7: Set(Process) disseminated𝑖 ← empty ⊲ processes who have disseminated a vector 8: upon disseminate(Vector vec): 9: 𝐻𝑖 ← hash(vec) 10: invoke slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast(vec) 11: upon slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='deliver(Vector vec′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Process 𝑃𝑗): 12: if 𝑃𝑗 ∉ disseminated𝑖: 13: disseminated𝑖 ← disseminated𝑖 ∪ {𝑃𝑗 } 14: vectors𝑖 [hash(vec′)] ← vec′ ⊲ cache vec′ 15: send ⟨stored,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' hash(vec′)⟩𝜎𝑖 to 𝑃𝑗 ⊲ acknowledge the reception by sending a signature to 𝑃𝑗 16: ⊲ acknowledgements are received 17: upon reception of Message 𝑚𝑗 = ⟨stored,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hash_Value 𝐻′⟩𝜎𝑗 such that 𝐻′ = 𝐻𝑖 from 𝑛 − 𝑡 distinct processes: 18: Storage_Proof sp ← Combine�{𝜎 | 𝜎 is a signature of a received stored message}� 19: invoke beb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast�⟨storage_proof, 𝐻𝑖, sp⟩� ⊲ disseminate the storage proof 20: ⊲ a storage proof is received 21: upon reception of Message 𝑚 = ⟨storage_proof, Hash_Value 𝐻′, Storage_Proof sp′⟩: 22: if sp′ is a valid (𝑛 − 𝑡)-threshold signature of ⟨stored, 𝐻′⟩: ⊲ check that the storage proof is valid 23: invoke beb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast�⟨storage_proof, 𝐻′, sp′⟩� ⊲ rebroadcast the storage proof 24: trigger obtain(𝐻′, sp′) 25: stop participating in vector dissemination and slow broadcast sending (and signing) a stored message for 𝐻 ′, all these correct processes have cached a vector vec′ (line 14), where hash(vec′) = 𝐻 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Next, we prove 𝛿-closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 5 satisfies 𝛿-closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑃first be a correct process which obtains a hash value and a storage proof at time 𝑡first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Before the aforementioned attainment, 𝑃first rebroadcasts the hash value and the storage proof (line 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, every correct process receives a hash value and storage proof by time max(GST,𝑡first) + 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ The following lemma proves that, if a correct process 𝑃𝑖 starts the dissemination of its vector at time 𝑡𝑖, then every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We emphasize that the max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿 time is not tight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we choose it due to the simplicity of the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a correct process 𝑃𝑖 starts the dissemination of its vector at time 𝑡𝑖, every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We separate the proof into two cases: There exists a correct process which obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this case, the statement of the lemma holds as every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿 due to the “rebroadcasting step” (line 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 25 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira There does not exist a correct process which obtains a hash value and a storage proof by time 𝑇 = max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, no process stops participating in vector dissemination by time 𝑇, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', no process executes line 25 by time 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Every correct process receives a slow_broadcast message from process 𝑃𝑖 by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 2𝛿, 𝑃𝑖 receives 𝑛 − 𝑡 partial signatures (line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿, every correct process receives a storage_proof message from 𝑃𝑖 (line 21), and obtains a hash value and a storage proof (line 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In this case, the statement of the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As the statement of the lemma holds in both cases, the proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ The next lemma proves that Algorithm 5 satisfies termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 5 satisfies termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Follows directly from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Next, we prove integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 5 satisfies integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Follows from the check at line 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Therefore, Algorithm 5 solves the vector dissemination problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 5 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, we prove that the communication complexity of Algorithm 5 is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Recall that the communication complexity denotes the number of bits sent by correct processes at and after GST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let no correct process start the dissemination of its vector after time GST + 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, the communication complexity of Algorithm 5 is 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑖 be the minimum index such that (1) process 𝑃𝑖 is correct, and (2) 𝑃𝑖 sends a slow_broadcast message at some time ≥ GST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If 𝑖 does not exist, the lemma trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑡𝑖 denote the time at which 𝑃𝑖 starts the dissemination of its vector (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' By assumption, 𝑡𝑖 ≤ GST+𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Every correct process obtains a hash value and a storage proof by time max(GST,𝑡𝑖) + 𝛿 · 𝑛𝑖 + 3𝛿 (by Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, as 𝑡𝑖 ≤ GST + 𝛿, every correct process obtains a hash value and a storage proof by time GST + 𝛿 · 𝑛𝑖 + 4𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, by time GST + 𝛿 · 𝑛𝑖 + 4𝛿, all correct processes stop sending slow_broadcast messages (due to line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑃𝑗 be a correct process such that 𝑗 > 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the slow broadcast primitive (Algorithm 4), 𝑃𝑗 has a “waiting step” of (at least) 𝛿 · 𝑛𝑖 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, during the [GST, GST + 𝛿 · 𝑛𝑖 + 4𝛿] period, 𝑃𝑗 can send only 𝑂(1) slow_broadcast messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, at most one correct process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', 𝑃𝑖) sends more than 𝑂(1) slow_broadcast messages during the [GST, GST + 𝛿 · 𝑛𝑖 + 4𝛿] period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' that process sends at most 𝑛 slow_broadcast messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As each message is of size 𝑂(𝑛) (since it carries a vector of 𝑛 − 𝑡 values), the communication complexity of Algorithm 5 is 𝑂(𝑛) · 𝑂(1) · 𝑂(𝑛) + 1 · 𝑂(𝑛) · 𝑂(𝑛) = 𝑂(𝑛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Vector Consensus with 𝑂(𝑛2 log𝑛) Communication Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we are ready to present our vector consensus algorithm (Algorithm 6) with subcubic communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Our algorithm consists of three building blocks: (1) vector dissemination (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), (2) Quad (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), and (3) add [29], an algorithm for asynchronous data dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In Algorithm 6, we rely on a specific instance of Quad in which (1) each proposal is a hash value, and (2) given a hash value 𝐻 and a (Quad) proof Σ,4 verify(𝐻, Σ) = true if and only if valid_SP(𝐻, Σ) = true (recall the vector dissemination problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Below, we briefly explain add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 4Do not confuse Quad proofs with storage proofs of the vector dissemination problem (Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 26 On the Validity of Consensus add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This algorithm solves the data dissemination [29] problem defined in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let 𝑀 be a data blob which is an input of (at least) 𝑡 +1 correct processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' other correct processes have ⊥ as their input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The data dissemination problem requires every correct process to eventually output 𝑀, and no other message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The key feature of add is that it solves the problem with 𝑂(𝑛2 log𝑛) communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' (For full details on add, refer to [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=') Description of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We give the description of Algorithm 6 from the perspective of a correct process 𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' When 𝑃𝑖 proposes its value (line 10), it disseminates the value (using the best-effort broadcast primitive) to all processes (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 receives proposals of 𝑛 − 𝑡 distinct processes (line 16), it constructs an input configuration (line 17), and starts disseminating it (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='5 When 𝑃𝑖 obtains a hash value 𝐻 and a storage proof sp (line 19), 𝑃𝑖 proposes (𝐻, sp) to Quad (line 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Observe that verify(𝐻, sp) = true (due to the integrity property of vector dissemination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 decides from Quad (line 22), it starts add (line 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Specifically, 𝑃𝑖 checks whether it has cached an input configuration whose hash value is 𝐻 ′ (line 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If so, 𝑃𝑖 inputs the input configuration to add;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' otherwise, 𝑃𝑖 inputs ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Once 𝑃𝑖 outputs an input configuration from add (line 25), it decides it (line 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 6 𝑂(𝑛2 log𝑛) Vector Consensus: Pseudocode (for process 𝑃𝑖) 1: Uses: 2: Best-Effort Broadcast [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance beb ⊲ broadcast with no guarantees if the sender is faulty 3: Vector Dissemination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance disseminator ⊲ see Algorithm 5 4: Quad [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance quad 5: add [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' instance add 6: upon init: 7: Integer received_proposals𝑖 ← 0 ⊲ the number of received proposals 8: Map(Process → V𝐼 ) proposals𝑖 ← empty ⊲ received proposals 9: Map(Process → Message) messages𝑖 ← empty ⊲ received proposal messages 10: upon propose(𝑣 ∈ V𝐼 ): 11: invoke beb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='broadcast�⟨proposal, 𝑣⟩𝜎𝑖 � ⊲ broadcast a signed proposal 12: upon reception of Message 𝑚 = ⟨proposal, 𝑣𝑗 ∈ V𝐼 ⟩𝜎𝑗 from process 𝑃𝑗 and received_proposals𝑖 < 𝑛 − 𝑡: 13: received_proposals𝑖 ← received_proposals𝑖 + 1 14: proposals𝑖 [𝑃𝑗 ] ← 𝑣𝑗 15: messages𝑖 [𝑃𝑗 ] ← 𝑚 16: if received_proposals𝑖 = 𝑛 − 𝑡: ⊲ received 𝑛 − 𝑡 proposals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' can start disseminating 17: Input_Configuration vector ← input configuration constructed from proposals𝑖 18: invoke disseminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='disseminate(vector) 19: upon disseminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='obtain�(Hash_Value H, Storage_Proof sp)�: 20: if have not yet proposed to Quad: 21: invoke quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='propose�(H, sp)� 22: upon quad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='decide�(Hash_Value H′, Storage_Proof sp′)�: 23: Input_Configuration vector′ ← a cached vector whose hash value is 𝐻′ ⊲ can be ⊥ 24: invoke add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='input(vector′) 25: upon add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='output�Input_Configuration vector′′�: 26: trigger decide(vector′′) 5Recall that this input configuration is actually a vector of 𝑛 − 𝑡 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 27 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira Proof of correctness and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by proving that Algorithm 6 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm 6 is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us prove that Algorithm 6 satisfies all properties of vector consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Agreement: Due to Agreement of Quad, no two correct processes decide different pairs from Quad (line 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, for all correct processes which input a non-⊥ input configuration to add (line 24), they input the same input configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, due to the specification of add, all correct processes output the same input configuration from add (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Agreement is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Termination: Every correct process broadcasts its proposal (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, every correct even- tually receives 𝑛 −𝑡 proposals (line 16), and starts the dissemination of an input configuration (line 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the termination property of vector dissemination (Lemma 11), every correct process eventually obtains a hash value and a storage proof (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, every correct process eventually proposes to Quad (line 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to Termination of Quad, every correct process eventually decides from Quad (line 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As the pair decided from Quad includes a storage proof (due to the specification of Quad), at least 𝑡 +1 correct processes have cached an input configuration whose hash value is decided from Quad (by the redundancy property of vector dissemination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, due to the specification of add, every correct process eventually outputs an input configuration from add (line 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Termination is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Vector Validity: Let vec′ be an input configuration of 𝑛 − 𝑡 proposals decided by a correct process (line 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, a storage proof sp′ such that valid_SP(hash(vec′), sp′) = true is obtained (due to the specification of add).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Therefore, vec′ is cached by (at least) 𝑡 + 1 correct processes (due to the redundancy property of vector dissemination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Before a correct process caches a vector (Algorithm 5), it verifies that it is associated with corresponding proposal messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' we omit this check for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As correct processes only send proposal messages for their proposals (line 11), Vector Validity is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ Lastly, we show the communication complexity of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The communication complexity of Algorithm 6 is 𝑂(𝑛2 log𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The communication complexity of a single best-effort broadcast instance is 𝑂(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Every correct process starts the dissemination of its vector by time GST + 𝛿 (as every correct process receives 𝑛 −𝑡 proposals by this time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, the communication complexity of vector dissemination is 𝑂(𝑛2) (by Theorem 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' The communication complexity of Quad is 𝑂(𝑛2) (see [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, the communication complexity of add is 𝑂(𝑛2 log𝑛) (see [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As Algorithm 6 is a composition of the aforementioned building blocks, its communication complexity is 𝑛 · 𝑂(𝑛) + 𝑂(𝑛2) + 𝑂(𝑛2) + 𝑂(𝑛2 log𝑛) = 𝑂(𝑛2 log𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' □ D EXTENDED FORMALISM In this section, we give intuition behind an extension of our formalism which is suitable for the analysis of blockchain-specific validity properties, such as External Validity [18, 20, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' External Validity stipulates that any decided value must satisfy a predetermined logical predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, the “difficulty” of this property is that the logical predicate (usually) verifies a cryptographic proof, which processes might not know “in advance” (see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In a nutshell, we make our original formalism more expressive by (1) making the input (V𝐼) and output (V𝑂) spaces “unknown” to the processes, and (2) taking into account “proposals” of faulty processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the rest of the paper: 28 On the Validity of Consensus We refer to the formalism introduced in the main body of the paper as the “original formalism”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We refer to the formalism we introduce below as the “extended formalism”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We start by giving an intuition behind our extended formalism (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Then, we introduce some preliminaries (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we define our extended formalism (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1 Intuition In the original formalism, processes “know” the entire input space V𝐼 and the entire output space V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, processes are able to “produce” any value which belongs to V𝐼 or V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, this assumption limits the expressiveness of our formalism as it is impossible to describe a Byzantine consensus problem in which input or output spaces are not “known”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us give an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Imagine a committee-based blockchain which establishes two roles: Clients are the users of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' They issue signed transactions to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Servers are the operating nodes of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Servers receive signed transactions issued by the clients, and solve the Byzantine consensus problem to agree on the exact order transactions are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As servers propose transactions signed by the clients and they do not have access to the private keys of the clients, servers do not “know” the input space V𝐼 nor the output space V𝑂 of the Byzantine consensus problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, our original formalism cannot describe the Byzantine consensus problem in the core of the aforementioned blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Extended vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' original formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' As highlighted above, the main difference between the two formalisms is that the extended one allows us to specify the “knowledge level” of the input and output spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the extended formalism, a process is able to “learn” output values by observing input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, we define a discovery function that defines which output values are learned given observed input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the committee-based blockchain example, once a server observes signed (by the issuing clients) transactions tx1 and tx2, it learns the following output values: (1) tx1, (2) tx2, (3) tx1||tx2, and (4) tx2||tx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='6 The second difference between the original and the extended formalism is that the extended formalism takes into account “proposals” of faulty processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Indeed, the original formalism does not enable us to define which values are admissible given the adversary’s knowledge of the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Think of the aforementioned example with a blockchain system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If no process (correct or faulty) obtains a transaction tx, tx cannot be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, if only a faulty process obtains a transaction tx, tx could be an admissible decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' This scenario can be described by the extended formalism, while it cannot by the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='2 Preliminaries We denote by V𝐼 the input space of Byzantine consensus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', processes propose values contained in V𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Similarly, V𝑂 denotes the output space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', processes decide values which belong to V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Membership functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define two membership functions: valid_input : {0, 1}∗ → {true, false}: Intuitively, the valid_input(·) function specifies whether a bit-sequence belongs to the input space V𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' valid_output : {0, 1}∗ → {true, false}: Intuitively, the valid_output(·) function specifies whether a bit-sequence belongs to the output space V𝑂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We assume that each process has access to these two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, each process can verify whether an arbitrary sequence of bits belongs to the input (V𝐼) or output (V𝑂) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the case of 6We denote by “||” the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 29 Pierre Civit, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, and Manuel Vidigueira a committee-based blockchain (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), the membership functions are simply signature- verification functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Discovery function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We define a function discover: 2V𝐼 → 2V𝑂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given a set of proposals 𝑉𝐼 ⊆ V𝐼, discover(𝑉𝐼) ⊆ V𝑂 specifies the set of decisions which are “discoverable” by 𝑉𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We assume that each process has access to the discover(·) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, for any two sets 𝑉 1 𝐼 ,𝑉 2 𝐼 with 𝑉 1 𝐼 ⊆ 𝑉 2 𝐼 , discover(𝑉 1 𝐼 ) ⊆ discover(𝑉 2 𝐼 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' in other words, “knowledge” of the output space can only be improved upon learning more input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Let us take a look at the committee-based blockchain example again (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' If a server obtains a proposal tx, it learns tx as a potential decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We model this “deduction” concept using the discover(·) function: discover�{tx}� = {tx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Adversary pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given an execution E, P(E) ⊆ V𝐼 defines the adversary pool in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Informally, the adversary pool represents the input values the adversary “knows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the example of a committee- based blockchain (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='1), the adversary pool is a set of signed transactions which the adversary “learns” from the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We underline that the adversary pool is an abstract concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Specifically, the adversary pool rep- resents the “starting knowledge” the adversary has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' However, the notion of the “starting knowledge” must be precisely defined once all particularities of the exact considered system are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Due to the sophisticated details as the aforementioned one, we believe that the formalism suitable for blockchain-specific validity properties deserves a standalone paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='3 Validity We start by restating the definition of process-proposal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A process-proposal pair is a pair (𝑃, 𝑣), where (1) 𝑃 ∈ Π is a process, and (2) 𝑣 ∈ V𝐼 is a proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given a process-proposal pair pp = (𝑃, 𝑣), proposal(pp) = 𝑣 denotes the proposal associated with pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' An input configuration is a tuple � pp1, pp2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=', pp𝑥, 𝜌 � of 𝑥 process-proposal pairs and a set 𝜌 ⊆ V𝐼, where (1) 𝑛 −𝑡 ≤ 𝑥 ≤ 𝑛, (2) every process-proposal pair is associated with a distinct process, and (3) if 𝑥 = 𝑛, 𝜌 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Intuitively, an input configuration represents an assignment of proposals to correct processes, as well as a “part” of the input space known to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, an input configuration � (𝑃1, 𝑣), (𝑃2, 𝑣), (𝑃3, 𝑣), {𝑣, 𝑣 ′, 𝑣 ′′} � describes an execution in which (1) only processes 𝑃1, 𝑃2, and 𝑃3 are correct, (2) processes 𝑃1, 𝑃2, and 𝑃3 propose the same value 𝑣, and (3) faulty processes know only 𝑣, 𝑣 ′, and 𝑣 ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We denote by I the set of all input configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every input configuration 𝑐 ∈ I, we denote by 𝑐[𝑖] the process-proposal pair associated with process 𝑃𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' if such a process-proposal pair does not exist, 𝑐[𝑖] = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Moreover, we define by pool(𝑐) the set of input values associated with 𝑐 (the “𝜌” field of 𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, 𝜋(𝑐) = {𝑃𝑖 ∈ Π | 𝑐[𝑖] ≠ ⊥} denotes the set of all processes included in 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, correct_proposals(𝑐) = {𝑣 ∈ V𝐼 | ∃𝑖 ∈ [1,𝑛] : 𝑐[𝑖] ≠ ⊥ ∧ proposal(𝑐[𝑖]) = 𝑣} denotes the set of all proposals of correct processes (as specified by 𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Given (1) an execution E ∈ execs(A), where A is an algorithm which exposes the propose(·)/decide(·) interface, and (2) an input configuration 𝑐 ∈ I, we say that E corresponds to 𝑐 if and only if (1) 𝜋(𝑐) = Corr A(E), (2) for every process 𝑃𝑖 ∈ Corr A(E), 𝑃𝑖’s proposal in E is proposal(𝑐[𝑖]), and (3) P(E) = pool(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' We denote by corresponding(E) = 𝑐 the input configuration to which E corresponds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' A validity property val is a function val : I → 2V𝑂 such that, for every input configuration 𝑐 ∈ I, val(c) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Algorithm A, where A exposes the propose(·)/decide(·) interface, satisfies a validity property val if and only if, in every execution E ∈ execs(A), no correct process decides a value 𝑣 ′ ∉ val�corresponding(E)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' That is, an algorithm satisfies a validity property if and only if correct processes decide only admissible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 30 On the Validity of Consensus Assumptions on executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Lastly, we introduce two assumptions that conclude our proposal for the extended formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every execution E of any algorithm A which solves the Byzantine consensus problem with some validity property, if a correct process 𝑃 decides a value 𝑣 ′ ∈ V𝑂 in E, then 𝑣 ′ ∈ discover�correct_proposals(𝑐) ∪ pool(𝑐)�, where corresponding(E) = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Assumption 1 states that correct processes can only decide values which are “discoverable” using all the proposals of correct processes and the knowledge of the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, if every correct process proposes the same value 𝑣 ∈ V𝐼 and the adversary pool contains only 𝑣 ′ ∈ V𝐼, then a correct process can only decide a value from discover({𝑣, 𝑣 ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Next, we introduce an assumption concerned only with the canonical executions (executions in which faulty processes do not take any computational step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For every canonical execution E of any algorithm A which solves the Byzantine consensus problem with some validity property, if a correct process 𝑃 decides a value 𝑣 ′ ∈ V𝑂 in E, then 𝑣 ′ ∈ discover�correct_proposals(𝑐)�, where corresponding(E) = 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Intuitively, Assumption 2 states that, if faulty processes are silent, correct processes can only decide values which can be discovered using their own proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In other words, correct processes cannot use “hidden” proposals (possessed by the “silent” adversary) to discover a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Finally, we underline that these two assumptions do not completely prevent “unreasonable” executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' For example, given these two assumptions, a (correct or faulty) process is still able to send a message with a value which cannot be discovered using the proposals of correct processes and the adversary pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Hence, an assumption that prevents such an execution should be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' Thus, due to the complexity we envision for the extended formalism, we leave it out of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' In the future, we will focus on this interesting and important problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE4T4oBgHgl3EQfJQzO/content/2301.04920v1.pdf'} diff --git a/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/2301.04696v1.pdf.txt b/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/2301.04696v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c63f9d41d45f0ac7b2744ceac63c550e9e66e1d9 --- /dev/null +++ b/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/2301.04696v1.pdf.txt @@ -0,0 +1,1068 @@ +ARXIV PREPRINT - FEBRUARY 2023 +1 +On Modeling Network Slicing Communication +Resources with SARSA Optimization +Eduardo S. Xavier, Nazim Agoulmine and Joberto S. B. Martins +Abstract—Network slicing is a crucial enabler to support the +composition and deployment of virtual network infrastructures +required by the dynamic behavior of networks like 5G/6G mobile +networks, IoT-aware networks, e-health systems, and industry +verticals like the internet of vehicles (IoV) and industry 4.0. +The communication slices and their allocated communication +resources are essential in slicing architectures for resource +orchestration and allocation, virtual network function (VNF) +deployment, and slice operation functionalities. The communi- +cation slices provide the communications capabilities required +to support slice operation, SLA guarantees, and QoS/ QoE +application requirements. Therefore, this contribution proposes a +networking slicing conceptual model to formulate the optimiza- +tion problem related to the sharing of communication resources +among communication slices. First, we present a conceptual +model of network slicing, we then formulate analytically some +aspects of the model and the optimization problem to address. +Next, we proposed to use a SARSA agent to solve the problem +and implement a proof of concept prototype. Finally, we present +the obtained results and discuss them. +Index Terms—Network Slicing, Communication Slice, Re- +source Allocation, Conceptual and Analytical Model, Machine +Learning, SARSA. +I. INTRODUCTION +Network slicing is a crucial enabler to support the composi- +tion and deployment of virtual network infrastructures required +by the dynamic behavior of networks like 5G/6G mobile +networks, IoT-aware networks, e-health systems, and industry +verticals like the internet of vehicles (IoV) and industry 4.0 [1] +[2] [3]. In general, the slicing process results from the need +to share resources among existing infrastructures to improve +performance, provide cost-efficient solutions, and optimize +operation [4]. +This technology is already used in the context of 5G +networks [1] [5] and provided as a service (slice-as-a-Service: +SlaaS) by network operators. This allows customs to create +their private virtual networks (slices) tailored to their specific +application domains and to develop their own business mod- +els. Network slicing is expanding its use in other scenarios +of telecommunication networks, content provider networks +(ISPs), experimental networks, and IoT systems, among others +[6]. +Xavier, Eduardo F. is with Salvador University (UNIFACS), Brazil - +eduardo.sidney@animaeducacao.com.br +Agoulmine, Nazim is with University of Paris Saclay, University of Evry +- IBISC Lab, France - nazim.agoulmine@univ-evry.fr +Martins, Joberto S. B. is with Salvador University (UNIFACS), Brazil - +joberto.martins@gmail.com +This work IS supported by ANIMA Institute and FAPESP - MCTIC - +Project 2018/23097-3. +Network slice instance life cycle process such as commis- +sioning, operating, and decommissioning [1] requires appro- +priate network communication resources. A communication +slice 1 eventually represents a set of communication resources +that can be used in the slicing process. It holds resources like +links, optical slots, virtual private networks (VPNs), and other +communication facilities necessary to provide the exchange +of information among logical slices, and architectural slicing +entities and for supporting the slicing process functionalities. +The communication slice resources significantly impact the +performance of the resulting sliced virtual network (SVN) +or virtual network operator (VNO). Among the most com- +mon network characteristics that impact the network slicing +process, we can mention delay-aware network slicing like in +5G deployments [7], quality of service (QoS) aware network +slicing [3], energy-aware network slicing [8], and, in general, +application-dependent and multi-domain network slicing [9]. +The objective of this paper is therefore to propose a concep- +tual model of slice communication and formulate analytically +some of its aspects. The model should be able to capture the +set of communication resources to support the optimization +of the allocation of communication resources to the different +slices on top of various underlying technologies (e.g. Elastic +Optical Networks - EON [10], MultiProtocol Label Switching +- MPLS [11], others) +This paper is organized as follows. Section II presents the +related work and Section III introduces the concept of multido- +main sliced virtual networks. Section IV presents a conceptual +and analytical model for a communication slice used in the +network slicing process. Section V presents a proof of concept +of using the models with a SARSA agent optimizing the +allocation of bandwidth resources for a communication slice. +Finally, Section VI presents the final considerations. +II. RELATED WORK +There have been a very significant number of state-of- +art research projects launched in the area during the last +decade such as SFI2 (Slicing Future Internet Infrastructures) +[12] [13], NECOS (Novel Enablers for Cloud Slicing) [14], +SELFNET [15] and MATILDA [16], standardization initiatives +launched by the IETF (Internet Engineering Task Force) [17], +3GPP (3rd Generation Partnership Project) [18], ITU (ITU-T +- Telecommunication Standardization) [19], ETSI (European +Telecommunications Standards Institute) [20] and ONF (Open +Networking Foundation) [21] and published surveys [2] [22] +1A specialized slice that provides communication services among network +slicing entities +arXiv:2301.04696v1 [cs.NI] 11 Jan 2023 + +ARXIV PREPRINT - FEBRUARY 2023 +2 +[23] [24] [25]. These different initiatives have focused on +different technical aspects, architectures, and slicing strategies, +and all require communication slices to operate and manage +the provided functionalities. +However, these slicing architectures, projects, and initiatives +did only address the conceptual and analytical modeling of the +basic structures and functionalities that compose the slicing +process in a preliminary way or did only indicate them as +future challenges to solve. To the best of our knowledge, the +conceptual and analytical modeling of communication slices +is a new contribution to the network slicing domain. +III. RESOURCES, SLICE AND SLICED VIRTUAL NETWORK +(SVN) +A multi-domain Sliced Virtual Network (SVN) as viewed +in Figure 1 is a multi-domain or a multi-tenant2 infrastructure +that is dynamically configured and deployed by requesting and +orchestrating resources from a pool of providers on domains. +Figure 1. +A Multi-Domain Sliced Virtual Network (SVN) and its Resources +A. The Slice +For the scope of this paper aiming at the slicing model and +deployment understanding, it is essential to conceptualize the +vision of a slice as a component of the sliced virtual network. +We define a slice as a specific resource, service, function, +or set of resources, services, and functions virtualized, shared, +and grouped using any software or hardware facility. The slice +2For the scope of this paper, a tenant can be a network domain, a service +provider, a business unit, or a specific multi-tier or single-application tier +providing resources for network slicing. +with its resources, services, and functions physically resides +in nodes or other physical deployments in domains. +As such, slice resource examples are virtual machines, +virtual switches with hosts deployed with OpenFlow, chunks +of bandwidth belonging to a physical link, slots of a fiber +EON deployment, LSP MPLS connections, shared spectrum +in 5G radio access networks (RAN), and others. Slice function +and service examples are virtual network functions (VNFs) de- +ployed over a network providing specific services or facilities +to the user. +Considering this slice basic concept, an SVN encompasses +resources, services, and functions with the necessary commu- +nication resources to interconnect them inside domains and +between domains as illustrated in Figure 2. +Figure 2. +A Generic SVN with Slices View +In general, resources belonging to the same SVN reside in +different domains and are physically or virtually attached to +nodes in their respective domains (Figure 2). +The network slicing architecture functionalities (resource +marketplace, resource broker, resource orchestrator, slice in- +stantiation, slice monitoring, and others) are distributed in +terms of the domains participating in the SVN deployment +and certainly, depend on the proposed architecture and the +deployed functional blocks of the network slicing architecture +(SELFNET, NECOS, SFI2, MATILDA, other). +B. Communication Resources and Communication Slice +In order to allow the execution of the network slicing pro- +cess and functionalities in any deployed slicing architecture, +it is necessary to allocate communication resources allowing +communication among the entities involved in the slicing pro- +cess. Furthermore, once the SVN is deployed, communication +resources are also necessary to support the communication +requirements of the applications running (slice operation). + +SVN andSlicedResourcesand +Rj1 +Rin +CommunicationsperDomain +Ri1 +Rin +Di +Domainx +Domainj +D +Domain; +Domainw +Domainz +RK +Rkn +Rx-ResourcefromDomainx +Cx-Communicationresource +fromDomainx +Domaink +SlicedVirtualNetwork(SVN) +D: +D +Dk +Z R,Di +ZR,Dj +ZR.Dk +Resources +Resources +Resources +Multi-Domain Network SlicingDOMAIN D: +SVNi slices at Nodez +DOMAIN D, +SVN2Slices at Nodes +inDOMAIND +in DOMAIN Dz +VNF +VM; +VNFi +VM; +SLICE; at Nodez +SLICE; at Nodes +VNF. +VM. +VNF.m +VMm +SLICEm at Nodez +SLICEm at NodeB +SVNi Slice at Nodew +SVN2Slice at Nodec +inDOMAIND +inDOMAINDz +VNFm +VMm +VNFm +VMm +SW, +SW. +SLICEm +natNodew +SLICEmatNodec +SVN2 Slice at Nodey +SVNi Slice at Nodep +inDOMAIN D +in DOMAIN Dz +slice +VNF. +VM. +VNFm +VMm +Communications +SLICEm at Nodey +SLICEmat NodepARXIV PREPRINT - FEBRUARY 2023 +3 +Figure 3. +Intradomain and Interdomain Communication Slices +The generic view of communication resources used by a +network slicing infrastructure to enable resource orchestration, +deployment, and slice operation is illustrated in Figure 3. +We assume that the slicing process to create a sliced +virtual network (SVN) involves single or multiple domains +(Dx, ..., Dz). Each domain is generically configured by a +single or a set of nodes (ni, ...nj) hosting resources and +domains that are interconnected by communication resources. +A communication slice is then defined as a set of commu- +nication resources orchestrated and allocated between slices, +nodes, network-slicing entities, and domains. As such, the +domain nodes (ni, ...nj) hosting resources and domains are +interconnected by communication slices (Cx, ...Cy). +We identify two types of communication slices that are +orchestrated and deployed with distinct configurations and +characteristics: +• Intradomain communication slices; and +• Interdomain communication slices. +In infrastructures composed of network domains, the mod- +eling assumes that a gateway concentrates all communications +between different domains. +We focus in this paper specifically on interdomain com- +munications and how to model it in terms of communication +slices. +IV. NETWORK SLICING INTERDOMAIN COMMUNICATIONS +The objective of a network slicing interdomain communi- +cation model is to formally structure and capture the needs in +terms of communications for the slicing process. It also allows +the identification of parameters leading to the optimization of +the resource allocation process. +A. Network Slicing Assumptions +We first introduce the following assumptions in the con- +text of network-slicing interdomain communications that are +necessary for our modeling and problem formulation: +• Each network domain is SDN-compatible; +• Each network domain gateway GW_Di (Figure 3) is an +SDN-enabled switch whose programmed behavior is to +route packets between domains; +Notation +Description +Dli +i +The domain i located in physical location li +RDli +i +Domain’s set of shareable resources at a physical location +R +Dli +i +i +A shareable resource at domain Dli +i +R_ISli +Di +The infrastructure and service resources +R_Cli +Di +The network communication resources +BDi,Dj +Bandwidth between domains +LDi,Dj +Packet loss between domains +DlDi,Dj +Delay between domains +Bni,nj +Bandwidth between nodes +Lni,nj +Packet loss between nodes +Dlni,nj +Delay between nodes +P_RCli +Dk,Dj +Set of communication’s link parameters between domains +Table I +NOTATION AND VARIABLES +• Each network domain implements monitoring mecha- +nisms to collect performance monitoring parameters; +• All intradomain and interdomain links are configurable +in terms of allocated resources; and +• All network domains support network resource identifi- +cation and has capabilities for resource allocation. +B. Network Slicing Model +Based on these assumptions, we can now specify an analyt- +ical model of multi-domain SVN considering a set of network +domains federating together their resources and infrastructures +to the slicing process: +ℵ =< Dli +i , Dlj +j , Dlk +k , ..., Dlz +z > +(1) +Where: +• Dli +i is a network infrastructure domain located at site li. +Each network infrastructure domain Dli +i has a set of share- +able resources such as: +RDli +i =< R +D +li +i +i +, R +D +li +i +j +, R +D +li +i +k +, ..., R +D +li +i +z +> +(2) +Where: +• RDli +i is the set of shareable resources provided by Di +and located at site li; +• R +D +li +i +i +is one particular shareable resource. +There are different types of resources at each network +infrastructure domain location Dli +i : +• Infrastructure appliance like virtual machines, access +points, and IoT devices; +• Computing services like virtual network functions (VNF), +storage and computing services; and +• Communications +services +like +physical +links, +LSPs +(MPLS Link Switched Paths), fiber lambdas, and 5G +connections. +For the purpose of the SVN model, we distinguish between +two types of resources: +• Infrastructure and service resources - R_ISli +Di; and +• Communications resources - R_Cli +Di. + +R_CDx Dy +Domain D +Domain Dx +SW_0 +GW Dx +DomainD +0 +R_ Cox [no,ni] +R_CDx [n2,n3] +-COMMUNICATION +INTERDOMAIN +R_Cox [ni,n2] +LINK 0-1 +R_ C Dx, Dk +sw_1 + sW_2 +Domain D +R_C Dx DI +INTRADOMAINCOMMUNICATIONS +Domain D;ARXIV PREPRINT - FEBRUARY 2023 +4 +Users (clients) request infrastructure, service, and commu- +nication resources that are orchestrated by a network slicing +software (NECOS, MATILDA, other) to create their sliced +virtual network (SVN) as illustrated in Figure 1. +The communication resources R_Cli +Di provide the intercon- +nection of infrastructure and service resources R_ISli +Di for +intradomain and inter-domain connections. As such, for the +SVN modeling there are two distinct communication resources +or communication slices (Figure 3): +• Intradomain communication slices used between internal +nodes of the domain: R_Cli +Di[nj,nk]; and +• Interndomain communication slices used between do- +mains: R_Cli +Di,Dk +The communication slices are characterized by as set of pa- +rameters related to interdomain (Equation 5) and intradomain +(Equation 4) communications: +P_RCDi,Dj =< BDi,Dj, LDi,Dj, DlDi,Dj > +(3) +P_RCni,nj =< Bni,nj, Lni,nj, Dlni,nj > +(4) +Where: +• BDi,Dj is the available bandwidth between domains Di +and Dj; +• LDi,Dj is the packet loss between domains Di and Dj; +• DlDi,Dj is the delay between domains Di and Dj. +• Bni,nj is the available bandwidth between nodes ni and +nj in a domain; +• Lni,nj is the packet loss between nodes ni and nj at a +domain; and +• Dlni,nj is the packet delay between nodes ni and nj at +a domain. +Figures 1 and 3 illustrate a generic view of the slicing +process and related interdomain communications. The network +slicing infrastructure setup from the point of view of commu- +nication resources is as follows: +• A set of domains (Di); +• A single communication slice (configurable link or an- +other communication resource) between domains; and +• A SDN switch (gateway) programmed to handling the +interdomain packet routing among domains. +The +interdomain +slice +communication +parameters +P_RCli +Dk,Dj are configured during the slicing commissioning +phase, as proposed in the 3GPP network slicing reference +architecture and model [26]. +An SVN will require resources of distinct domains to be +allocated end-to-end: +SLDi +k +=< R +D +li +i +i +, R +D +li +i +j +, RD +li +y +k +, ..., RD +li +y +z +> +(5) +The communication slice modeling assumes that each do- +main contributes to a set of different resources that are located +in various physical sites (domains). +The model is agnostic to the issue of traffic distinction +between packets generated with the slices already instantiated +Figure 4. +Openflow Switch Handling Operation and Management Slicing +Generated Packets +(slice operation) and packets generated by the network slicing +management software installed (orchestrator, resource market- +place, monitoring, others). +The slicing-related interdomain traffic between domains is +handled by an SDN switch as illustrated in Figure 4. +In summary, the interdomain traffic at the gateway is com- +posed of the packets generated (operation and management) by +all resources belonging to the domain Di having as destination +the domain Dj. +The slicing communication model assumes that domains +have only one network connection together. In other words, the +domains do not act as intermediate domains switching packets +in the path to a destination domain. +For the interdomain packets at the gateway, the following +definitions hold (Figure 4): +• All packets belonging to a set of resources R +D +li +i +i +at +domain Di with the same performance parameters con- +straint use a specific queue Qn; +• N switch queues handle the packet generated by the +shareable resources at domain Di; +• The switch queues have SDN resources control capa- +bilities controlled by SDN Controllers [27] for resource +control; +• A priority is assigned to each output queue; and +• Each queue has a threshold level control parameter PQn. +The priority and threshold level assigned to the queues are +used to support for optimization (e.g. optimization controller +as shown in the following section). +In summary, the model assumes that packets generated from +any sliced resource with similar performance constraints are +grouped in the same controlled queue in the gateway. +The following hypotheses are considered for the control of +the intradomain packets and the gateway queues as highlighted +in Figures 3 and 5): +• Intradomain communications will be based on existing +underlying communication technologies (MPLS LSPs +connections, EON fiber slots, other); + +GATEWAY - SW, DOMAIN; +SWZINTERDOMAINPACKETS +OPENFLOWOUTPUTQUEUES +DTODOMAIND +Q1 +Q2 +Qn +Pqn +Pqi +Pqz +PORT1 +PORTk +PORT, +TOUT +RiD,tk +RkD,tk +SW,INTERDOMAINPACKETS +To and from Domain Di +RmD,k +R,D,tkARXIV PREPRINT - FEBRUARY 2023 +5 +Figure 5. +Interdomain Communication Slice and Gateway at Domain i +• A gateway handles all the inbound and outbound inter- +domain traffics, +• In a domain, each node hosting sharing resources for the +slicing creates a path to the gateway, +• Each path associated with a resource provided by a node +is associated with a particular queue in the gateway. +The intradomain slice communication analytical model is +not the focus of this paper, and these premises make clear +its interrelation with the interdomain modeling and allows the +independent modeling of it. +The optimization problem to solve here is the sharing of the +communication resources between the different slices taking +into account the QoS requirement of each slice. This means +scheduling the packet originating from the different slices +towards the different available queues in the gateway. This +a complex engineering problem that is difficult to solve in an +analytical way considering all the parameters that need to be +taken into account. For that, we propose to investigate the use +of a Reinforcement Learning SARSA agent which is explained +in the following section. +V. SARSA AGENT TO OPTIMIZE RESOURCES SHARING +The interdomain communication slice model is now applied +to the network slicing deployment setup illustrated in Figure +5 in which we have: +• A multidomain slicing infrastructure with n domains; +• A single communication slice between domains; and +• A SDN-capable switch (gateway) handling bidirectional +interdomain packets between the domains. +In terms of the proof of concept, each interdomain commu- +nication slice has a reinforcement learning SARSA agent aim- +ing to optimize the allocation of communication resources. The +RL-SARSA agent acts during slice operation to dynamically +keep performance parameters accordingly to management- +defined objectives. +The +interdomain +slice +communication +parameters +(P_RCli +Dk,Dj) +are +configured +during +the +slicing +commissioning +phase +and +are +dynamically +adjusted +by +the SARSA agent during the slice operation phase. +A. SARSA Agent Model and Configuration +The objective of the SARSA agent is to control the queue +flushing transmission rates to preserve the performance param- +eters defined by the manager while sharing unused resources. +The slice communication queues (Qi) are configured as +follows: +• Three queues corresponding to three performance param- +eters controlled by the agent; +• Each configured queue threshold (Thi) corresponds to +the performance parameter assigned to the queue and +served to packets generated by sliced resources with this +requirement, and +• Each queue Qi has two states: below threshold (BT) and +above threshold (AT). +The actions defined for the queues in the AT state are to +increase the transmission rate, reduce the transmission rate, +and do nothing. Each executed state/action has a defined +reward. +The SARSA agent and communication slice parameters and +initial conditions for running are as follows: +• Agent configuration parameters: +– Epsilon-greedy policy ϵ = 8%; +– Learning rate α = 20%; and +– Discount factor γ = 80% +• Threshold limit (triggers agent action) = 50% +• Agent actions: bandwidth increased or reduced by 10% +• Maximum number of attempts = 500 +• Queue priorities are: p1, p2 and p3 with p1 > p2 > p3. +SARSA Q-values are therefore updated based on the Equa- +tions 6: +Q(xt, at) ← Q(xt, at) + α[rt+1 + γQ(xt+1, at+1) +−Q(xt, at)] +(6) + +GATEWAY-SW, +DOMAIN:TO DOMAIN +Q1 +"o +SARSA +Thn +Th +Pqn +DOMAIN +Agent +Pai +Th +Pq2 +InterDomain +Communication +IntraDon +Setof +Communication Slices +DOMAINK +- +R;Di +Lk +INTERDOMAIN +INTRADOMAIN +TRAFFIC +TRAFFIC +- +IntraDomainPath +1 +- +SetofCommunication Slicesm1 +- +R.D. +Lm +- +DOMAINM +DOMAINARXIV PREPRINT - FEBRUARY 2023 +6 +B. Implementation and tests +The simulation environment was configured on a Linux +(Ubuntu 22.04.1 LTS) Intel(R) Core(TM) i5-3470 CPU @ +3.20GHz desktop. The Visual Studio Code v.1.73.0 and Python +v3.10.6 are used to execute the tests and the statistical analysis. +Each test run scenario has a minimum process cycle of 104 +packet production for each queue with a Poisson distribution. +The SARSA agent is called each time any queue reaches +its configured threshold. The SARSA agent processes up +to 500 episodes in search of a new configuration of the +flushing bandwidth distribution among queues to keep buffer +occupation in the configured threshold limit. +C. The Slice Communication Evaluation Results +A series of tests have been undertaken. It aims to overload +the queues to evaluate the behavior of the agent. The three +defined scenarios are the following: +• Scenario 1 - One of the queues is overloaded; +• Scenario 2 - Two queues are overloaded; and +• Scenario 3 - All queues are overloaded. +The dynamics of the overloaded queues are configured as +follows: +• First set traffic 30% above the queue defined limit for 10 +minutes; +• Increase to 50% above its defined limit for additional 10 +minutes; +• Increase to 80% above its defined limit for additional 10 +minutes; and +• Increase to 100% above its defined limit for additional +10 minutes. +Figures 6 and 7 illustrate the SARSA agent’s behavior for +scenario one. Figure 6 plots the state of the queues while they +are being saturated with overload traffic of packets. The queue +transmission rate (flushing rate) configured by the SARSA +agent is illustrated in Figure 7. We observe that the total +available bandwidth for the link is distributed and reconfigured +among the queues according to the dynamic need to flush +packets from a specific queue and keep queue occupation +below the defined threshold. +For scenario two, the behavior of the SARSA agent is +illustrated in Figure 8. In this scenario, two queues may +overload, and, as observed in scenario one, the SARSA agent +reconfigures the queue’s transmission rate to keep buffer oc- +cupation below the defined threshold. The agent can deal with +simultaneous overload for the simulation-defined parameters +by keeping queue occupation as required. +Finally, the behavior of the SARSA agent for scenario 3 +is illustrated in Figure 9 and is equivalent to its behavior on +scenario two. +VI. FINAL CONSIDERATIONS +This paper presents a conceptual model of network slicing +and present an analytical model to allocate communication +resources between slide process. The conceptual model is +along with a SARSA agent that optimize the allocation of com- +munication resources among slices. The SARSA agent uses +the conceptual model to formulate the required communication +ressources of each slice. A proof of concept implementation +of the SARSA agent aims to demonstrate that the SARSA +agent contributes to dynamically adjusting and controlling +the slice communication parameters between domains. The +proposed conceptual model demonstrates the feasibility and +ease of handling different types of communication resources +for optimizing the communication slice. Future work includes +the leverage of the conceptual model with the integration of +intradomain and interdomain models and the new formulation +of the distributed optimization problem to solve by a federation +of SARSA agents. +REFERENCES +[1] +Shalitha Wijethilaka and Madhusanka Liyanage. “Sur- +vey on Network Slicing for Internet of Things Real- +ization in 5G Networks”. 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In: IEEE Latin America Transac- +tions 18.5 (Apr. 2020), pp. 853–860. + +Scenario 2: Two queues are overloaded +Queue Fill Volume +38828893290 +Q1 +Queue Ql +Threshold +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +38828893290 +Q2 +Queue Q2 +Threshold +38828893290 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Queue Q3 +Threshold +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Measurements during execution \ No newline at end of file diff --git a/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/load_file.txt b/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee05ee875d2191e04637231251f683e5c981ee33 --- /dev/null +++ b/LNE3T4oBgHgl3EQfvwtG/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf,len=544 +page_content='ARXIV PREPRINT - FEBRUARY 2023 1 On Modeling Network Slicing Communication Resources with SARSA Optimization Eduardo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Xavier, Nazim Agoulmine and Joberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Martins Abstract—Network slicing is a crucial enabler to support the composition and deployment of virtual network infrastructures required by the dynamic behavior of networks like 5G/6G mobile networks, IoT-aware networks, e-health systems, and industry verticals like the internet of vehicles (IoV) and industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The communication slices and their allocated communication resources are essential in slicing architectures for resource orchestration and allocation, virtual network function (VNF) deployment, and slice operation functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The communi- cation slices provide the communications capabilities required to support slice operation, SLA guarantees, and QoS/ QoE application requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Therefore, this contribution proposes a networking slicing conceptual model to formulate the optimiza- tion problem related to the sharing of communication resources among communication slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' First, we present a conceptual model of network slicing, we then formulate analytically some aspects of the model and the optimization problem to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Next, we proposed to use a SARSA agent to solve the problem and implement a proof of concept prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Finally, we present the obtained results and discuss them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Index Terms—Network Slicing, Communication Slice, Re- source Allocation, Conceptual and Analytical Model, Machine Learning, SARSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' INTRODUCTION Network slicing is a crucial enabler to support the composi- tion and deployment of virtual network infrastructures required by the dynamic behavior of networks like 5G/6G mobile networks, IoT-aware networks, e-health systems, and industry verticals like the internet of vehicles (IoV) and industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='0 [1] [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In general, the slicing process results from the need to share resources among existing infrastructures to improve performance, provide cost-efficient solutions, and optimize operation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' This technology is already used in the context of 5G networks [1] [5] and provided as a service (slice-as-a-Service: SlaaS) by network operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' This allows customs to create their private virtual networks (slices) tailored to their specific application domains and to develop their own business mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Network slicing is expanding its use in other scenarios of telecommunication networks, content provider networks (ISPs), experimental networks, and IoT systems, among others [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Xavier, Eduardo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' is with Salvador University (UNIFACS), Brazil - eduardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='sidney@animaeducacao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='br Agoulmine, Nazim is with University of Paris Saclay, University of Evry IBISC Lab, France - nazim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='agoulmine@univ-evry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='fr Martins, Joberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' is with Salvador University (UNIFACS), Brazil - joberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='martins@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='com This work IS supported by ANIMA Institute and FAPESP - MCTIC - Project 2018/23097-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Network slice instance life cycle process such as commis- sioning, operating, and decommissioning [1] requires appro- priate network communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A communication slice 1 eventually represents a set of communication resources that can be used in the slicing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' It holds resources like links, optical slots, virtual private networks (VPNs), and other communication facilities necessary to provide the exchange of information among logical slices, and architectural slicing entities and for supporting the slicing process functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The communication slice resources significantly impact the performance of the resulting sliced virtual network (SVN) or virtual network operator (VNO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Among the most com- mon network characteristics that impact the network slicing process, we can mention delay-aware network slicing like in 5G deployments [7], quality of service (QoS) aware network slicing [3], energy-aware network slicing [8], and, in general, application-dependent and multi-domain network slicing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The objective of this paper is therefore to propose a concep- tual model of slice communication and formulate analytically some of its aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The model should be able to capture the set of communication resources to support the optimization of the allocation of communication resources to the different slices on top of various underlying technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Elastic Optical Networks - EON [10], MultiProtocol Label Switching MPLS [11], others) This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Section II presents the related work and Section III introduces the concept of multido- main sliced virtual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Section IV presents a conceptual and analytical model for a communication slice used in the network slicing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Section V presents a proof of concept of using the models with a SARSA agent optimizing the allocation of bandwidth resources for a communication slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Finally, Section VI presents the final considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' RELATED WORK There have been a very significant number of state-of- art research projects launched in the area during the last decade such as SFI2 (Slicing Future Internet Infrastructures) [12] [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' NECOS (Novel Enablers for Cloud Slicing) [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SELFNET [15] and MATILDA [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' standardization initiatives launched by the IETF (Internet Engineering Task Force) [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 3GPP (3rd Generation Partnership Project) [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ITU (ITU-T Telecommunication Standardization) [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ETSI (European Telecommunications Standards Institute) [20] and ONF (Open Networking Foundation) [21] and published surveys [2] [22] 1A specialized slice that provides communication services among network slicing entities arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='04696v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='NI] 11 Jan 2023 ARXIV PREPRINT - FEBRUARY 2023 2 [23] [24] [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' These different initiatives have focused on different technical aspects, architectures, and slicing strategies, and all require communication slices to operate and manage the provided functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' However, these slicing architectures, projects, and initiatives did only address the conceptual and analytical modeling of the basic structures and functionalities that compose the slicing process in a preliminary way or did only indicate them as future challenges to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' To the best of our knowledge, the conceptual and analytical modeling of communication slices is a new contribution to the network slicing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' RESOURCES, SLICE AND SLICED VIRTUAL NETWORK (SVN) A multi-domain Sliced Virtual Network (SVN) as viewed in Figure 1 is a multi-domain or a multi-tenant2 infrastructure that is dynamically configured and deployed by requesting and orchestrating resources from a pool of providers on domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A Multi-Domain Sliced Virtual Network (SVN) and its Resources A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The Slice For the scope of this paper aiming at the slicing model and deployment understanding, it is essential to conceptualize the vision of a slice as a component of the sliced virtual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' We define a slice as a specific resource, service, function, or set of resources, services, and functions virtualized, shared, and grouped using any software or hardware facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The slice 2For the scope of this paper, a tenant can be a network domain, a service provider, a business unit, or a specific multi-tier or single-application tier providing resources for network slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' with its resources, services, and functions physically resides in nodes or other physical deployments in domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' As such, slice resource examples are virtual machines, virtual switches with hosts deployed with OpenFlow, chunks of bandwidth belonging to a physical link, slots of a fiber EON deployment, LSP MPLS connections, shared spectrum in 5G radio access networks (RAN), and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Slice function and service examples are virtual network functions (VNFs) de- ployed over a network providing specific services or facilities to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Considering this slice basic concept, an SVN encompasses resources, services, and functions with the necessary commu- nication resources to interconnect them inside domains and between domains as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A Generic SVN with Slices View In general, resources belonging to the same SVN reside in different domains and are physically or virtually attached to nodes in their respective domains (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The network slicing architecture functionalities (resource marketplace, resource broker, resource orchestrator, slice in- stantiation, slice monitoring, and others) are distributed in terms of the domains participating in the SVN deployment and certainly, depend on the proposed architecture and the deployed functional blocks of the network slicing architecture (SELFNET, NECOS, SFI2, MATILDA, other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Communication Resources and Communication Slice In order to allow the execution of the network slicing pro- cess and functionalities in any deployed slicing architecture, it is necessary to allocate communication resources allowing communication among the entities involved in the slicing pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Furthermore, once the SVN is deployed, communication resources are also necessary to support the communication requirements of the applications running (slice operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SVN andSlicedResourcesand Rj1 Rin CommunicationsperDomain Ri1 Rin Di Domainx Domainj D Domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Domainw Domainz RK Rkn Rx-ResourcefromDomainx Cx-Communicationresource fromDomainx Domaink SlicedVirtualNetwork(SVN) D: D Dk Z R,Di ZR,Dj ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Dk Resources Resources Resources Multi-Domain Network SlicingDOMAIN D: SVNi slices at Nodez DOMAIN D, SVN2Slices at Nodes inDOMAIND in DOMAIN Dz VNF VM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VNFi VM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SLICE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' at Nodez SLICE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' at Nodes VNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='m VMm SLICEm at Nodez SLICEm at NodeB SVNi Slice at Nodew SVN2Slice at Nodec inDOMAIND inDOMAINDz VNFm VMm VNFm VMm SW, SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SLICEm natNodew SLICEmatNodec SVN2 Slice at Nodey SVNi Slice at Nodep inDOMAIN D in DOMAIN Dz slice VNF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VNFm VMm Communications SLICEm at Nodey SLICEmat NodepARXIV PREPRINT - FEBRUARY 2023 3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Intradomain and Interdomain Communication Slices The generic view of communication resources used by a network slicing infrastructure to enable resource orchestration, deployment, and slice operation is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' We assume that the slicing process to create a sliced virtual network (SVN) involves single or multiple domains (Dx, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=', Dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each domain is generically configured by a single or a set of nodes (ni, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='nj) hosting resources and domains that are interconnected by communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A communication slice is then defined as a set of commu- nication resources orchestrated and allocated between slices, nodes, network-slicing entities, and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' As such, the domain nodes (ni, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='nj) hosting resources and domains are interconnected by communication slices (Cx, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Cy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' We identify two types of communication slices that are orchestrated and deployed with distinct configurations and characteristics: Intradomain communication slices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Interdomain communication slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In infrastructures composed of network domains, the mod- eling assumes that a gateway concentrates all communications between different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' We focus in this paper specifically on interdomain com- munications and how to model it in terms of communication slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' NETWORK SLICING INTERDOMAIN COMMUNICATIONS The objective of a network slicing interdomain communi- cation model is to formally structure and capture the needs in terms of communications for the slicing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' It also allows the identification of parameters leading to the optimization of the resource allocation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Network Slicing Assumptions We first introduce the following assumptions in the con- text of network-slicing interdomain communications that are necessary for our modeling and problem formulation: Each network domain is SDN-compatible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each network domain gateway GW_Di (Figure 3) is an SDN-enabled switch whose programmed behavior is to route packets between domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Notation Description Dli i The domain i located in physical location li RDli i Domain’s set of shareable resources at a physical location R Dli i i A shareable resource at domain Dli i R_ISli Di The infrastructure and service resources R_Cli Di The network communication resources BDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Dj Bandwidth between domains LDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Dj Packet loss between domains DlDi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Dj Delay between domains Bni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='nj Bandwidth between nodes Lni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='nj Packet loss between nodes Dlni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='nj Delay between nodes P_RCli Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Dj Set of communication’s link parameters between domains Table I NOTATION AND VARIABLES Each network domain implements monitoring mecha- nisms to collect performance monitoring parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' All intradomain and interdomain links are configurable in terms of allocated resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and All network domains support network resource identifi- cation and has capabilities for resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Network Slicing Model Based on these assumptions, we can now specify an analyt- ical model of multi-domain SVN considering a set of network domains federating together their resources and infrastructures to the slicing process: ℵ =< Dli i , Dlj j , Dlk k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=', Dlz z > (1) Where: Dli i is a network infrastructure domain located at site li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each network infrastructure domain Dli i has a set of share- able resources such as: RDli i =< R D li i i , R D li i j , R D li i k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=', R D li i z > (2) Where: RDli i is the set of shareable resources provided by Di and located at site li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' R D li i i is one particular shareable resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' There are different types of resources at each network infrastructure domain location Dli i : Infrastructure appliance like virtual machines, access points, and IoT devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Computing services like virtual network functions (VNF), storage and computing services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Communications services like physical links, LSPs (MPLS Link Switched Paths), fiber lambdas, and 5G connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' For the purpose of the SVN model, we distinguish between two types of resources: Infrastructure and service resources - R_ISli Di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Communications resources - R_Cli Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' R_CDx Dy Domain D Domain Dx SW_0 GW Dx DomainD 0 R_ Cox [no,ni] R_CDx [n2,n3] COMMUNICATION INTERDOMAIN R_Cox [ni,n2] LINK 0-1 R_ C Dx, Dk sw_1 sW_2 Domain D R_C Dx DI INTRADOMAINCOMMUNICATIONS Domain D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='ARXIV PREPRINT - FEBRUARY 2023 4 Users (clients) request infrastructure, service, and commu- nication resources that are orchestrated by a network slicing software (NECOS, MATILDA, other) to create their sliced virtual network (SVN) as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The communication resources R_Cli Di provide the intercon- nection of infrastructure and service resources R_ISli Di for intradomain and inter-domain connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' As such, for the SVN modeling there are two distinct communication resources or communication slices (Figure 3): Intradomain communication slices used between internal nodes of the domain: R_Cli Di[nj,nk];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Interndomain communication slices used between do- mains: R_Cli Di,Dk The communication slices are characterized by as set of pa- rameters related to interdomain (Equation 5) and intradomain (Equation 4) communications: P_RCDi,Dj =< BDi,Dj, LDi,Dj, DlDi,Dj > (3) P_RCni,nj =< Bni,nj, Lni,nj, Dlni,nj > (4) Where: BDi,Dj is the available bandwidth between domains Di and Dj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' LDi,Dj is the packet loss between domains Di and Dj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' DlDi,Dj is the delay between domains Di and Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Bni,nj is the available bandwidth between nodes ni and nj in a domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Lni,nj is the packet loss between nodes ni and nj at a domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Dlni,nj is the packet delay between nodes ni and nj at a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Figures 1 and 3 illustrate a generic view of the slicing process and related interdomain communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The network slicing infrastructure setup from the point of view of commu- nication resources is as follows: A set of domains (Di);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A single communication slice (configurable link or an- other communication resource) between domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and A SDN switch (gateway) programmed to handling the interdomain packet routing among domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The interdomain slice communication parameters P_RCli Dk,Dj are configured during the slicing commissioning phase, as proposed in the 3GPP network slicing reference architecture and model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' An SVN will require resources of distinct domains to be allocated end-to-end: SLDi k =< R D li i i , R D li i j , RD li y k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=', RD li y z > (5) The communication slice modeling assumes that each do- main contributes to a set of different resources that are located in various physical sites (domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The model is agnostic to the issue of traffic distinction between packets generated with the slices already instantiated Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Openflow Switch Handling Operation and Management Slicing Generated Packets (slice operation) and packets generated by the network slicing management software installed (orchestrator, resource market- place, monitoring, others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The slicing-related interdomain traffic between domains is handled by an SDN switch as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In summary, the interdomain traffic at the gateway is com- posed of the packets generated (operation and management) by all resources belonging to the domain Di having as destination the domain Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The slicing communication model assumes that domains have only one network connection together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In other words, the domains do not act as intermediate domains switching packets in the path to a destination domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' For the interdomain packets at the gateway, the following definitions hold (Figure 4): All packets belonging to a set of resources R D li i i at domain Di with the same performance parameters con- straint use a specific queue Qn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' N switch queues handle the packet generated by the shareable resources at domain Di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The switch queues have SDN resources control capa- bilities controlled by SDN Controllers [27] for resource control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A priority is assigned to each output queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Each queue has a threshold level control parameter PQn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The priority and threshold level assigned to the queues are used to support for optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' optimization controller as shown in the following section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In summary, the model assumes that packets generated from any sliced resource with similar performance constraints are grouped in the same controlled queue in the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The following hypotheses are considered for the control of the intradomain packets and the gateway queues as highlighted in Figures 3 and 5): Intradomain communications will be based on existing underlying communication technologies (MPLS LSPs connections, EON fiber slots, other);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' GATEWAY - SW, DOMAIN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SWZINTERDOMAINPACKETS OPENFLOWOUTPUTQUEUES DTODOMAIND Q1 Q2 Qn Pqn Pqi Pqz PORT1 PORTk PORT, TOUT RiD,tk RkD,tk SW,INTERDOMAINPACKETS To and from Domain Di RmD,k R,D,tkARXIV PREPRINT - FEBRUARY 2023 5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Interdomain Communication Slice and Gateway at Domain i A gateway handles all the inbound and outbound inter- domain traffics, In a domain, each node hosting sharing resources for the slicing creates a path to the gateway, Each path associated with a resource provided by a node is associated with a particular queue in the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The intradomain slice communication analytical model is not the focus of this paper, and these premises make clear its interrelation with the interdomain modeling and allows the independent modeling of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The optimization problem to solve here is the sharing of the communication resources between the different slices taking into account the QoS requirement of each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' This means scheduling the packet originating from the different slices towards the different available queues in the gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' This a complex engineering problem that is difficult to solve in an analytical way considering all the parameters that need to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' For that, we propose to investigate the use of a Reinforcement Learning SARSA agent which is explained in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SARSA AGENT TO OPTIMIZE RESOURCES SHARING The interdomain communication slice model is now applied to the network slicing deployment setup illustrated in Figure 5 in which we have: A multidomain slicing infrastructure with n domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A single communication slice between domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and A SDN-capable switch (gateway) handling bidirectional interdomain packets between the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In terms of the proof of concept, each interdomain commu- nication slice has a reinforcement learning SARSA agent aim- ing to optimize the allocation of communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The RL-SARSA agent acts during slice operation to dynamically keep performance parameters accordingly to management- defined objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The interdomain slice communication parameters (P_RCli Dk,Dj) are configured during the slicing commissioning phase and are dynamically adjusted by the SARSA agent during the slice operation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SARSA Agent Model and Configuration The objective of the SARSA agent is to control the queue flushing transmission rates to preserve the performance param- eters defined by the manager while sharing unused resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The slice communication queues (Qi) are configured as follows: Three queues corresponding to three performance param- eters controlled by the agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each configured queue threshold (Thi) corresponds to the performance parameter assigned to the queue and served to packets generated by sliced resources with this requirement, and Each queue Qi has two states: below threshold (BT) and above threshold (AT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The actions defined for the queues in the AT state are to increase the transmission rate, reduce the transmission rate, and do nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each executed state/action has a defined reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The SARSA agent and communication slice parameters and initial conditions for running are as follows: Agent configuration parameters: – Epsilon-greedy policy ϵ = 8%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' – Learning rate α = 20%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and – Discount factor γ = 80% Threshold limit (triggers agent action) = 50% Agent actions: bandwidth increased or reduced by 10% Maximum number of attempts = 500 Queue priorities are: p1, p2 and p3 with p1 > p2 > p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SARSA Q-values are therefore updated based on the Equa- tions 6: Q(xt, at) ← Q(xt, at) + α[rt+1 + γQ(xt+1, at+1) −Q(xt, at)] (6) GATEWAY-SW, DOMAIN:TO DOMAIN Q1 "o SARSA Thn Th Pqn DOMAIN Agent Pai Th Pq2 InterDomain Communication IntraDon Setof Communication Slices DOMAINK R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='Di Lk INTERDOMAIN INTRADOMAIN TRAFFIC TRAFFIC IntraDomainPath 1 SetofCommunication Slicesm1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Lm DOMAINM DOMAINARXIV PREPRINT - FEBRUARY 2023 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Implementation and tests The simulation environment was configured on a Linux (Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='1 LTS) Intel(R) Core(TM) i5-3470 CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='20GHz desktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The Visual Studio Code v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='0 and Python v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='6 are used to execute the tests and the statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Each test run scenario has a minimum process cycle of 104 packet production for each queue with a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The SARSA agent is called each time any queue reaches its configured threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The SARSA agent processes up to 500 episodes in search of a new configuration of the flushing bandwidth distribution among queues to keep buffer occupation in the configured threshold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The Slice Communication Evaluation Results A series of tests have been undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' It aims to overload the queues to evaluate the behavior of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The three defined scenarios are the following: Scenario 1 - One of the queues is overloaded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Scenario 2 - Two queues are overloaded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Scenario 3 - All queues are overloaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The dynamics of the overloaded queues are configured as follows: First set traffic 30% above the queue defined limit for 10 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Increase to 50% above its defined limit for additional 10 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Increase to 80% above its defined limit for additional 10 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' and Increase to 100% above its defined limit for additional 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Figures 6 and 7 illustrate the SARSA agent’s behavior for scenario one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Figure 6 plots the state of the queues while they are being saturated with overload traffic of packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The queue transmission rate (flushing rate) configured by the SARSA agent is illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' We observe that the total available bandwidth for the link is distributed and reconfigured among the queues according to the dynamic need to flush packets from a specific queue and keep queue occupation below the defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' For scenario two, the behavior of the SARSA agent is illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In this scenario, two queues may overload, and, as observed in scenario one, the SARSA agent reconfigures the queue’s transmission rate to keep buffer oc- cupation below the defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The agent can deal with simultaneous overload for the simulation-defined parameters by keeping queue occupation as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Finally, the behavior of the SARSA agent for scenario 3 is illustrated in Figure 9 and is equivalent to its behavior on scenario two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' FINAL CONSIDERATIONS This paper presents a conceptual model of network slicing and present an analytical model to allocate communication resources between slide process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The conceptual model is along with a SARSA agent that optimize the allocation of com- munication resources among slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The SARSA agent uses the conceptual model to formulate the required communication ressources of each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' A proof of concept implementation of the SARSA agent aims to demonstrate that the SARSA agent contributes to dynamically adjusting and controlling the slice communication parameters between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' The proposed conceptual model demonstrates the feasibility and ease of handling different types of communication resources for optimizing the communication slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Future work includes the leverage of the conceptual model with the integration of intradomain and interdomain models and the new formulation of the distributed optimization problem to solve by a federation of SARSA agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' REFERENCES [1] Shalitha Wijethilaka and Madhusanka Liyanage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “Sur- vey on Network Slicing for Internet of Things Real- ization in 5G Networks”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Xiao and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Krunz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “Dynamic Network Slicing for Scalable Fog Computing Systems With Energy Harvesting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: IEEE Journal on Selected Areas in Communications 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='12 (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2640–2654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Samdanis, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Costa-Perez, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Sciancalepore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “From Network Sharing to Multi-Tenancy: The 5G Net- work Slice Broker”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: IEEE Communications Maga- zine 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='7 (July 2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 32–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [10] Gilvan Durães, Rafael Reale, Romildo Bezerra, Alexan- dre Fontinele, André Soares, and Joberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Martins.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' MPLS Fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' CISCO Press, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ISBN: 1-58705-197-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Dias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Rezende, L N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Ciuffo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Machado, Flavio Silva, Tereza C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' de Brito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Redigolo, Joberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Martins, Leobino Sampaio, and Antonio Abelem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “SFI2 - Slicing Future Internet Infrastructures project”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: Proceedings of the The Global Experimentation for Future Internet (GEFI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Coimbra, Portugal, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [13] Joberto S B Martins, Tereza C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Carvalho, Flavio Silva, and Rodrigo Moreira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' SFI2 Network Slicing Reference Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Technical Report TR03/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Universi- dade de São Paulo - USP, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [17] IETF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Framework for IETF Network Slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' RFC- Re- quest for Comments draft-ietf-teas-ietf-network-slice- framework-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Internet Engineering Task Force, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [18] 3GPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 5G-Evolution-3GPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Technical Report Release 16-17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 3GPP, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 1–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [19] Telecommunication Standardization ITU-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Framework of Network Virtualization for Future Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ITU-T Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='3011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ITU-T - Telecommunication Stan- dardization, Jan.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' European Telecommunications Standards Institute, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [21] ONF - Open Networking Foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Applying SDN Architecture to 5G Slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Technical Report TR-526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' ONF - Open Networking Foundations, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [22] Shunliang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “An Overview of Network Slicing for 5G”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: IEEE Wireless Communications 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='3 (June 2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 111–117.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “Network Slicing and Softwarization: A Sur- vey on Principles, Enabling Technologies, and Solu- tions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: IEEE Communications Surveys Tutorials 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='3 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2429–2453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [26] 3GPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 3rd Generation Partnership Project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Technical Specification Group Services and System Aspects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Man- agement and Orchestration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Concepts, Use Cases and Requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Technical Specification 3GPP TS 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='530 V15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 3GPP, 2019, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' [27] Eliseu Torres, Rafael Reale, Leobino Sampaio, and Joberto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Martins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' “A SDN/OpenFlow Framework for Dynamic Resource Allocation based on Bandwidth Allocation Model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' In: IEEE Latin America Transac- tions 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content='5 (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' 853–860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} +page_content=' Scenario 2: Two queues are overloaded Queue Fill Volume 38828893290 Q1 Queue Ql Threshold 0 2000 4000 6000 8000 10000 12000 14000 38828893290 Q2 Queue Q2 Threshold 38828893290 0 2000 4000 6000 8000 10000 12000 14000 Queue Q3 Threshold 0 2000 4000 6000 8000 10000 12000 14000 Measurements during execution' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE3T4oBgHgl3EQfvwtG/content/2301.04696v1.pdf'} diff --git a/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.pkl b/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..4e9c31a018f0b486afae77bb213faf6e2f4b9ee3 --- /dev/null +++ b/NNAyT4oBgHgl3EQf6_pQ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5811a127f826a32a9a357cd34e48d14ab399b5a6f89b6de79568da31dd59c84d +size 171162 diff --git a/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/2301.11450v1.pdf.txt b/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/2301.11450v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..88d061e4de26e56790382dce8cca2d8680175a66 --- /dev/null +++ b/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/2301.11450v1.pdf.txt @@ -0,0 +1,1434 @@ +SMILE: Robust Network Localization via Sparse and Low-Rank +Matrix Decomposition +Lillian Clark +lilliamc@usc.edu +Electrical and Computer Engineering +University of Southern California +Los Angeles, California, USA +Sampad Mohanty +sbmohant@usc.edu +Computer Science +University of Southern California +Los Angeles, California, USA +Bhaskar Krishnamachari +bkrishna@usc.edu +Electrical and Computer Engineering +University of Southern California +Los Angeles, California, USA +ABSTRACT +Motivated by collaborative localization in robotic sensor networks, +we consider the problem of large-scale network localization where +location estimates are derived from inter-node radio signals. Well- +established methods for network localization commonly assume +that all radio links are line-of-sight and subject to Gaussian noise. +However, the presence of obstacles which cause non-line-of-sight +attenuation present distinct challenges. To enable robust network +localization, we present Sparse Matrix Inference and Linear Em- +bedding (SMILE), a novel approach which draws on both the well- +known Locally Linear Embedding (LLE) algorithm and recent ad- +vances in sparse plus low-rank matrix decomposition. We demon- +strate that our approach is robust to noisy signal propagation, severe +attenuation due to non-line-of-sight, and missing pairwise measure- +ments. Our experiments include simulated large-scale networks, an +11-node sensor network, and an 18-node network of mobile robots +and static anchor radios in a GPS-denied limestone mine. Our find- +ings indicate that SMILE outperforms classical multidimensional +scaling (MDS) which ignores the effect of non-line of sight (NLOS), +as well as outperforming state-of-the-art robust network localiza- +tion algorithms that do account for NLOS attenuation including a +graph convolutional network-based approach. We demonstrate that +this improved accuracy is not at the cost of complexity, as SMILE +sees reduced computation time for very large networks which is +important for position estimation updates in a dynamic setting, e.g +for mobile robots. +KEYWORDS +network localization, graph signal processing, low-rank matrix +approximation +1 +INTRODUCTION +Robotic sensor networks operating in GPS-denied environments +can benefit from collaborative localization [26]. When distance +measurements between network nodes (e.g. mobile robots or static +beacons) are available, the network localization problem seeks to +exactly recover the positions of each node in space. Intuitively, +network localization algorithms leverage pairwise links to constrain +the position estimate of every node in the network. If the locations +of some nodes are known, these nodes are considered anchors. +When sufficient conditions on the number of anchors and the graph +induced by the distance measurements are met, we can recover the +positions of all nodes [16]. +Conference’17, July 2017, Washington, DC, USA +2023. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +Figure 1: Illustration of network localization with NLOS. +The objective is to accurately determine the localization of +agents (grey) based on the known positions of anchors (yel- +low) given the constrains of noisy LOS links (green) and +links with additional NLOS attenuation (red). +However, in realistic scenarios the distance measurements are +typically noisy. For example, radio signals can be used to estimate +distance based on received signal strength, but signals are subject +to noise from the wireless channel [31]. When this noise is assumed +to be Gaussian with zero-mean1, the redundancy offered by many +links in a highly connected network serves to mitigate the problem. +This explains why localization performance improves with the +number of anchors used as reference points. +For many applications, including robotic exploration of unknown +environments, signals see significant degradation from the presence +of walls and obstacles. These non-line-of-sight (NLOS) links are +difficult to model without an a priori map but have a significant +affect on the relationship between received signal strength (RSS) +and distance [10, 14]. Therefore, the ability to infer whether a pair +of transmitters and receivers is NLOS can greatly improve network +localization performance. +Additionally, when network nodes are spread over long distances, +as may also be the case for robotic exploration, it is possible for the +packets used to measure RSS to be dropped. When signal strength +measurements are unavailable at long distances, our large-scale +network localization approach must be robust to missing pairwise +measurements. +Contribution: In this work, we examine the network localiza- +tion problem in the presence of noisy signals where pairwise mea- +surements may be NLOS or missing. We make two key observations: +(1) the Euclidean distance matrix of the true node positions (in 𝑑 +dimensions) will have rank 𝑑 + 2, which we show in Sec. 3, and (2) +1Gaussian noise in dB is also referred to as log-normal fading. +arXiv:2301.11450v1 [eess.SP] 26 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari +the matrix capturing the additional NLOS attenuation is commonly +sparse and non-negative, i.e., there are a limited number of walls +which can only degrade signal strength. Given these observations, +our approach draws on recent advances in sparse plus low-rank de- +composition to first extract the positively biased NLOS attenuation, +which we refer to as sparse matrix inference. After extracting the +NLOS attenuation, we leverage a popular method for dimension- +ality reduction, locally linear embedding (LLE), to recover a set of +weights which allow linear reconstruction to determine the exact +positions of all nodes given the anchors’ positions. +Specifically, we extend the discrete optimization approach for +sparse matrix recovery which is presented in [3] to (1) handle miss- +ing pairwise measurements by imputation and (2) approximate +the unknown sparsity structure of the sparse component of the +matrix. Additionally, we provide details for solving for node po- +sitions using the LLE weight matrix directly (using least squares +method) without the need for additional eigen-decomposition and +coordinate frame alignment used conventionally. Our algorithm +combines recent advances in sparse plus low rank decomposition +and well established and interpretable algorithms like MDS and +LLE and outperforms them in the face of NLOS attenuation. +Evaluation: We demonstrate that SMILE significantly improves +localization accuracy over baseline methods which ignore the af- +fect of NLOS attenuation. Further, we draw parallels between this +approach and another method for graph signal processing which +has recently been demonstrated as promising, namely Graph Con- +volutional Networks (GCNs) [36]. We compare the performance of +SMILE and a state-of-the-art GCN implementation on large-scale +simulated networks and demonstrate an improvement in localiza- +tion accuracy and reduced computation time for network of more +than 1000 nodes. Finally, we evaluate performance on two real- +world networks: an outdoor wireless sensor network with 11 nodes, +and 5 mobile robots and 13 static radios in a GPS-denied limestone +mine with significant NLOS attenuation. +The paper is organized as follows. In Section 2 we provide a brief +overview of seminal and recent work in network localization, and +in Section 3 we formally define the problem and notation. SMILE is +introduced and explained in detail in Section 4, including sparse +matrix inference and subsequent steps for position estimation. We +also draw parallels between our approach and the existing graph +learning-based method. In Section 5 we provide implementation +details as well as experimental results on real and simulated data. +We give concluding remarks in Section 6. +2 +BACKGROUND AND RELATED WORK +Network localization: Network localization is well-researched, +and is commonly formulated as a least squares problem [2, 9, 24]. +Multidimensional scaling (MDS) and its extensions are popular +methods for solving this problem [29]. Classical MDS uses the dis- +tance matrix to compute a matrix of scalar products, typically called +the Gram matrix, that captures pairwise correlation of the posi- +tion vectors. The principal components from eigen-decomposition +of this matrix are then used to recover relative node positions [1]. +Rather than compute over the entire distance matrix, Locally Linear +Embedding (LLE) [27, 30] applies principal component analysis to +Table 1: List of Abbreviations +Abbr. +Description +EDM +Euclidean Distance Matrix +LLE +Locally Linear Embedding +MDS +Multidimensional Scaling +NLOS +Non Line of Sight +PSVD +Partial SVD +RMSE +Root Mean Squared Error +SDP +Semi Definite Programming +SVD +Singular Value Decomposition +small neighborhoods, which improves performance when the reduc- +tion from the noisy (high-dimensional) data to the low-dimensional +true positions is non-linear, and has shown promise in sensor net- +work localization [18]. +Exploiting sparsity: If the data is well-described by a particular +statistical model (e.g. Gaussian or log-normal), we can instead form +the maximum likelihood estimation problem [25], and solve using +semi-definite programming (SDP) methods [4, 5]. Recent works +extend SDP methods to consider non-Gaussian noise [38] and, more +specifically, NLOS noise [8, 19, 22]. Jin et al. demonstrate that when +the NLOS noise has a certain structure, namely non-negative and +sparse, a sparsity-promoting term in the objective function can +improve the performance of SDP approaches [19]. However, SDP +methods suffer with respect to complexity and are intractable for +very large networks [36]. +Sparse and low rank decomposition: Another approach is to +recover the matrix of NLOS attenuation directly. In recent years, +sparse and low-rank matrix recovery has drawn attention due to its +relevance in signal processing, statistics, and machine learning [34]. +Bertsimas et al. recently proposed a discrete optimization approach +to sparse and low-rank recovery which uses alternating minimiza- +tion [3]. Our work leverages this method, extends it to the case of +missing measurements and unknown sparsity, and demonstrates +that it can serve as an important component of a robust network +localization algorithm. +Graph convolutional networks: To effectively exploit the rela- +tional information of graph-structured data, graph neural networks +(GNNs) have recently become a popular method for approaching +optimization problems in wireless networks [21]. Yan et al. recently +demonstrated promising results in the application of Graph Con- +volutional Networks (GCNs) to the network localization problem. +Their approach maintains accurate localization despite NLOS at- +tenuation, and is scalable to large-scale networks at an affordable +computation cost. However, we will see that the learned model +is unable to exactly recover positions for a completely observed +distance matrix in the absence of noise. In this work we propose a +novel network localization algorithm which builds on the principles +of multidimensional scaling for exact recovery in the absence of +noise, exploits the sparsity of NLOS attenuation for improved local- +ization accuracy, and scales to very large networks at an affordable +computation cost. + +SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition +Conference’17, July 2017, Washington, DC, USA +3 +PROBLEM FORMULATION +Let 𝐴 be the set of anchors whose positions are known, where +|𝐴| = 𝑛𝐴. Let 𝐵 be the set of agents whose positions are unknown, +where |𝐵| = 𝑛𝐵. Let 𝐶 = 𝐴 ∪ 𝐵 be the set of nodes which includes +both anchors and agents, with 𝑛𝐴 + 𝑛𝐵 = 𝑛. We define the (𝑛 × 𝑛) +true distance matrix D as +D[𝑖, 𝑗] = ||p𝑖 − p𝑗 || +where p𝑖 = [𝑥𝑖,𝑦𝑖]𝑇 is the position of node 𝑖. We establish the +convention that the first 𝑛𝐴 rows and 𝑛𝐴 columns pertain to the +anchors. D◦2 is the Euclidean distance matrix (EDM, containing +squared distances i.e D squared entry-wise), which is zero along the +diagonal and symmetric [17]. If matrix P ∈ R𝑛×2 with P[𝑖, :] = p𝑇 +𝑖 +be the position matrix, we can see that D◦2 has rank 4. In general, +the EDM of a configuration of points embedded in R𝑑 has rank at +most 𝑑 + 2 (and exactly 𝑑 + 2 if the points are in general position +as opposed to more special or coincidental cases, e.g., points in +3D which lie on a line). This is briefly justified below, with further +details provided in [13]. +D[𝑖, 𝑗]2 = ||p𝑖 − p𝑗 ||2 += ⟨p𝑖 − p𝑗, p𝑖 − p𝑗⟩ += ⟨p𝑖, p𝑖⟩ + ⟨p𝑗, p𝑗⟩ − 2⟨p𝑖, p𝑗⟩ += p𝑇 +𝑖 p𝑖 + p𝑇 +𝑗 p𝑗 − 2p𝑇 +𝑖 p𝑗 +=⇒ D◦2 = diag(PPT)1T +������������������������ +𝑟𝑎𝑛𝑘−1 ++ 1diag(PPT) +𝑇 +������������������������ +𝑟𝑎𝑛𝑘−1 +− 2PPT +���� +𝑟𝑎𝑛𝑘−2 +Since𝑟𝑎𝑛𝑘(A+B) ≤ 𝑟𝑎𝑛𝑘(A)+𝑟𝑎𝑛𝑘(B) and𝑟𝑎𝑛𝑘(CCT) = 𝑟𝑎𝑛𝑘(CTC) = +𝑟𝑎𝑛𝑘(C), we have D◦2 as low-rank with rank = 4. +Let O be the matrix that contains the distances that we observe, +where O[𝑖, 𝑗] is a function of the strength of radio signal transmitted +by node 𝑖 and received by node 𝑗. In general, our observation can +be captured as +O = Ω ◦ [D + N + S] +where Ω is the observation mask and takes on values of 1 when a +measurement is available and 0 otherwise (for instance, when the +transmitter and receiver are out of range). N is an asymmetric matrix +capturing noise in the observations, which we assume is Gaussian +and zero-mean. S captures the additional NLOS attenuation, and as +in [19] we assume S is non-negative and sparse. We also assume +S is symmetric which relies on assumptions that an attenuating +obstacle will affect a radio link in both directions equally. +We make the realistic assumption that nodes are in general +position, meaning that in 2D they do not lie on a straight line. We +also assume that an upper bound on the distance between any two +nodes, 𝑑max, is known or can be approximated. Our objective is +finding an estimate for the locations of all nodes ˆP = [ˆp1, ..., ˆp𝑛]𝑇 +which is consistent with the prior information - (1) the observations +in O as well as (2) the known anchor positions PA = [p1, ..., p𝑛𝐴]𝑇 . +4 +SMILE +In this section we propose Sparse Matrix Inference and Linear +Embedding, our novel large-scale network localization algorithm, +which is illustrated in Fig. 2. Details of the algorithm are presented +Anchor positions +X +Observations +Complete the +Euclidean +distance matrix +Sparse matrix inference +Construct +locally linear +weight matrix +Solve sparse +sub-problem +Solve low-rank +sub-problem +E +Y +Y +(sparse NLOS) +X +Predicted locations +W +Figure 2: An overview of Sparse Matrix Inference and Linear +Embedding (SMILE). +in Algorithm 1. In subsection 4.1, we discuss our method of extract- +ing NLOS attenuation via sparse matrix inference, which results in +a low-rank approximation of the Euclidean distance matrix. Then in +subsection 4.2, we discuss our method of transforming this matrix +into an estimate of the locations of all nodes. +4.1 +Sparse Matrix Inference +For a given input matrix E ∈ R𝑛×𝑛, for which E = X + Y, sparse +matrix inference seeks to find the low-rank component X ∈ R𝑛×𝑛 +and sparse component Y ∈ R𝑛×𝑛 which solves: +min +X,Y 𝑓 (X, Y) = ||E − X − Y||2 +𝐹 + 𝜆||X||2 +𝐹 + 𝜇||Y||2 +𝐹 +s.t. +rank(X) ≤ 𝛼, ||Y||0 ≤ 𝛽 +(1) +where ||X||𝐹 denotes the Frobenius norm. In this subsection we +describe how to decompose E X and Y using alternating minimiza- +tion, and how to use this technique on the problem defined in Sec. +3. +Firstly, we square the observed matrix (line 2). Temporarily as- +suming N = 0 and no missing observations, i.e Ω = 11T, +E = O◦2 = (D + S)◦2 += +D◦2 +���� +low-rank ++ S◦2 + 2D ◦ S +���������������������� +sparse +(2) +where the expression in parentheses is non-negative and sparse +(because S is non-negative and sparse), and D◦2 is low-rank. Thus +the problem is amenable to the sparse matrix inference framework. +Let us consider the case with noise 𝑁 ≠ 0 and no missing obser- +vations +E = O◦2 = ( +˜D +���� +D + N +S)◦2 = ( ˜D + S)◦2 += ˜D◦2 + S◦2 + ˜D ◦ S +������������������ +sparse +(3) +The last expression is exactly the same as the formulation in 2 +except that ˜D◦2 = D◦2 + N◦2 + 2N ◦ D may no longer be low rank +due to the addition of N◦2 + 2N ◦ D where entries of the first term +are now i.i.d from a Chi-Squared Distribution X2 +1 . However, in our +ablation study, we empirically verify that as long as the standard +deviation of the normal noise in entries of N is not too large, using +the estimated low-rank matrix from the sparse matrix plus low-rank + +Conference’17, July 2017, Washington, DC, USA +Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari +Table 2: SMILE parameters +Parameter +Symbol +𝑘 +Number of neighbors +ˆ𝛽𝑖 +Initial sparsity estimate +𝜂 +Step size +𝑇 +Number of random initializations +𝜆 +Low-rank matrix regularizer +𝜇 +Sparse matrix regularlizer +𝜖 +Inner loop tolerance +𝜖𝛽 +Outer loop tolerance +inference, SMILE is able to recover P fairly faithfully under RMSE +loss. +Missing measurements: The approach presented in [3] as- +sumes a complete input matrix. While more complex approaches +to matrix completion exist, for example by finding the sum total +length of the shortest path between two nodes [32, 33], we take a +simpler approach. When Ω ≠ 11T, we complete the observation +matrix by filling in missing values with 𝑑max (line 1). Even in the +face of missing observations, we show that the problem remains +amenable to the sparse matrix inference framework as long as the +missing observations are not too large of a fraction of the total +number of observations in O. +E = ˜O◦2 = [Ω ◦ O]◦2 += [Ω ◦ (D + N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 += [Ω ◦ D + Ω ◦ (N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 += [D − (1 − Ω) ◦ D + Ω ◦ (N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 += [D + Ω ◦ N + Ω ◦ S + (1 − Ω) ◦ (𝑑𝑚𝑎𝑥I − D) +������������������������������������������������������������������������ +˜𝑆 (sparse) +]◦2 += [D + Ω ◦ N +���������������� +˜D ++˜S]◦2 = [ ˜D + ˜S]◦2 += ˜D◦2 + ˜S◦2 + 2 ˜D ◦ ˜S +���������������������� +sparse +(4) +Thus, ˜O decomposes similarly to O as long as (1 − Ω) + S is still +sparse, i.e. the total number of NLOS and missing measurements is +low. ˜D◦2 is still amenable to SMILE like in the previous case. +Alternating minimization: As presented in [3], we alternate +between solving two sub-problems. For a given X, we estimate Y +by composing a sparse matrix with non-zero entries at the largest +indices of (E − X). This is described further in Algorithm 2. Then +for a given Y, we estimate X by reducing the rank of (E − Y). This +repeats until the value of the objective function in Eq. 1 has con- +verged, corresponding to the inner loop (lines 11-18). We initialize +X with a random, low-rank matrix and repeat this process for𝑇 > 0 +random initialization, ultimately selecting the decomposition which +minimizes the objective function (lines 5-19). +Unknown sparsity: The approach presented in [3] assumes the +sparsity of Y is known, however in realistic settings this may not be +true. We observed empirically that for a matrix with known compo- +nents, solving the sparse matrix inference problem for increasing +Algorithm 1 SMILE +Input: O (observation matrix), P𝐴 (anchor positions), 𝛼 (desired +rank) +Output: ˆPB (agent location estimates) +1: ˜O = O + (1 + Ω)𝑑max +2: E = ˜O◦2 +3: ˆ𝛽 ← ˆ𝛽𝑖 +4: while Δ𝑓 /𝑓 > 𝜖𝛽 do +5: +for 𝑡 in {1, ...,𝑇 } do +6: +X′ ∈ R𝑛×𝑛 ← random +7: +U, Σ, V𝑇 = PSVD(X′, 𝛼) +8: +X = UΣV𝑇 +9: +Y ∈ R𝑛×𝑛 ← 0 +10: +𝑓 = ||E − X − Y||2 +𝐹 + 𝜆||X||2 +𝐹 + 𝜇||Y||2 +𝐹 +11: +while Δ𝑓 /𝑓 > 𝜖 do +12: +Y′ = compose_sparse(E − X, ˆ𝛽) +13: +Y = +1 +1+𝜇 Y′ ◦ (E − X) +14: +X′ = +1 +1+𝜆 (E − Y) +15: +U, Σ, V𝑇 = PSVD(X′, 𝛼) +16: +X = UΣV𝑇 +17: +𝑓 = ||E − X − Y||2 +𝐹 + 𝜆||X||2 +𝐹 + 𝜇||Y||2 +𝐹 +18: +end while +19: +end for +20: +ˆ𝛽 ← ˆ𝛽 + 𝜂 +21: end while +22: W ∈ R𝑛×𝑛 ← 0 +23: for each node 𝑖 do +24: +find 𝑘 nearest neighbors 𝑁𝑁 (𝑖) +25: +for each pair of neighbors (𝑗, 𝑙) do +26: +H ∈ R𝑘×𝑘 where H[𝑗,𝑙] = 1 +2 (X[𝑖,𝑙] + X[𝑗,𝑖] − X[𝑗,𝑙]) +27: +solve Hw = 1 for w +28: +end for +29: +W𝑖 ← w/� w at indices of neighbors +30: end for +31: m = (I − W)𝐴P𝐴 +32: solve (W − I)𝐵 ˆP𝐵 = m for ˆP𝐵 +33: return ˆPB +Algorithm 2 compose_sparse +Input: M ∈ R𝑛×𝑛 (matrix), 𝛽 (sparsity) +Output: B (binary matrix) +1: sorted indices = argsort(M) +2: one indices = sorted indices[-𝛽:] +3: B ∈ R𝑛×𝑛 ← 0 +4: B[one indices] = 1 +5: return B +estimate ˆ𝛽 causes the objective function to decrease, and around +the true value of 𝛽 it converges (Fig. 3. This motivates the outer +loop (lines 3-21). To reduce computation time, we can initialize ˆ𝛽 +with the approximate sparsity, increase the search step size 𝜂, or +increase the converge tolerance 𝜖𝛽. However this may come at the +cost of localization accuracy. + +SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition +Conference’17, July 2017, Washington, DC, USA +0 +20000 +40000 +60000 +80000 +100000 120000 140000 +0 +1 +2 +3 +4 +5 +Objective function f +1e8 +Figure 3: The final value of the sparse matrix inference ob- +jective function 𝑓 (Eq. 1) when we assume different levels of +sparsity ˆ𝛽, plotted alongside the true sparsity 𝛽 (dotted line) +for various simulated matrices E ∈ R500×500. +4.2 +Linear Embedding +Next we find the set of k nearest neighbors 𝑁𝑁 (𝑖) for each node +𝑖 using the estimated EDM gotten from sparse matrix inference. +For each node, we compute a 𝑘 × 𝑘 matrix H which captures the +pairwise similarity between the nodes in 𝑁𝑁 (𝑖) ∪ {𝑖} (line 25) +using the corresponding sub-matrix in the estimated EDM. H is +calculated using the same approach used for classical MDS, but for +local neighborhoods [1]. From H we can recover the desired weight +matrix W (lines 26-27), first by solving for w and then inserting +it at the row position corresponding to the node index of 𝑖 with +the entries in w aligned with the column belonging to the nearest +neighbours and zeros for non-neighbours. At this point, each row +of W corresponds to a node and captures how the node’s position +can be expressed as a linear combination of its neighbors. This is +a meaningful representation of the positions of all nodes, but is +not yet a node embedding. Typically, LLE then computes a sparse +matrix (I − W)𝑇 (I − W) whose eigenvectors corresponding to the +two smallest non-zero eigenvalues result in a solution up to rotation +and translation. +Using the anchors’ positions: In this setting, we can compute +a solution using W and the anchor location P𝐴 directly. This is +possible because, by construction, WP = P. This is essentially a +system of 𝑛 equations with 𝑛𝐵 < 𝑛 unknowns. Treating W − I ∈ +R𝑛×𝑛 as a block matrix, let (I − W)𝐴 = (I − W)[:, : nA] ∈ R𝑛×𝑛𝐴 +be the sub-matrix corresponding to the anchors and (W − I)𝐵 = +(I − W)[:, −nB :] ∈ R𝑛×𝑛𝐵 be the sub-matrix corresponding to the +agents, where I ∈ R𝑛×𝑛 is the identity matrix. Similarly treating +𝑃 ∈ R𝑛×2 as a block matrix, let 𝑃𝐴 = 𝑃 [: 𝑛𝐴, :] ∈ R𝑛𝐴×2 and 𝑃𝐵 = +𝑃 [−𝑛𝐵, :] ∈ R𝑛𝐵×2 correspond to the block matrices for positions +of anchors and agents respectively. With some manipulation we +get, +WP = P =⇒ +𝑇 +���������� +(W − I) P = 0 =⇒ TP = 0 +=⇒ +�TA +| +TB +�  +PA +− +PB + += 0 +=⇒ TAPA + TBPB = 0 =⇒ TAPA = −TBPB +=⇒ (I − W)𝐴P𝐴 +���������������������� +m, known += (W − I)𝐵 +�������������� +known +P𝐵 +���� +unknown +(5) +(lines 29-30) and from the last expression we can estimate a solu- +tion ˆPB for the unknown agent positions using the least squares +minimization. +4.3 +Comparison with Graph Convolutional +Networks +We highlight several interesting parallels between SMILE and GCN. +Briefly, the approach presented by Yan et al. computes the position +estimates +ˆP = A′𝜙(A′(A ⊙ O)Z(1))Z(2) +(6) +where A is the thresholded adjacency matrix for a given threshold +𝜃𝐺𝐶𝑁 , A′ is the row-normalized augmented thresholded adjacency +matrix, 𝜙(·) is a nonlinear activation function, and Z(1), Z(2) are +learned weights. Note that the graph signal is the observed matrix. +(1) 𝜃𝐺𝐶𝑁 and 𝑘: This approach introduces a threshold such +that edges between nodes in the graph are present only if +the observed distance is less than the threshold. Decreasing +this threshold is similar to decreasing the number of nearest +neighbors, and both have the benefit of noise truncation. +However, as the threshold is increased, GCN experiences +over-smoothing due to the aggregation step. Intuitively, if +every node is connected to every other node, the aggregation +step causes all node embedding to collapse at a single point. +This means that more information, i.e., additional pairwise +measurements, actually hurts the algorithm. SMILE does not +experience this issue, and we will see in Fig. 6 that increasing +𝑘 improves performance at the cost of increased runtime. +(2) Low-pass filtering and PSVD: Repeated multiplication by +the normalized adjacency matrix acts as a low pass filter [23]. +This reduces the rank of the graph signal, similar to the +process of extracting the low-rank component of the noisy +euclidean distance matrix. However, repeated multiplication +by the normalized adjacency matrix reduces the rank of the +observed matrix by some amount, which corresponds to dis- +tances (rather than squared distances). Our approach applies +rank reduction to exactly reduce the Euclidean distance ma- +trix to rank 𝑑 + 2, informed by the principles of the problem +formulation. +(3) Convolution and linear embedding: Each convolution +layer multiplies the adjacency matrix and input matrix by a +set of weights. Intuitively, this makes the node features at +the next layer a weighted sum of the neighbor features. The +learned GCN weights are analogous to the set of weights W +in SMILE which allow linear reconstruction of agents from + +Conference’17, July 2017, Washington, DC, USA +Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari +the anchors positions. However, the repeated convolutions +and nonlinear activation prevent a straightforward analysis. +The interpretability of W is an advantage of our approach. +Qualitatively, we expect SMILE to outperform GCN because (1) +it is not subject to oversmoothing and makes careful and productive +use of additional pairwise measurements, (2) it finds a Euclidean +distance matrix with the exact expected rank, and (3) it relies on +principled methods to determine a weight matrix relating nodes to +their neighbors, which is thus interpretable. In the next section, we +compare these approaches quantitatively. +5 +RESULTS +In this section we evaluate the performance of SMILE with respect +to localization accuracy and computation time. We also consider +performance under different noise settings, as an ideal method is +both robust to high-levels noise and able to exactly recover positions +when possible. Localization accuracy is measured by the root-mean- +squared-error (RMSE) given by ||P𝐵 − ˆP𝐵||𝐹 . Robustness considers +the affect of Guassian noise, NLOS attenuation, and missing pair- +wise measurements. The experiments consider networks in 2D, thus +the desired rank of EDM is 𝛼 = 4. The SMILE parameters are set +to 𝑘 = 50, ˆ𝛽𝑖 = 5 𝑛2 +100,𝜂 = 𝑛2 +100,𝑇 = 1, 𝜆 = 0.01, 𝜇 = 0.1,𝜖 = 0.001, and +𝜖𝛽 = 0.01. For comparison, we train a two-layer GCN according +to [36] with distance threshold 1.2. More details of our implemen- +tation are available online 2. +5.1 +Simulation results +Our simulated scenarios consider 𝑛 nodes randomly placed over a +5m × 5m square area, with the first 𝑛𝐴 nodes considered anchors. +Noise N is drawn from a zero-mean Gaussian, N[𝑖, 𝑗] ∼ N (0, 𝜎2). +NLOS noise is drawn from a uniform distribution, S[𝑖, 𝑗] = S[𝑗,𝑖] ∼ +U[0, 10] with probability 𝑝NLOS and 0 otherwise. Both matrices +are zero along the diagonal, N[𝑖,𝑖] = S[𝑖,𝑖] = 0. The observation +mask limits our observed distance measurements by a threshold 𝜃 +such that Ω[𝑖, 𝑗] = 1 if O[𝑖, 𝑗] ≤ 𝜃 and 0 otherwise. +Comparison with state of the art: Firstly we consider the +setting where 𝑛 = 500,𝑛𝐴 = 50, 𝜎 = 0.1, 𝑝NLOS = +1 +10, and 𝜃 = +𝑑max, for which an example dataset is available3. Fig. 4 illustrates +the performance of SMILE, which achieves an RMSE of 0.06 in +5.22 seconds, and GCN which achieves an RMSE of 0.11 in 5.73 +seconds. GCN’s performance is consistent with that reported in [36], +which reports RMSE for various other methods, including SDP with +sparsity promoting regularization [19] which achieves an RMSE +of 0.26. SMILE achieves the highest reported localization accuracy, +and Fig. 5 illustrates that not only is the error low on average but +the error density has a smaller tail. +In Fig. 6, we investigate the localization accuracy and compu- +tation time as we vary 𝑘, the number of neighbors used for linear +embedding. We observe that RMSE remains consistent for 𝑛𝐴 ≥ 50. +While the minimum of 0.048 is reached at 𝑘 = 130, this comes at +the expense of increased computation time. Note that selecting 𝑘 is +analogous to setting the GCN threshold, however we do not observe +the degradation in performance that GCN is prone to when the +threshold is too high. This comes from the aggregation component +2https://github.com/ANRGUSC/smile-network-localization +3https://github.com/Yanzongzi/GNN-For-localization +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +GCN (0.11) +true anchor +pred anchor +true node +pred node +0 +1 +2 +3 +4 +5 +SMILE (0.06) +true anchor +pred anchor +true node +pred node +Figure 4: Localization with GCN (Yan et al. [36]) and novel +SMILE for a 500 node network with 50 anchors. GCN +achieves RMSE 0.11 and SMILE achieves RMSE 0.06. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Error +0 +20 +40 +60 +80 +100 +120 +Frequency +GCN +SMILE +Figure 5: Error density for GCN and SMILE on the data from +Fig. 4, where SMILE results in a more desirable distribution. +0 +20 +40 +60 +80 +100 +120 +140 +Number of neighbors +0.1 +0.2 +0.3 +0.4 +0.5 +RMSE +8 +9 +10 +11 +12 +Runtime (sec) +Figure 6: RMSE (solid line) and runtime (dashed line) trade- +off as we vary the number of neighbors 𝑘. Each point is the +average of 10 trials. +of convolution, which causes over-smoothing if a node has too +many neighbors and results in the position estimates collapsing to +a point. +Robustness: Fig. 7 demonstrates the performance of SMILE and +GCN as 𝑝NLOS varies from 0 to 1 +2. From this, we observe that SMILE +outperforms GCN for up to 30% NLOS links. Beyond this, the as- +sumption that S is sparse is no longer true, and performance suffers +as expected. The same is true when 𝜃 falls below 3.5, and (1 − Ω) is +no longer sparse, i.e not enough entries in the observations matrix + +SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition +Conference’17, July 2017, Washington, DC, USA +0 +10 +20 +30 +40 +50 +pNLOS +0.1 +0.2 +RMSE +SMILE +GCN +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.10 +0.15 +RMSE +SMILE +GCN +3 +4 +5 +6 +7 +8 +0.0 +0.5 +1.0 +RMSE +SMILE +GCN +Figure 7: Performance of SMILE and GCN as the probabil- +ity of NLOS links, standard deviation of Gaussian noise, and +threshold for distances which can be measured are varied +(𝑇 = 2). +˜O. Notably, we do not observe a clear trend in the performance +of GCN as 𝜎 increases. We posit that this is because the learned +approach does not rely on exact decomposition, even in the absence +of noise. +To test this, we compare the performance of GCN and SMILE +on an ideal dataset and consider the results in Fig. 8. While the +performance of both methods is better in this ideal setting, we +observe that GCN cannot exactly recover positions. While being +robust to Gaussian noise and sparse non-Gaussian noise, SMILE is +also accurate in ideal scenarios. +Complexity: As we increase the number of nodes, accuracy +of both methods increases (accuracy similarly increases with the +percentage of anchors). However, complexity increases with the +size of the network. Because our approach is iterative, we measure +complexity numerically (computation time) in lieu of analytically +as it is difficult to predict when the sparse matrix inference will con- +verge. Fig. 9 shows that the computation time of both SMILE and +GCN remain reasonable as 𝑛 increases. For least squares optimiza- +tion and SDP, we have seen that the compute time for very large +networks becomes intractable [36]. For the 1000 node network, the +time to predict ˆP using SMILE is 29.84 seconds while the time to +train and predict ˆP using GCN is 46.90 seconds. Note that this is +a fair comparison because the learned approach requires training +a specific model for each network; the same learned weights are +not applicable to a new network, and neither is the SMILE weight +matrix H. The low compute times for SMILE are likely because the +linear embedding component depends on the number of neighbors +(rather than the number of nodes) and the sparse matrix inference +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +GCN ideal (0.12) +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +SMILE ideal (0.0) +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +GCN noisy (0.19) +0 +2 +4 +0 +1 +2 +3 +4 +5 +SMILE noisy (0.1) +actual a +predicted a +actual +predicted +Figure 8: Performance of SMILE and GCN in an ideal setting +(𝑝NLOS = 0, 𝜎 = 0) and a noisy setting (𝑝NLOS = +1 +10, 𝜎 = 0.5). +SMILE is robust to noise without sacrifice performance in +ideal settings. +component adapts the step size to the number of nodes. We high- +light that scalability is not at the cost of localization accuracy. For +𝑛 = 1500, SMILE achieves RMSE of 0.05 while GCN achieves RMSE +of 0.08. +Ablation: Thus far we have compared our approach to an exist- +ing solution, but it is also interesting to consider the role of each +component of SMILE and, in particular, whether simpler existing +approaches from the literature are sufficient. Fig. 10 considers a +dataset with 𝜎 = 0.3, 𝑝NLOS = +1 +10, and 𝜃 = 5. We compare the +performance of (A) no rank reduction, (B) rank reduction via PSVD, +and (C) sparse and low-rank matrix recovery. In conjunction, we +consider (1) classical MDS with the Kabsch algorithm for coordinate +system registration [1] and (2) LLE-based embedding using anchor +positions. Specifically, the top left plot represents a naive baseline: +multidimensional scaling assuming Gaussian zero-mean noise. The +bottom right plot represents SMILE. +We observe several key takeaways from this figure. +(1) De-noising: Moving from left to right, each column of this +plot contains increasing more sophisticated noise reduction +and decreases the final RMSE. In particular, sparse matrix +inference increases localization accuracy by almost an order +of magnitude. +(2) Nearest neighbors: Methods in the top row use all avail- +able links in eigen-decomposition, while methods in the +bottom row use only the nearest neighbors to determine +weights for linear reconstruction. This local-neighborhood +approach consistently improves localization accuracy. Note + +Conference’17, July 2017, Washington, DC, USA +Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari +0 +200 +400 +600 +800 +1000 +1200 +1400 +Number of Nodes +0.2 +0.4 +0.6 +0.8 +RMSE +SMILE +GCN +0 +200 +400 +600 +800 +1000 +1200 +1400 +Number of Nodes +0 +10 +20 +30 +40 +Runtime (sec) +SMILE +GCN +Figure 9: Compute time for different sizes of networks for +GCN and SMILE. Previous work has shown that learned ap- +proaches (GCN, MLP, NTK) scale better than optimization +approaches (LS, ECM, SDP) [36]. SMILE outperforms GCN +for very large networks. +that if 𝜃 ≥ 𝑑max, i.e. no measurements are missing, the per- +formance of (1) and (2) are similar. This is likely because +LLE’s strength comes from relying more on nearby measure- +ments, and our approach to matrix completion puts missing +measurements at effectively long distances. In setting where +long-distance measurements are unreliable, which is com- +monly the case in realistic wireless communication [31], +using nearest neighbors is advantageous. +(3) SMILE: The combination of both sparse matrix inference +and linear embedding is more robust to NLOS attenuation +and missing measurements than either component in isola- +tion. +5.2 +Experimental results +In this section we apply SMILE to two real-world small scale datasets, +and discuss its performance, our findings, and directions for future +work. +Sensor network: First we consider a network of 11 Mica2 motes +placed randomly in a parking lot [37] 4. The maximum distance +between any two nodes is 10.57m. We consider received signal +strength (RSS) measurements between pairs (RSS is averaged over +20 packets). To estimate distance, we use the following log-distance +path loss model +𝑅𝑆𝑆(𝑑) = 𝑃tx − 𝑃𝐿(𝑑0) − 10𝛾 log10( 𝑑 +𝑑0 +) + N (0, 𝜎2) +(7) +where 𝑃tx is the transmit power, 𝑃𝐿(𝑑0) is the path loss at a refer- +ence distance, and 𝛾 is the path loss exponent [31]. For this data, +use the model 𝛾 = 2.9 and 𝑃tx − 𝑃𝐿(𝑑0) = −49.12𝑑𝐵 for 𝑑0 = 1. +Given the ground truth location estimates, Fig. 11 shows that the +4http://anrg.usc.edu/www/download_files/RSSLocalizationDataSet_11nodes.txt +noise in this dataset has mean 0.35m with standard deviation 2.94. +Note that even though we believe this data to be in open space (all +LOS), seven links see a measurement error of greater than 5m. +Figure 12 shows the performance of GCN and SMILE on this +network, where we update the SMILE parameters to 𝜆 = 0.05, 𝜇 = +0.05, and𝑇 = 10 because the network is smaller. If we assume four of +these nodes are anchors, GCN achieves RMSE of 1.67 with threshold +5.2m, and SMILE achieves a comparable RMSE of 1.63 with 𝑘 = 3, +guessing 𝛽 = 11. These results are the average performance of 10 +trials, and the best parameters for each algorithm were selected +empirically. +Robotic network: Secondly, we consider measurements from +a robotic network with 18 nodes, 13 of which are stationary bea- +cons at known locations, and 5 of which are mobile (legged and +wheeled) robots5. Each robot carries a Streamcaster 4200 from Sil- +vus Technologies, which are also used as beacons. We consider +a set of pairwise RSS measurements from a single timestamp of +robotic exploration. They nodes are spread over a large area in an +underground limestone mine with distances between nodes of up +to 341 meters. Ground truth positions are available via a simultane- +ous localization and mapping algorithm, and we assume these are +accurate [7]. Estimates of whether links are LOS is also available +via a LiDAR-based predictive model [10]. Fig. 13 plots the error +density for LOS, NLOS, and missing links from this data, where we +note that unfortunately only 14% of measurements are available +and LOS. We augment this real dataset with simulated data such +that missing measurements are sampled from a zero-mean Gauss- +ian with standard deviation equivalent to the observed LOS noise +(𝜎 = 36.13). +Fig. 14 show the results of our approach on this realistic dataset, +where edges indicate error in distance. GCN achieves RMSE of 45.8 +on this simulated data with threshold 110, while SMILE achieves an +improved RMSE of 34.22 for 𝑘 = 17. These results are the average +performance of 10 trials, and the best parameters for each algorithm +were selected empirically. We observe that the average localization +error for unknown nodes is roughly the standard deviation of the +noise on LOS links, while the standard deviation of NLOS error is +59.60 (mean 30.14m). This indicates that SMILE is able to achieve +localization accuracy comparable to a distance-based model on a +single LOS link for all nodes, even those which have many NLOS +neighbors, and shows promising performance. +During robotic exploration, these anchors were deployed au- +tonomously [15, 28]. This means anchors are only located in places +the robots have already visited, while the exploration objective +encourages the robots to move away from these anchors. In fact, +the exploring robots then tend to be outside the convex hull de- +fined by the anchors. This configuration appears to stress network +localization, especially for smaller networks with significant noise. +Prior work exists in robotic motion which minimizes localization +uncertainty [6, 11], and network localization algorithms which +specifically address this geometric setting may be an interesting +direction for future work. +5https://github.com/NeBula-Autonomy + +SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition +Conference’17, July 2017, Washington, DC, USA +0 +2 +4 +6 +6 +4 +2 +0 +2 +4 +6 +MDS (1.49) +0 +2 +4 +0 +1 +2 +3 +4 +5 +PSVD + MDS (0.25) +0 +2 +4 +0 +1 +2 +3 +4 +5 +SMI + MDS (0.19) +2 +0 +2 +4 +6 +2 +0 +2 +4 +6 +8 +LLE (1.16) +0 +2 +4 +0 +1 +2 +3 +4 +5 +PSVD + LLE (0.19) +0 +2 +4 +0 +1 +2 +3 +4 +5 +SMILE (0.12) +actual a +predicted a +actual +predicted +Figure 10: Ablation study with 𝜎 = 0.3,𝜃 = 5. Top row, left to right: classical MDS achieves RMSE 1.49, de-noising via PSVD +improves RMSE to 0.25, and sparse matrix inference further improves RMSE to 0.19. Bottom row, left to right: LLE with direct +position recovery achieves RMSE 1.16, de-noising via PSVD improves RMSE to 0.19, and and sparse matrix inference further +improves RMSE to 0.12. +5 +0 +5 +10 +15 +Measurement error (m) +0 +2 +4 +6 +8 +10 +Frequency +Figure 11: Measurement error for 11 node wireless sensor +network outdoors, comparing distance from RSS (Eq. 7) with +ground truth. +5.3 +Discussion +One advantage of SMILE is that we compute Y which approximates +the sparsity structure of S. Therefore, we can use this method to +estimate which links are NLOS and where walls, obstacles, or other +potentially adversarial sources of attenuation may be present. This +means SMILE could be useful as a complementary modality for +simultaneous localisation and mapping (SLAM) algorithm for multi- +robot systems, and has potential applications in robotic exploration +for disaster mitigation and military applications [10]. +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +10 +12 +14 +GCN (1.6) +true anchor +pred anchor +true node +pred node +0 +2 +4 +6 +8 +SMILE (1.63) +true anchor +pred anchor +true node +pred node +Figure 12: Localization accuracy of GCN (RMSE of 1.6) and +SMILE (RMSE of 1.63) on the sensor network. Edges in +the left graph represent observed distances less than 𝜃GCn. +Edges in the right graph connect each node 𝑖 to its nearest +neighbors 𝑁𝑁 (𝑖). +While it is impossible to unique localize a network in the ab- +sence of anchors (given the possibility of translations and rotations), +anchor-less localization problems have significant overlap with +other useful signal processing problems. In face, SMILE could be +directly applied to anchor-less localization problems by replacing +lines 29-30 in the algorithm with the embedding step of conven- +tional LLE, as LLE typically does not require anchors [30]. Thus, + +Conference’17, July 2017, Washington, DC, USA +Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari +100 +50 +0 +50 +100 +150 +200 +250 +Measurement error (m) +0 +2 +4 +6 +8 +10 +12 +14 +Frequency +LOS +NLOS +Missing +Figure 13: Measurement error for 18 node robotic network +in GPS-denied environment. +100 +50 +0 +50 +100 +150 +200 +200 +150 +100 +50 +0 +50 +100 +GCN (45.8) +100 +50 +0 +50 +100 +150 +200 +SMILE (34.22) +Figure 14: Localization accuracy of GCN (RMSE 45.8) and +SMILE (RMSE 34.22) on the robotic network. Edges indicate +error in distance. +our approach may be applicable to problems outside the sensor net- +works domain. For instance, consider a matrix of user data, where +each row corresponds to a specific user and a small number of +users obfuscate their data with the addition of non-negative noise. +SMILE could be applied to the problem of determining how similar +users are, by extracting the sparse obfuscation matrix and finding +an embedding of users in low-dimensional space. +We considered a modification to LLE in which we replace the +𝑘 nearest neighbors with the 𝑛𝐴 anchors as reference points. We +expected this would minimize the risk of errors propagating, as +each of the anchors is at a known location. Interestingly, we did +not see an improvement in performance. Similarly, we considered a +modification to the sparse matrix inference approach in which we +project X and Y on to the space of matrices which are non-negative +and symmetric. We expected this would improve performance as D +and S are non-negative and symmetric. Preliminary experiments did +not demonstrate a significant improvement in localization accuracy, +and in some cases accuracy suffered. Low-rank matrix approxima- +tion which constrains the solution to have the characteristics of +Euclidean distance matrix, without sacrificing localization accuracy, +is a direction for further study. +Following [36], we considered the probability of each link being +NLOS as independent. However, in realistic settings there will be +some correlation based on the structure of the environment itself. +For example, in Fig. 1 we illustrate 20 nodes in an indoor setting +with three rooms and one hallway. Nodes in the smaller rooms +are more subject to NLOS attenuation, given their proximity to +two walls. Currently, we estimate that links with the weakest RSS +are most likely to be NLOS. However it may instead be the case +that certain nodes are more likely to have NLOS links, is in Fig. 1. +Exploiting patterns like this in the structure of S is an interesting +direction for future work. +6 +CONCLUSION +We present Sparse Matrix Inference and Linear embedding, a novel +network localization algorithm which is robust to noisy, NLOS, +and missing measurements. Our approach outperforms the state of +the art on large, simulated networks, achieving high localization +accuracy with low computation times. We see promising results +for small networks in real-world settings, and in the future we’d +like to collect real-world data for larger networks. Other direc- +tions for future work include extensions to make the approach +distributed [4, 12], or consider time-series RSSI measurements and +Bayesian estimation [20, 35]. +ACKNOWLEDGMENTS +Thanks to Kiran Yedavalli for the sensor network data and Team +CoSTAR for the robotic network data. 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IEEE Transactions on Signal Processing 63, 6 (2015), +1448–1463. + diff --git a/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/load_file.txt b/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a17b0d98e4f211048e87a7a0ce1e2be2409f53f --- /dev/null +++ b/NtFJT4oBgHgl3EQfHCyo/content/tmp_files/load_file.txt @@ -0,0 +1,667 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf,len=666 +page_content='SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition Lillian Clark lilliamc@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='edu Electrical and Computer Engineering University of Southern California Los Angeles, California, USA Sampad Mohanty sbmohant@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='edu Computer Science University of Southern California Los Angeles, California, USA Bhaskar Krishnamachari bkrishna@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='edu Electrical and Computer Engineering University of Southern California Los Angeles, California, USA ABSTRACT Motivated by collaborative localization in robotic sensor networks, we consider the problem of large-scale network localization where location estimates are derived from inter-node radio signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Well- established methods for network localization commonly assume that all radio links are line-of-sight and subject to Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, the presence of obstacles which cause non-line-of-sight attenuation present distinct challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' To enable robust network localization, we present Sparse Matrix Inference and Linear Em- bedding (SMILE), a novel approach which draws on both the well- known Locally Linear Embedding (LLE) algorithm and recent ad- vances in sparse plus low-rank matrix decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We demon- strate that our approach is robust to noisy signal propagation, severe attenuation due to non-line-of-sight, and missing pairwise measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our experiments include simulated large-scale networks, an 11-node sensor network, and an 18-node network of mobile robots and static anchor radios in a GPS-denied limestone mine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our find- ings indicate that SMILE outperforms classical multidimensional scaling (MDS) which ignores the effect of non-line of sight (NLOS), as well as outperforming state-of-the-art robust network localiza- tion algorithms that do account for NLOS attenuation including a graph convolutional network-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We demonstrate that this improved accuracy is not at the cost of complexity, as SMILE sees reduced computation time for very large networks which is important for position estimation updates in a dynamic setting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='g for mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' KEYWORDS network localization, graph signal processing, low-rank matrix approximation 1 INTRODUCTION Robotic sensor networks operating in GPS-denied environments can benefit from collaborative localization [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' When distance measurements between network nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' mobile robots or static beacons) are available, the network localization problem seeks to exactly recover the positions of each node in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Intuitively, network localization algorithms leverage pairwise links to constrain the position estimate of every node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' If the locations of some nodes are known, these nodes are considered anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' When sufficient conditions on the number of anchors and the graph induced by the distance measurements are met, we can recover the positions of all nodes [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='nnnnnnn Figure 1: Illustration of network localization with NLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The objective is to accurately determine the localization of agents (grey) based on the known positions of anchors (yel- low) given the constrains of noisy LOS links (green) and links with additional NLOS attenuation (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, in realistic scenarios the distance measurements are typically noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For example, radio signals can be used to estimate distance based on received signal strength, but signals are subject to noise from the wireless channel [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' When this noise is assumed to be Gaussian with zero-mean1, the redundancy offered by many links in a highly connected network serves to mitigate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This explains why localization performance improves with the number of anchors used as reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For many applications, including robotic exploration of unknown environments, signals see significant degradation from the presence of walls and obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' These non-line-of-sight (NLOS) links are difficult to model without an a priori map but have a significant affect on the relationship between received signal strength (RSS) and distance [10, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Therefore, the ability to infer whether a pair of transmitters and receivers is NLOS can greatly improve network localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Additionally, when network nodes are spread over long distances, as may also be the case for robotic exploration, it is possible for the packets used to measure RSS to be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' When signal strength measurements are unavailable at long distances, our large-scale network localization approach must be robust to missing pairwise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Contribution: In this work, we examine the network localiza- tion problem in the presence of noisy signals where pairwise mea- surements may be NLOS or missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We make two key observations: (1) the Euclidean distance matrix of the true node positions (in 𝑑 dimensions) will have rank 𝑑 + 2, which we show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 3, and (2) 1Gaussian noise in dB is also referred to as log-normal fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='11450v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='SP] 26 Jan 2023 Conference’17, July 2017, Washington, DC, USA Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari the matrix capturing the additional NLOS attenuation is commonly sparse and non-negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=', there are a limited number of walls which can only degrade signal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Given these observations, our approach draws on recent advances in sparse plus low-rank de- composition to first extract the positively biased NLOS attenuation, which we refer to as sparse matrix inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' After extracting the NLOS attenuation, we leverage a popular method for dimension- ality reduction, locally linear embedding (LLE), to recover a set of weights which allow linear reconstruction to determine the exact positions of all nodes given the anchors’ positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Specifically, we extend the discrete optimization approach for sparse matrix recovery which is presented in [3] to (1) handle miss- ing pairwise measurements by imputation and (2) approximate the unknown sparsity structure of the sparse component of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Additionally, we provide details for solving for node po- sitions using the LLE weight matrix directly (using least squares method) without the need for additional eigen-decomposition and coordinate frame alignment used conventionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our algorithm combines recent advances in sparse plus low rank decomposition and well established and interpretable algorithms like MDS and LLE and outperforms them in the face of NLOS attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Evaluation: We demonstrate that SMILE significantly improves localization accuracy over baseline methods which ignore the af- fect of NLOS attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Further, we draw parallels between this approach and another method for graph signal processing which has recently been demonstrated as promising, namely Graph Con- volutional Networks (GCNs) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We compare the performance of SMILE and a state-of-the-art GCN implementation on large-scale simulated networks and demonstrate an improvement in localiza- tion accuracy and reduced computation time for network of more than 1000 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Finally, we evaluate performance on two real- world networks: an outdoor wireless sensor network with 11 nodes, and 5 mobile robots and 13 static radios in a GPS-denied limestone mine with significant NLOS attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In Section 2 we provide a brief overview of seminal and recent work in network localization, and in Section 3 we formally define the problem and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE is introduced and explained in detail in Section 4, including sparse matrix inference and subsequent steps for position estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We also draw parallels between our approach and the existing graph learning-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In Section 5 we provide implementation details as well as experimental results on real and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We give concluding remarks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORK Network localization: Network localization is well-researched, and is commonly formulated as a least squares problem [2, 9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Multidimensional scaling (MDS) and its extensions are popular methods for solving this problem [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Classical MDS uses the dis- tance matrix to compute a matrix of scalar products, typically called the Gram matrix, that captures pairwise correlation of the posi- tion vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The principal components from eigen-decomposition of this matrix are then used to recover relative node positions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Rather than compute over the entire distance matrix, Locally Linear Embedding (LLE) [27, 30] applies principal component analysis to Table 1: List of Abbreviations Abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Description EDM Euclidean Distance Matrix LLE Locally Linear Embedding MDS Multidimensional Scaling NLOS Non Line of Sight PSVD Partial SVD RMSE Root Mean Squared Error SDP Semi Definite Programming SVD Singular Value Decomposition small neighborhoods, which improves performance when the reduc- tion from the noisy (high-dimensional) data to the low-dimensional true positions is non-linear, and has shown promise in sensor net- work localization [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Exploiting sparsity: If the data is well-described by a particular statistical model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Gaussian or log-normal), we can instead form the maximum likelihood estimation problem [25], and solve using semi-definite programming (SDP) methods [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Recent works extend SDP methods to consider non-Gaussian noise [38] and, more specifically, NLOS noise [8, 19, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' demonstrate that when the NLOS noise has a certain structure, namely non-negative and sparse, a sparsity-promoting term in the objective function can improve the performance of SDP approaches [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, SDP methods suffer with respect to complexity and are intractable for very large networks [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Sparse and low rank decomposition: Another approach is to recover the matrix of NLOS attenuation directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In recent years, sparse and low-rank matrix recovery has drawn attention due to its relevance in signal processing, statistics, and machine learning [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Bertsimas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' recently proposed a discrete optimization approach to sparse and low-rank recovery which uses alternating minimiza- tion [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our work leverages this method, extends it to the case of missing measurements and unknown sparsity, and demonstrates that it can serve as an important component of a robust network localization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Graph convolutional networks: To effectively exploit the rela- tional information of graph-structured data, graph neural networks (GNNs) have recently become a popular method for approaching optimization problems in wireless networks [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' recently demonstrated promising results in the application of Graph Con- volutional Networks (GCNs) to the network localization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Their approach maintains accurate localization despite NLOS at- tenuation, and is scalable to large-scale networks at an affordable computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, we will see that the learned model is unable to exactly recover positions for a completely observed distance matrix in the absence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In this work we propose a novel network localization algorithm which builds on the principles of multidimensional scaling for exact recovery in the absence of noise, exploits the sparsity of NLOS attenuation for improved local- ization accuracy, and scales to very large networks at an affordable computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition Conference’17, July 2017, Washington, DC, USA 3 PROBLEM FORMULATION Let 𝐴 be the set of anchors whose positions are known, where |𝐴| = 𝑛𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Let 𝐵 be the set of agents whose positions are unknown, where |𝐵| = 𝑛𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Let 𝐶 = 𝐴 ∪ 𝐵 be the set of nodes which includes both anchors and agents, with 𝑛𝐴 + 𝑛𝐵 = 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We define the (𝑛 × 𝑛) true distance matrix D as D[𝑖, 𝑗] = ||p𝑖 − p𝑗 || where p𝑖 = [𝑥𝑖,𝑦𝑖]𝑇 is the position of node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We establish the convention that the first 𝑛𝐴 rows and 𝑛𝐴 columns pertain to the anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' D◦2 is the Euclidean distance matrix (EDM, containing squared distances i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e D squared entry-wise), which is zero along the diagonal and symmetric [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' If matrix P ∈ R𝑛×2 with P[𝑖, :] = p𝑇 𝑖 be the position matrix, we can see that D◦2 has rank 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In general, the EDM of a configuration of points embedded in R𝑑 has rank at most 𝑑 + 2 (and exactly 𝑑 + 2 if the points are in general position as opposed to more special or coincidental cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=', points in 3D which lie on a line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is briefly justified below, with further details provided in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' D[𝑖, 𝑗]2 = ||p𝑖 − p𝑗 ||2 = ⟨p𝑖 − p𝑗, p𝑖 − p𝑗⟩ = ⟨p𝑖, p𝑖⟩ + ⟨p𝑗, p𝑗⟩ − 2⟨p𝑖, p𝑗⟩ = p𝑇 𝑖 p𝑖 + p𝑇 𝑗 p𝑗 − 2p𝑇 𝑖 p𝑗 =⇒ D◦2 = diag(PPT)1T ������������������������ 𝑟𝑎𝑛𝑘−1 + 1diag(PPT) 𝑇 ������������������������ 𝑟𝑎𝑛𝑘−1 − 2PPT ���� 𝑟𝑎𝑛𝑘−2 Since𝑟𝑎𝑛𝑘(A+B) ≤ 𝑟𝑎𝑛𝑘(A)+𝑟𝑎𝑛𝑘(B) and𝑟𝑎𝑛𝑘(CCT) = 𝑟𝑎𝑛𝑘(CTC) = 𝑟𝑎𝑛𝑘(C), we have D◦2 as low-rank with rank = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Let O be the matrix that contains the distances that we observe, where O[𝑖, 𝑗] is a function of the strength of radio signal transmitted by node 𝑖 and received by node 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In general, our observation can be captured as O = Ω ◦ [D + N + S] where Ω is the observation mask and takes on values of 1 when a measurement is available and 0 otherwise (for instance, when the transmitter and receiver are out of range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' N is an asymmetric matrix capturing noise in the observations, which we assume is Gaussian and zero-mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' S captures the additional NLOS attenuation, and as in [19] we assume S is non-negative and sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We also assume S is symmetric which relies on assumptions that an attenuating obstacle will affect a radio link in both directions equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We make the realistic assumption that nodes are in general position, meaning that in 2D they do not lie on a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We also assume that an upper bound on the distance between any two nodes, 𝑑max, is known or can be approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our objective is finding an estimate for the locations of all nodes ˆP = [ˆp1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=', ˆp𝑛]𝑇 which is consistent with the prior information - (1) the observations in O as well as (2) the known anchor positions PA = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=', p𝑛𝐴]𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4 SMILE In this section we propose Sparse Matrix Inference and Linear Embedding, our novel large-scale network localization algorithm, which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Details of the algorithm are presented Anchor positions X Observations Complete the Euclidean distance matrix Sparse matrix inference Construct locally linear weight matrix Solve sparse sub-problem Solve low-rank sub-problem E Y Y (sparse NLOS) X Predicted locations W Figure 2: An overview of Sparse Matrix Inference and Linear Embedding (SMILE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1, we discuss our method of extract- ing NLOS attenuation via sparse matrix inference, which results in a low-rank approximation of the Euclidean distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Then in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2, we discuss our method of transforming this matrix into an estimate of the locations of all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 Sparse Matrix Inference For a given input matrix E ∈ R𝑛×𝑛, for which E = X + Y, sparse matrix inference seeks to find the low-rank component X ∈ R𝑛×𝑛 and sparse component Y ∈ R𝑛×𝑛 which solves: min X,Y 𝑓 (X, Y) = ||E − X − Y||2 𝐹 + 𝜆||X||2 𝐹 + 𝜇||Y||2 𝐹 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' rank(X) ≤ 𝛼, ||Y||0 ≤ 𝛽 (1) where ||X||𝐹 denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In this subsection we describe how to decompose E X and Y using alternating minimiza- tion, and how to use this technique on the problem defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Firstly, we square the observed matrix (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Temporarily as- suming N = 0 and no missing observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e Ω = 11T, E = O◦2 = (D + S)◦2 = D◦2 ���� low-rank + S◦2 + 2D ◦ S ���������������������� sparse (2) where the expression in parentheses is non-negative and sparse (because S is non-negative and sparse), and D◦2 is low-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Thus the problem is amenable to the sparse matrix inference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Let us consider the case with noise 𝑁 ≠ 0 and no missing obser- vations E = O◦2 = ( ˜D ���� D + N +S)◦2 = ( ˜D + S)◦2 = ˜D◦2 + S◦2 + ˜D ◦ S ������������������ sparse (3) The last expression is exactly the same as the formulation in 2 except that ˜D◦2 = D◦2 + N◦2 + 2N ◦ D may no longer be low rank due to the addition of N◦2 + 2N ◦ D where entries of the first term are now i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='d from a Chi-Squared Distribution X2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' in our ablation study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' we empirically verify that as long as the standard deviation of the normal noise in entries of N is not too large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' using the estimated low-rank matrix from the sparse matrix plus low-rank Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' USA Lillian Clark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Sampad Mohanty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' and Bhaskar Krishnamachari Table 2: SMILE parameters Parameter Symbol 𝑘 Number of neighbors ˆ𝛽𝑖 Initial sparsity estimate 𝜂 Step size 𝑇 Number of random initializations 𝜆 Low-rank matrix regularizer 𝜇 Sparse matrix regularlizer 𝜖 Inner loop tolerance 𝜖𝛽 Outer loop tolerance inference,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE is able to recover P fairly faithfully under RMSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Missing measurements: The approach presented in [3] as- sumes a complete input matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' While more complex approaches to matrix completion exist, for example by finding the sum total length of the shortest path between two nodes [32, 33], we take a simpler approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' When Ω ≠ 11T, we complete the observation matrix by filling in missing values with 𝑑max (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Even in the face of missing observations, we show that the problem remains amenable to the sparse matrix inference framework as long as the missing observations are not too large of a fraction of the total number of observations in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' E = ˜O◦2 = [Ω ◦ O]◦2 = [Ω ◦ (D + N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 = [Ω ◦ D + Ω ◦ (N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 = [D − (1 − Ω) ◦ D + Ω ◦ (N + S) + (1 − Ω)𝑑𝑚𝑎𝑥]◦2 = [D + Ω ◦ N + Ω ◦ S + (1 − Ω) ◦ (𝑑𝑚𝑎𝑥I − D) ������������������������������������������������������������������������ ˜𝑆 (sparse) ]◦2 = [D + Ω ◦ N ���������������� ˜D +˜S]◦2 = [ ˜D + ˜S]◦2 = ˜D◦2 + ˜S◦2 + 2 ˜D ◦ ˜S ���������������������� sparse (4) Thus, ˜O decomposes similarly to O as long as (1 − Ω) + S is still sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' the total number of NLOS and missing measurements is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' ˜D◦2 is still amenable to SMILE like in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Alternating minimization: As presented in [3], we alternate between solving two sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For a given X, we estimate Y by composing a sparse matrix with non-zero entries at the largest indices of (E − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is described further in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Then for a given Y, we estimate X by reducing the rank of (E − Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This repeats until the value of the objective function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 1 has con- verged, corresponding to the inner loop (lines 11-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We initialize X with a random, low-rank matrix and repeat this process for𝑇 > 0 random initialization, ultimately selecting the decomposition which minimizes the objective function (lines 5-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Unknown sparsity: The approach presented in [3] assumes the sparsity of Y is known, however in realistic settings this may not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We observed empirically that for a matrix with known compo- nents, solving the sparse matrix inference problem for increasing Algorithm 1 SMILE Input: O (observation matrix), P𝐴 (anchor positions), 𝛼 (desired rank) Output: ˆPB (agent location estimates) 1: ˜O = O + (1 + Ω)𝑑max 2: E = ˜O◦2 3: ˆ𝛽 ← ˆ𝛽𝑖 4: while Δ𝑓 /𝑓 > 𝜖𝛽 do 5: for 𝑡 in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='𝑇 } do 6: X′ ∈ R𝑛×𝑛 ← random 7: U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' V𝑇 = PSVD(X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 𝛼) 8: X = UΣV𝑇 9: Y ∈ R𝑛×𝑛 ← 0 10: 𝑓 = ||E − X − Y||2 𝐹 + 𝜆||X||2 𝐹 + 𝜇||Y||2 𝐹 11: while Δ𝑓 /𝑓 > 𝜖 do 12: Y′ = compose_sparse(E − X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' ˆ𝛽) 13: Y = 1 1+𝜇 Y′ ◦ (E − X) 14: X′ = 1 1+𝜆 (E − Y) 15: U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' V𝑇 = PSVD(X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 𝛼) 16: X = UΣV𝑇 17: 𝑓 = ||E − X − Y||2 𝐹 + 𝜆||X||2 𝐹 + 𝜇||Y||2 𝐹 18: end while 19: end for 20: ˆ𝛽 ← ˆ𝛽 + 𝜂 21: end while 22: W ∈ R𝑛×𝑛 ← 0 23: for each node 𝑖 do 24: find 𝑘 nearest neighbors 𝑁𝑁 (𝑖) 25: for each pair of neighbors (𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 𝑙) do 26: H ∈ R𝑘×𝑘 where H[𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='𝑙] = 1 2 (X[𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='𝑙] + X[𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='𝑖] − X[𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='𝑙]) 27: solve Hw = 1 for w 28: end for 29: W𝑖 ← w/� w at indices of neighbors 30: end for 31: m = (I − W)𝐴P𝐴 32: solve (W − I)𝐵 ˆP𝐵 = m for ˆP𝐵 33: return ˆPB Algorithm 2 compose_sparse Input: M ∈ R𝑛×𝑛 (matrix),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 𝛽 (sparsity) Output: B (binary matrix) 1: sorted indices = argsort(M) 2: one indices = sorted indices[-𝛽:] 3: B ∈ R𝑛×𝑛 ← 0 4: B[one indices] = 1 5: return B estimate ˆ𝛽 causes the objective function to decrease,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' and around the true value of 𝛽 it converges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This motivates the outer loop (lines 3-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' To reduce computation time, we can initialize ˆ𝛽 with the approximate sparsity, increase the search step size 𝜂, or increase the converge tolerance 𝜖𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However this may come at the cost of localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition Conference’17, July 2017, Washington, DC, USA 0 20000 40000 60000 80000 100000 120000 140000 0 1 2 3 4 5 Objective function f 1e8 Figure 3: The final value of the sparse matrix inference ob- jective function 𝑓 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 1) when we assume different levels of sparsity ˆ𝛽, plotted alongside the true sparsity 𝛽 (dotted line) for various simulated matrices E ∈ R500×500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 Linear Embedding Next we find the set of k nearest neighbors 𝑁𝑁 (𝑖) for each node 𝑖 using the estimated EDM gotten from sparse matrix inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For each node, we compute a 𝑘 × 𝑘 matrix H which captures the pairwise similarity between the nodes in 𝑁𝑁 (𝑖) ∪ {𝑖} (line 25) using the corresponding sub-matrix in the estimated EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' H is calculated using the same approach used for classical MDS, but for local neighborhoods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' From H we can recover the desired weight matrix W (lines 26-27), first by solving for w and then inserting it at the row position corresponding to the node index of 𝑖 with the entries in w aligned with the column belonging to the nearest neighbours and zeros for non-neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' At this point, each row of W corresponds to a node and captures how the node’s position can be expressed as a linear combination of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is a meaningful representation of the positions of all nodes, but is not yet a node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Typically, LLE then computes a sparse matrix (I − W)𝑇 (I − W) whose eigenvectors corresponding to the two smallest non-zero eigenvalues result in a solution up to rotation and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Using the anchors’ positions: In this setting, we can compute a solution using W and the anchor location P𝐴 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is possible because, by construction, WP = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is essentially a system of 𝑛 equations with 𝑛𝐵 < 𝑛 unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Treating W − I ∈ R𝑛×𝑛 as a block matrix, let (I − W)𝐴 = (I − W)[:, : nA] ∈ R𝑛×𝑛𝐴 be the sub-matrix corresponding to the anchors and (W − I)𝐵 = (I − W)[:, −nB :] ∈ R𝑛×𝑛𝐵 be the sub-matrix corresponding to the agents, where I ∈ R𝑛×𝑛 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Similarly treating 𝑃 ∈ R𝑛×2 as a block matrix, let 𝑃𝐴 = 𝑃 [: 𝑛𝐴, :] ∈ R𝑛𝐴×2 and 𝑃𝐵 = 𝑃 [−𝑛𝐵, :] ∈ R𝑛𝐵×2 correspond to the block matrices for positions of anchors and agents respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' With some manipulation we get, WP = P =⇒ 𝑇 ���������� (W − I) P = 0 =⇒ TP = 0 =⇒ �TA | TB � \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 PA − PB \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = 0 =⇒ TAPA + TBPB = 0 =⇒ TAPA = −TBPB =⇒ (I − W)𝐴P𝐴 ���������������������� m, known = (W − I)𝐵 �������������� known P𝐵 ���� unknown (5) (lines 29-30) and from the last expression we can estimate a solu- tion ˆPB for the unknown agent positions using the least squares minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3 Comparison with Graph Convolutional Networks We highlight several interesting parallels between SMILE and GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Briefly, the approach presented by Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' computes the position estimates ˆP = A′𝜙(A′(A ⊙ O)Z(1))Z(2) (6) where A is the thresholded adjacency matrix for a given threshold 𝜃𝐺𝐶𝑁 , A′ is the row-normalized augmented thresholded adjacency matrix, 𝜙(·) is a nonlinear activation function, and Z(1), Z(2) are learned weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Note that the graph signal is the observed matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (1) 𝜃𝐺𝐶𝑁 and 𝑘: This approach introduces a threshold such that edges between nodes in the graph are present only if the observed distance is less than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Decreasing this threshold is similar to decreasing the number of nearest neighbors, and both have the benefit of noise truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, as the threshold is increased, GCN experiences over-smoothing due to the aggregation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Intuitively, if every node is connected to every other node, the aggregation step causes all node embedding to collapse at a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This means that more information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=', additional pairwise measurements, actually hurts the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE does not experience this issue, and we will see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 6 that increasing 𝑘 improves performance at the cost of increased runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (2) Low-pass filtering and PSVD: Repeated multiplication by the normalized adjacency matrix acts as a low pass filter [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This reduces the rank of the graph signal, similar to the process of extracting the low-rank component of the noisy euclidean distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, repeated multiplication by the normalized adjacency matrix reduces the rank of the observed matrix by some amount, which corresponds to dis- tances (rather than squared distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our approach applies rank reduction to exactly reduce the Euclidean distance ma- trix to rank 𝑑 + 2, informed by the principles of the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (3) Convolution and linear embedding: Each convolution layer multiplies the adjacency matrix and input matrix by a set of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Intuitively, this makes the node features at the next layer a weighted sum of the neighbor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The learned GCN weights are analogous to the set of weights W in SMILE which allow linear reconstruction of agents from Conference’17, July 2017, Washington, DC, USA Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari the anchors positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, the repeated convolutions and nonlinear activation prevent a straightforward analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The interpretability of W is an advantage of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Qualitatively, we expect SMILE to outperform GCN because (1) it is not subject to oversmoothing and makes careful and productive use of additional pairwise measurements, (2) it finds a Euclidean distance matrix with the exact expected rank, and (3) it relies on principled methods to determine a weight matrix relating nodes to their neighbors, which is thus interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In the next section, we compare these approaches quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5 RESULTS In this section we evaluate the performance of SMILE with respect to localization accuracy and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We also consider performance under different noise settings, as an ideal method is both robust to high-levels noise and able to exactly recover positions when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Localization accuracy is measured by the root-mean- squared-error (RMSE) given by ||P𝐵 − ˆP𝐵||𝐹 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Robustness considers the affect of Guassian noise, NLOS attenuation, and missing pair- wise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The experiments consider networks in 2D, thus the desired rank of EDM is 𝛼 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The SMILE parameters are set to 𝑘 = 50, ˆ𝛽𝑖 = 5 𝑛2 100,𝜂 = 𝑛2 100,𝑇 = 1, 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='01, 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1,𝜖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='001, and 𝜖𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For comparison, we train a two-layer GCN according to [36] with distance threshold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' More details of our implemen- tation are available online 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 Simulation results Our simulated scenarios consider 𝑛 nodes randomly placed over a 5m × 5m square area, with the first 𝑛𝐴 nodes considered anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Noise N is drawn from a zero-mean Gaussian, N[𝑖, 𝑗] ∼ N (0, 𝜎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' NLOS noise is drawn from a uniform distribution, S[𝑖, 𝑗] = S[𝑗,𝑖] ∼ U[0, 10] with probability 𝑝NLOS and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Both matrices are zero along the diagonal, N[𝑖,𝑖] = S[𝑖,𝑖] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The observation mask limits our observed distance measurements by a threshold 𝜃 such that Ω[𝑖, 𝑗] = 1 if O[𝑖, 𝑗] ≤ 𝜃 and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Comparison with state of the art: Firstly we consider the setting where 𝑛 = 500,𝑛𝐴 = 50, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1, 𝑝NLOS = 1 10, and 𝜃 = 𝑑max, for which an example dataset is available3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4 illustrates the performance of SMILE, which achieves an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='06 in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='22 seconds, and GCN which achieves an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='11 in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='73 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' GCN’s performance is consistent with that reported in [36], which reports RMSE for various other methods, including SDP with sparsity promoting regularization [19] which achieves an RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE achieves the highest reported localization accuracy, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5 illustrates that not only is the error low on average but the error density has a smaller tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 6, we investigate the localization accuracy and compu- tation time as we vary 𝑘, the number of neighbors used for linear embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We observe that RMSE remains consistent for 𝑛𝐴 ≥ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' While the minimum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='048 is reached at 𝑘 = 130, this comes at the expense of increased computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Note that selecting 𝑘 is analogous to setting the GCN threshold, however we do not observe the degradation in performance that GCN is prone to when the threshold is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This comes from the aggregation component 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='com/ANRGUSC/smile-network-localization 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='com/Yanzongzi/GNN-For-localization 0 1 2 3 4 5 0 1 2 3 4 5 GCN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='11) true anchor pred anchor true node pred node 0 1 2 3 4 5 SMILE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='06) true anchor pred anchor true node pred node Figure 4: Localization with GCN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' [36]) and novel SMILE for a 500 node network with 50 anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' GCN achieves RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='11 and SMILE achieves RMSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5 Error 0 20 40 60 80 100 120 Frequency GCN SMILE Figure 5: Error density for GCN and SMILE on the data from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 4, where SMILE results in a more desirable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 Number of neighbors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5 RMSE 8 9 10 11 12 Runtime (sec) Figure 6: RMSE (solid line) and runtime (dashed line) trade- off as we vary the number of neighbors 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Each point is the average of 10 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' of convolution, which causes over-smoothing if a node has too many neighbors and results in the position estimates collapsing to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Robustness: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 7 demonstrates the performance of SMILE and GCN as 𝑝NLOS varies from 0 to 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' From this, we observe that SMILE outperforms GCN for up to 30% NLOS links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Beyond this, the as- sumption that S is sparse is no longer true, and performance suffers as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The same is true when 𝜃 falls below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5, and (1 − Ω) is no longer sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e not enough entries in the observations matrix SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition Conference’17, July 2017, Washington, DC, USA 0 10 20 30 40 50 pNLOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 RMSE SMILE GCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='15 RMSE SMILE GCN 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='0 RMSE SMILE GCN Figure 7: Performance of SMILE and GCN as the probabil- ity of NLOS links, standard deviation of Gaussian noise, and threshold for distances which can be measured are varied (𝑇 = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' ˜O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Notably, we do not observe a clear trend in the performance of GCN as 𝜎 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We posit that this is because the learned approach does not rely on exact decomposition, even in the absence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' To test this, we compare the performance of GCN and SMILE on an ideal dataset and consider the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' While the performance of both methods is better in this ideal setting, we observe that GCN cannot exactly recover positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' While being robust to Gaussian noise and sparse non-Gaussian noise, SMILE is also accurate in ideal scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Complexity: As we increase the number of nodes, accuracy of both methods increases (accuracy similarly increases with the percentage of anchors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, complexity increases with the size of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Because our approach is iterative, we measure complexity numerically (computation time) in lieu of analytically as it is difficult to predict when the sparse matrix inference will con- verge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 9 shows that the computation time of both SMILE and GCN remain reasonable as 𝑛 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For least squares optimiza- tion and SDP, we have seen that the compute time for very large networks becomes intractable [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For the 1000 node network, the time to predict ˆP using SMILE is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='84 seconds while the time to train and predict ˆP using GCN is 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='90 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Note that this is a fair comparison because the learned approach requires training a specific model for each network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' the same learned weights are not applicable to a new network, and neither is the SMILE weight matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The low compute times for SMILE are likely because the linear embedding component depends on the number of neighbors (rather than the number of nodes) and the sparse matrix inference 0 1 2 3 4 5 0 1 2 3 4 5 GCN ideal (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='12) 0 1 2 3 4 5 0 1 2 3 4 5 SMILE ideal (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='0) 0 1 2 3 4 5 0 1 2 3 4 5 GCN noisy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='19) 0 2 4 0 1 2 3 4 5 SMILE noisy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='1) actual a predicted a actual predicted Figure 8: Performance of SMILE and GCN in an ideal setting (𝑝NLOS = 0, 𝜎 = 0) and a noisy setting (𝑝NLOS = 1 10, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE is robust to noise without sacrifice performance in ideal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' component adapts the step size to the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We high- light that scalability is not at the cost of localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For 𝑛 = 1500, SMILE achieves RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='05 while GCN achieves RMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Ablation: Thus far we have compared our approach to an exist- ing solution, but it is also interesting to consider the role of each component of SMILE and, in particular, whether simpler existing approaches from the literature are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 10 considers a dataset with 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3, 𝑝NLOS = 1 10, and 𝜃 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We compare the performance of (A) no rank reduction, (B) rank reduction via PSVD, and (C) sparse and low-rank matrix recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In conjunction, we consider (1) classical MDS with the Kabsch algorithm for coordinate system registration [1] and (2) LLE-based embedding using anchor positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Specifically, the top left plot represents a naive baseline: multidimensional scaling assuming Gaussian zero-mean noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The bottom right plot represents SMILE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We observe several key takeaways from this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (1) De-noising: Moving from left to right, each column of this plot contains increasing more sophisticated noise reduction and decreases the final RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In particular, sparse matrix inference increases localization accuracy by almost an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (2) Nearest neighbors: Methods in the top row use all avail- able links in eigen-decomposition, while methods in the bottom row use only the nearest neighbors to determine weights for linear reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This local-neighborhood approach consistently improves localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Note Conference’17, July 2017, Washington, DC, USA Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari 0 200 400 600 800 1000 1200 1400 Number of Nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='8 RMSE SMILE GCN 0 200 400 600 800 1000 1200 1400 Number of Nodes 0 10 20 30 40 Runtime (sec) SMILE GCN Figure 9: Compute time for different sizes of networks for GCN and SMILE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Previous work has shown that learned ap- proaches (GCN, MLP, NTK) scale better than optimization approaches (LS, ECM, SDP) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE outperforms GCN for very large networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' that if 𝜃 ≥ 𝑑max, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' no measurements are missing, the per- formance of (1) and (2) are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This is likely because LLE’s strength comes from relying more on nearby measure- ments, and our approach to matrix completion puts missing measurements at effectively long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In setting where long-distance measurements are unreliable, which is com- monly the case in realistic wireless communication [31], using nearest neighbors is advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' (3) SMILE: The combination of both sparse matrix inference and linear embedding is more robust to NLOS attenuation and missing measurements than either component in isola- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2 Experimental results In this section we apply SMILE to two real-world small scale datasets, and discuss its performance, our findings, and directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Sensor network: First we consider a network of 11 Mica2 motes placed randomly in a parking lot [37] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' The maximum distance between any two nodes is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='57m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We consider received signal strength (RSS) measurements between pairs (RSS is averaged over 20 packets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' To estimate distance, we use the following log-distance path loss model 𝑅𝑆𝑆(𝑑) = 𝑃tx − 𝑃𝐿(𝑑0) − 10𝛾 log10( 𝑑 𝑑0 ) + N (0, 𝜎2) (7) where 𝑃tx is the transmit power, 𝑃𝐿(𝑑0) is the path loss at a refer- ence distance, and 𝛾 is the path loss exponent [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For this data, use the model 𝛾 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='9 and 𝑃tx − 𝑃𝐿(𝑑0) = −49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='12𝑑𝐵 for 𝑑0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Given the ground truth location estimates, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 11 shows that the 4http://anrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='edu/www/download_files/RSSLocalizationDataSet_11nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='txt noise in this dataset has mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='35m with standard deviation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Note that even though we believe this data to be in open space (all LOS), seven links see a measurement error of greater than 5m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Figure 12 shows the performance of GCN and SMILE on this network, where we update the SMILE parameters to 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='05, 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='05, and𝑇 = 10 because the network is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' If we assume four of these nodes are anchors, GCN achieves RMSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='67 with threshold 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='2m, and SMILE achieves a comparable RMSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='63 with 𝑘 = 3, guessing 𝛽 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' These results are the average performance of 10 trials, and the best parameters for each algorithm were selected empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Robotic network: Secondly, we consider measurements from a robotic network with 18 nodes, 13 of which are stationary bea- cons at known locations, and 5 of which are mobile (legged and wheeled) robots5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Each robot carries a Streamcaster 4200 from Sil- vus Technologies, which are also used as beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We consider a set of pairwise RSS measurements from a single timestamp of robotic exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' They nodes are spread over a large area in an underground limestone mine with distances between nodes of up to 341 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Ground truth positions are available via a simultane- ous localization and mapping algorithm, and we assume these are accurate [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Estimates of whether links are LOS is also available via a LiDAR-based predictive model [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 13 plots the error density for LOS, NLOS, and missing links from this data, where we note that unfortunately only 14% of measurements are available and LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We augment this real dataset with simulated data such that missing measurements are sampled from a zero-mean Gauss- ian with standard deviation equivalent to the observed LOS noise (𝜎 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 14 show the results of our approach on this realistic dataset, where edges indicate error in distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' GCN achieves RMSE of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='8 on this simulated data with threshold 110, while SMILE achieves an improved RMSE of 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='22 for 𝑘 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' These results are the average performance of 10 trials, and the best parameters for each algorithm were selected empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We observe that the average localization error for unknown nodes is roughly the standard deviation of the noise on LOS links, while the standard deviation of NLOS error is 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='60 (mean 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='14m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This indicates that SMILE is able to achieve localization accuracy comparable to a distance-based model on a single LOS link for all nodes, even those which have many NLOS neighbors, and shows promising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' During robotic exploration, these anchors were deployed au- tonomously [15, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This means anchors are only located in places the robots have already visited, while the exploration objective encourages the robots to move away from these anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In fact, the exploring robots then tend to be outside the convex hull de- fined by the anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This configuration appears to stress network localization, especially for smaller networks with significant noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Prior work exists in robotic motion which minimizes localization uncertainty [6, 11], and network localization algorithms which specifically address this geometric setting may be an interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='com/NeBula-Autonomy SMILE: Robust Network Localization via Sparse and Low-Rank Matrix Decomposition Conference’17, July 2017, Washington, DC, USA 0 2 4 6 6 4 2 0 2 4 6 MDS (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='49) 0 2 4 0 1 2 3 4 5 PSVD + MDS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='25) 0 2 4 0 1 2 3 4 5 SMI + MDS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='19) 2 0 2 4 6 2 0 2 4 6 8 LLE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='16) 0 2 4 0 1 2 3 4 5 PSVD + LLE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='19) 0 2 4 0 1 2 3 4 5 SMILE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='12) actual a predicted a actual predicted Figure 10: Ablation study with 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3,𝜃 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Top row, left to right: classical MDS achieves RMSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='49, de-noising via PSVD improves RMSE to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='25, and sparse matrix inference further improves RMSE to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Bottom row, left to right: LLE with direct position recovery achieves RMSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='16, de-noising via PSVD improves RMSE to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='19, and and sparse matrix inference further improves RMSE to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5 0 5 10 15 Measurement error (m) 0 2 4 6 8 10 Frequency Figure 11: Measurement error for 11 node wireless sensor network outdoors, comparing distance from RSS (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 7) with ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='3 Discussion One advantage of SMILE is that we compute Y which approximates the sparsity structure of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Therefore, we can use this method to estimate which links are NLOS and where walls, obstacles, or other potentially adversarial sources of attenuation may be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This means SMILE could be useful as a complementary modality for simultaneous localisation and mapping (SLAM) algorithm for multi- robot systems, and has potential applications in robotic exploration for disaster mitigation and military applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 0 2 4 6 8 0 2 4 6 8 10 12 14 GCN (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='6) true anchor pred anchor true node pred node 0 2 4 6 8 SMILE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='63) true anchor pred anchor true node pred node Figure 12: Localization accuracy of GCN (RMSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='6) and SMILE (RMSE of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='63) on the sensor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Edges in the left graph represent observed distances less than 𝜃GCn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Edges in the right graph connect each node 𝑖 to its nearest neighbors 𝑁𝑁 (𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' While it is impossible to unique localize a network in the ab- sence of anchors (given the possibility of translations and rotations), anchor-less localization problems have significant overlap with other useful signal processing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' In face, SMILE could be directly applied to anchor-less localization problems by replacing lines 29-30 in the algorithm with the embedding step of conven- tional LLE, as LLE typically does not require anchors [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Thus, Conference’17, July 2017, Washington, DC, USA Lillian Clark, Sampad Mohanty, and Bhaskar Krishnamachari 100 50 0 50 100 150 200 250 Measurement error (m) 0 2 4 6 8 10 12 14 Frequency LOS NLOS Missing Figure 13: Measurement error for 18 node robotic network in GPS-denied environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 100 50 0 50 100 150 200 200 150 100 50 0 50 100 GCN (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='8) 100 50 0 50 100 150 200 SMILE (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='22) Figure 14: Localization accuracy of GCN (RMSE 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='8) and SMILE (RMSE 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content='22) on the robotic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Edges indicate error in distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' our approach may be applicable to problems outside the sensor net- works domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For instance, consider a matrix of user data, where each row corresponds to a specific user and a small number of users obfuscate their data with the addition of non-negative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' SMILE could be applied to the problem of determining how similar users are, by extracting the sparse obfuscation matrix and finding an embedding of users in low-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We considered a modification to LLE in which we replace the 𝑘 nearest neighbors with the 𝑛𝐴 anchors as reference points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We expected this would minimize the risk of errors propagating, as each of the anchors is at a known location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Interestingly, we did not see an improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Similarly, we considered a modification to the sparse matrix inference approach in which we project X and Y on to the space of matrices which are non-negative and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We expected this would improve performance as D and S are non-negative and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Preliminary experiments did not demonstrate a significant improvement in localization accuracy, and in some cases accuracy suffered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Low-rank matrix approxima- tion which constrains the solution to have the characteristics of Euclidean distance matrix, without sacrificing localization accuracy, is a direction for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Following [36], we considered the probability of each link being NLOS as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However, in realistic settings there will be some correlation based on the structure of the environment itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 1 we illustrate 20 nodes in an indoor setting with three rooms and one hallway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Nodes in the smaller rooms are more subject to NLOS attenuation, given their proximity to two walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Currently, we estimate that links with the weakest RSS are most likely to be NLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' However it may instead be the case that certain nodes are more likely to have NLOS links, is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Exploiting patterns like this in the structure of S is an interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 6 CONCLUSION We present Sparse Matrix Inference and Linear embedding, a novel network localization algorithm which is robust to noisy, NLOS, and missing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Our approach outperforms the state of the art on large, simulated networks, achieving high localization accuracy with low computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' We see promising results for small networks in real-world settings, and in the future we’d like to collect real-world data for larger networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' Other direc- tions for future work include extensions to make the approach distributed [4, 12], or consider time-series RSSI measurements and Bayesian estimation [20, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' ACKNOWLEDGMENTS Thanks to Kiran Yedavalli for the sensor network data and Team CoSTAR for the robotic network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' This work was funded in part by NASA Space Technology Research Fellowship Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 80NSSC19K1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' REFERENCES [1] Jonathan Bachrach and Christopher Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} +page_content=' 2005.' metadata={'source': 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(2015), 1448–1463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQfHCyo/content/2301.11450v1.pdf'} diff --git a/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/2301.04048v1.pdf.txt b/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/2301.04048v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e683d627720a7540e373867a9cdcf5bbe6d3905 --- /dev/null +++ b/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/2301.04048v1.pdf.txt @@ -0,0 +1,736 @@ +arXiv:2301.04048v1 [math.OC] 10 Jan 2023 +A Sufficient Condition for the +Super-linearization of Polynomial Systems +M.-A. Belabbas∗ and +Xudong Chen† +Abstract +We provide in this paper a sufficient condition for a polynomial +dynamical system ˙x(t) = f(x(t)) to be super-linearizable, i.e., to be +such that all its trajectories are linear projections of the trajectories +of a linear dynamical system. The condition is expressed in terms of +the hereby introduced weighted dependency graph G, whose nodes vi +correspond to variables xi and edges vivj have weights ∂fj +∂xi . We show +that if the product of the edge weights along any cycle in G is a constant, +then the system is super-linearizable. The proof is constructive, and we +provide an algorithm to obtain super-linearizations and illustrate it on +an example. +1 +Introduction +The idea of linearizing system dynamics via embeddings dates back at least +to the works of Carleman [1] and Koopman [2, 3]. +These embeddings are +still actively studied a century later, and have found applications in nonlinear +control [4,5], and data-driven methods in control [6,7]. +We derive in this paper a sufficient condition under which a polynomial +system can be globally linearized by embedding it into a higher, yet finite- +dimensional vector space. In particular, the contribution of this paper is to +provide a generalized converse of the result established in [8]. We elaborate +on this below. +To proceed, we consider the following dynamical system: +˙x = f(x) +(1) +∗M.-A. Belabbas is with the Electrical and Computer Engineering Department and +the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign. +Email: +{belabbas@illinois.edu. +†X. Chen is with the Department of Electrical, Computer, and Energy Engineering, +University of Colorado Boulder. Email: xudong.chen@colorado.edu. +M.-A. Belabbas and X. Chen contributed equally to the manuscript in all categories. +1 + +where x ∈ Rn. This system is said to admit a super-linearization (see Defini- +tion 1 below) if there exist m ≥ 0 functions, called observables, which when +adjoined to the original system would permit its linearization. A typical ex- +ample [9] is the following two-dimensional system +� +˙x = −x + y2 +˙y = −y +(2) +Adding the observable w := y2, whose total time derivative is given by ˙w = +2y ˙y = −2y2 = −2w, we obtain the three-dimensional linear system: + + + + + +˙x = −x + w +˙y = −y +˙w = −2w. +(3) +Observe that the variables on which the nonlinear part of the dynamics (2) +depend (here, the variable y) evolve in a linear, autonomous (i.e., indepen- +dent from x) manner. In a recent paper [8], we showed that if a polynomial +system admits a so-called balanced super-linearization with only one visible +observable [10], then there exists a linear change of variables under which the +nonlinear part of the dynamics depends solely on variables evolving linearly +and autonomously. The dynamics resulting from the change of variable are +termed the canonical form [8] for the polynomial system (explicitly, the canoni- +cal form is given in (6) below), and its existence provides a necessary condition +for the super-linearization of that special class of polynomial systems. +Conversely, we exhibit in this paper a sufficient condition for the super- +linearization of general polynomial systems, without any restriction on the +number of visible observables. In particular, the result of this paper, combined +with the ones of [8], provide a necessary and sufficient condition for a class of +polynomial systems with a single visible observable to be super-linearizable. +The remainder of the paper is organized as follows: We describe the relevant +terminology and notation at the end of this section. We present the main result +in Section 2 and its proof in Section 3. The paper ends with a summary and +outlook. +Terminology and notation used. +We let G = (V, E) be a directed graph +(possibly with self-loops), with V the node set and E the edge set. We use +e = vivj to denote a directed edge of G from node vi to node vj (if vi = vj, +then e is a self-loop). A walk is a sequence of nodes w = vi1vi2 . . . vik such that +viℓviℓ+1 is an edge of G for each ℓ = 1, . . . , k − 1. The length of a walk is the +number of edges it traverses. A path is a walk which does not visit a node +more than once. We call the depth of G the length of the longest path in G. +2 + +For a dynamical system ˙x(t) = f(x(t)), we denote by etfx0 the solution +of the system at time t with initial state x0. For a vector field g : Rn → Rn +and a differentiable vector-valued function p : Rn → Rk, we denote the Lie +derivative of p along g by Lgp := ∂p +∂xg. +2 +Statement of the Result +We start by defining what it means for system (1) to be super-linearizable. Let +m ≥ 0 be an integer, and Π : Rn+m → Rn be the canonical projection onto +the first n variables, namely, we have for z ∈ Rn+m that Π(z) = (z1, . . . , zn). +We reproduce the following definition from [10]: +Definition 1 (Super-linearization). The vector field f : Rn → Rn is super- +linearizable to the system ˙z = Az + D with A ∈ R(n+m)×(n+m) and D ∈ Rn+m +if there exists an injective map p : Rn → Rm so that for all x0 ∈ Rn, the +following holds: +Π +� +et(Az+D)z0 +� += etfx0 with z0 = (x0, p(x0)). +(4) +We call the functions p : Rn → Rm the observables. +The data of A, D and p is referred to as a super-linearization of f. We can +express the relation (4) as the following commutative diagram +Rn +Rn +Rn+m +Rn+m +etf +(id, p) +et(Az+D) +Π +We next introduce the following notion: +Definition 2 (Weighted dependency graph). Let f : Rn → Rn be a differen- +tiable vector field. The weighted dependency graph (WDG) G = (V, E, γ) +of f is a weighted directed graph (with self-loop) on n nodes v1, . . . , vn. For +every ordered pair (vi, vj), we define the scalar function: +γij(x) := ∂fj(x) +∂xi +for 1 ≤ i, j ≤ n. +There is an edge vivj in G if γij ̸= 0, and its weight is γij. +We illustrate the definition on the following example: +3 + +v3 +v4 +v5 +v2 +v1 +−1 +2x3 +2x1x2 +x2 +1 +2x2 +1 +x2 +2 +2x1x2 +−1 +1 +Figure 1: The weighted dependency graph of system (5). +Example 1. Consider the following polynomial system: + + + + + + + + + + + + + + + +˙x1 = x2 +˙x2 = −x1 +˙x3 = x2 +2 +˙x4 = x3 + x1x2 +2 +˙x5 = −x5 + x2 +3 + x2 +1x2. +(5) +Its weighted dependency graph is depicted in Figure 1. +Next, for each directed walk w = vi1 . . . vik in G, we let +γw := +k−1 +� +j=1 +γijij+1. +In the sequel, we will assume that f is a polynomial vector field. It should be +clear that γw(x), for any walk w, is then a polynomial function in x. Also, we +assume, without loss of generality, that G is weakly connected (otherwise, the +original system can be decoupled into sub-systems of lower dimensions and +our result, stated below, can be applied to each sub-system independently). +The main result of this paper is as follows: +Theorem 1. For a polynomial system ˙x(t) = f(x(t)), let G be the associated +weighted dependency graph. If γc is a constant for every cycle c of G, then f +is super-linearizable. +The sufficient condition stated in the Theorem implies that the Jacobian +of f is constant. +Note that the weighted dependency graph of system (5), +4 + +depicted in Figure 1, satisfies the sufficient condition of Theorem 1, and thus +system (5) is super-linearizable. As an illustration of the proof technique used, +we will provide toward the end a super-linearization of this system. +3 +Proof of Theorem 1 and an Algorithm +3.1 +Proof of the Theorem +We start with a simple proposition, dealing with systems where the variables +on which the nonlinear part of the dynamics depend evolve linearly and au- +tonomously, and show that such systems are super-linearizable. This result +provides a converse of the result of [8], and will be used as a building block to +establish the general case. +Proposition 2. Suppose that the system ˙x(t) = f(x(t)) takes the following +form: +� +˙x′(t) = A′x′(t) + D +˙x′′(t) = A′′x′′(t) + g(x′(t)), +(6) +where x = (x′; x′′), D is a constant vector, and g is a polynomial; then, sys- +tem (6) is super-linearizable. +Proof. Let n′ be the dimension of x′ and d be the degree of g. Let Pd be the +vector space of all polynomials in x′ with real coefficients, whose dimension is +r := dim Pd = +�n′ + d +d +� +. +Next, for convenience, we let f ′(x′) := A′x′ + D. Since f ′ is affine, Lf′φ ⊆ Pd +for any φ ∈ Pd and, hence, Lf′ : Pd → Pd is a linear automorphism. Let the +minimal polynomial associated with Lf′ be given by +sN + αN−1sN−1 + · · · + α0 +for some N ≤ r. In particular, for any φ ∈ Pd, we have that +(LN +f′φ) + αN−1(LN−1 +f′ +φ) + · · · + α0φ = 0. +Now, define +p(x) = + + +p1(x) +p2(x) +... +pN(x) + + := + + +g(x′) +Lf′g(x′) +... +LN +f′g(x′) + + . +(7) +5 + +u1 +u2 +u3 +u4 +Figure 2: The skeleton graph S of the WDG G of system (5), depicted in +Figure 1. Note that π−1(u1) = {v1, v2}, π−1(u2) = {v3}, π−1(u3) = {v4}, and +π−1(u4) = {v5}. +It then follows that the time derivative of p(x(t)) is +d +dt + + +p1 +p2 +... +pN−1 +pN + + += + + +0 +I +0 +· · · +0 +0 +0 +I +· · · +0 +... +... +... +... +... +0 +0 +· · · +0 +I +−α0I +−α1I +· · · +−αN−2I +−αN−1I + + + + +p1 +p2 +... +pN−1 +pN + + +. +(8) +This completes the proof. +We next introduce two notions that are necessary for enabling the recursive +use of Proposition 2 in the proof of the main theorem. The first is the notion +of strong component decomposition. +Definition 3 (Strong component decomposition). Let G be a weakly connected +digraph. The subgraphs Gi = (Vi, Ei), for 1 ≤ i ≤ q, form a strong component +decomposition of G if the following items hold: +1. The Vi’s partition the vertex set as V = ⊔q +i=1Vi; +2. Each Gi is a subgraph induced by Vi and is strongly connected; +3. Any strongly connected subgraph G′ of G is a subgraph of some Gi, for +i ∈ {1, . . . , q}. +By treating the strongly connected components Gi as single nodes, we +obtain the second notion, namely the one of skeleton graph S of G: +Definition 4. Let G = (V, E) be a weakly connected digraph, and let G1, . . . , Gq +be the strong component decomposition of G. The skeleton graph S = (U, F) +is a digraph on q nodes u1, . . . , uq, corresponding to G1, . . . , Gq. There is no +self-loop in S. There is an edge uiuj, for ui ̸= uj, only if there exist a node vi′ +in Gi and a node vj′ in Gj such that vi′vj′ is an edge in G. Further, we denote +by π : V → U the map that sends nodes vi′ in Vi to ui. +6 + +(a) +u1 +u2 +u3 +(b) +u1 +u2 +(c) +(d) +Figure 3: Illustration of GS′: (a) A weakly connected digraph G = (V, E), +with three strongly connected components highlighted in blue, red, and green, +respectively; (b) The skeleton graph S = (U, F) of G, with U = {u1, u2, u3} +and F = {u1u2, u1u3, u2u3}; (c) A subgraph S′ of S; and (d) The corresponding +subgraph GS′ of G. +We illustrate the definition in Figure 2. +A subgraph S′ = (U′, F ′) of S induces a subgraph of G, obtained by only +keeping the nodes of G contained in the strong components represented by +nodes of S′; precisely, to S′, we attach the subgraph GS′ of G induced by +π−1(U′). See Figure 3 for an illustration. Note that the skeleton graph S is +acyclic because otherwise, it will contradict the third item of Definition 3. +Let ℓ be the depth of the graph S; we now introduce a node set decompo- +sition, termed the depth decomposition, of S: +U = ⊔ℓ +m=0Um. +(9) +Starting with U0, we simply let it be the subset of nodes of U without incoming +edges. Since S is acyclic, U0 is non-empty. Now to each node uj in U −U0, we +assign the set Pj of paths from nodes in U0 to uj. It should be clear that Pj is +non-empty. We define the depth of the node uj, denoted by depth(uj), to be +the maximal length of all paths in Pj, i.e., +depth(uj) := max{length(w) | w ∈ Pj}. +The subset Um is then the collection of all nodes in S of depth m. +The +subsets Um, for 0 ≤ m ≤ ℓ, are all nonempty, pairwise disjoint, and their +union is U. +With the preliminaries above, we establish Theorem 1 +7 + +Proof of Theorem 1. Let G = (V, E, γ) be the weighted dependency graph of +the polynomial vector field f. Let S = (U, F) be the associated skeleton graph +(obtained using Definition 4 and ignoring the weights γ of G), and ℓ be the +depth of S. Because G is weakly connected by assumption, so is S. The proof +will be carried out by induction on ℓ. +Base case ℓ = 0: +In this case, since S is weakly connected, it is a single node. +It follows that G is strongly connected. Next, we claim that all the weights γij +for the edges vivj of G are constant. To see this, for each edge vivj in G, we let +c = vi1vi2 · · · vikvi1 be a cycle in G that contains this edge, with vi1vi2 = vivj. +By the hypothesis of Theorem 1, it holds that γc is constant. We have that +γvivjγvi2···vikvi1 = γc. +Since γc is a constant and since both γvivj and γvi2···vik vi1 are polynomials (over +R), it must hold that they are also constants. This establishes the claim. As a +consequence, the vector field f is an affine function. This completes the proof +for the base case. +Inductive step: +We assume that the statement holds for ℓ ≥ 0 and prove it +for (ℓ + 1). Let ⊔ℓ+1 +m=0Um be the node set decomposition of U introduced in (9). +Consider the subgraph S′ of S induced by the nodes in ⊔ℓ +m=0Um, and S′′ the +subgraph of S induced by nodes in Uℓ+1. +It should be clear that S′ is itself an acyclic digraph whose depth is ℓ, +and that S′′ is a union of isolated nodes. To see that the latter statement +holds, it suffices to observe that if S′′ has an edge, then it necessarily has +nodes with different depths. We let x′(t) be the vector with entries taken from +x(t) corresponding to nodes in GS′ and x′′(t) be the vector corresponding to +GS′′. By construction of S′, the dynamics of x′(t) do not depend on x′′(t) and, +hence, we can write the said dynamics as ˙x′(t) = f ′(x′(t)). On the one hand, +by applying the induction hypothesis to each connected component of S′, we +have that f ′ is super-linearizable. We set p′ to be the associated observables, +on which the super-linearization relies. +On the other hand, the dynamics of x′′(t) may depend on both x′(t) and +x′′(t), i.e., ˙x′′(t) = f ′′(x′(t), x′′(t)) for f ′′ a polynomial vector field. +Since +each connected component of GS′′ is strongly connected, every edge in GS′′ +belongs to a cycle in GS′′. By the hypothesis of Theorem 1 and by the same +arguments given in the base case, we then have that all the edge weights in +GS′′ are constants. This implies that f ′′(x′, x′′) is affine in x′′ (note that edge +weights in GS′′ only take into account differentiation of f ′′ with respect to x′′, +i.e., the variables corresponding to nodes GS′′). Combining the above, the +dynamics can be expressed as +� +˙z′(t) = A′z′(t) + D +˙x′′(t) = A′′x′′(t) + g(z′(t)), +(10) +8 + +where, owing to Proposition 2, z′ := (x′; p′), A′ and A′′ are constant matrices, +D is a constant vector, and g is a polynomial vector field. By Proposition 2, +system (10) is super-linearizable. This completes the proof. +Remark 1. Using arguments similar to the ones of the proof of Theorem 1, +one can establish the following fact (with proof omitted): suppose that f is +a smooth vector field; then, the system ˙x = f(x) admits a super-linearization +˙z = Az +D, where z = +� +x1, . . . , xn, p1, . . . , pm +� +as described below (10), if and +only if there exist an integer N > 0 and coefficients ck ∈ R, for k = 0, . . . , N−1, +such that +LN +f f = +N−1 +� +k=0 +αk Lk +ff. +(11) +From that vantage point, the main result of this paper can be restated as +follows: if f is a polynomial vector field and if f satisfies the condition of +Theorem 1, then f satisfies (11) and is thus super-linearizable. +3.2 +Algorithm for Super-linearization +The steps outlined in the proof of Theorem 1 can be formalized as an algorithm, +which we will present below. +For ease of presentation, we introduce some +notations. +Let G be the WDG of a given system ˙x = f(x) and S be the corresponding +skeleton graph. Let U = ⊔ℓ +m=0Um be the depth decomposition, Sm be the +subgraph of S induced by Um. With a slight abuse of notation, we will use +xm to denote the “sub-vector” of x with entries corresponding to the nodes in +GSm, and let fm(x) be defined such that ˙xm(t) = fm(x(t)). +The algorithm for super-linearization is as follows: +Input: +A polynomial map f : Rn → Rn for the system ˙x(t) = f(x(t)). +Step 1: +Compute the WDG G of the system and terminate if G does not +satisfy the conditions of Theorem 1. +Step 2: +Compute the skeleton graph S = (U, F), its depth ℓ, and the depth +decomposition U = ⊔ℓ +m=0Um. +Step 3: +Set ℓ′ := 0 and z0 := x0. While ℓ′ < ℓ, repeat: +3.1: Perform the super-linearization of the following system: +� +˙zℓ′(t) = Aℓ′zℓ′(t) + Dℓ′, +˙xℓ′+1(t) = fℓ′+1(x(t)). +(12) +9 + +and obtain the super-linearized dynamics of (12) +˙zℓ′+1(t) = Aℓ′+1zℓ′+1(t) + Dℓ′+1 +(13) +with observables pℓ′+1. +3.2: Increase ℓ′ by 1. +Output: +The data (Aℓ, Dℓ, pℓ) as a super-linearization of the original system. +Remark 2. We elaborate below on a few points of Step 3.1 in the Algorithm: +1. When ℓ′ = 0, (12) implies that the dynamics of x0 are necessarily affine. +It is indeed the case, and was argued in the proof of Theorem 1 (the base +case). +2. In (12), the dynamics of xℓ′+1 depend only on x0, . . . , xℓ′+1 and, more- +over, linearly in xℓ′+1 as was argued in the proof of Theorem 1 (the +inductive step). Note that zℓ′+1 contains the variables x0, . . . , xℓ′+1 and +the observables pℓ′+1. +3. In order to obtain the super-linearized dynamics (13), one can follow, e.g., +the steps of the proof of Proposition 2. The fact that (12) is in the same +form as (6) is argued in the second item of this remark. More specifically, +the first step is then to determine the degree d of the polynomial vector +field fℓ′+1. Next, upon choosing a basis for Pd, determine the matrix +of the linear operator L ¯fℓ′ : Pd → Pd where ¯fℓ′(z) := Aℓ′z + Dℓ′ and +compute the minimal polynomial of this matrix. Finally, introduce the +observables p as given in (7); they obey the linear dynamics (8). +There exist other ways to obtain a super-linearization of the system; we +will in fact follow a slightly different approach in the example next. +We illustrate the algorithm on the polynomial system given in Example 1. +Recall that the WDG G of the system is given in Figure 1, and the correspond- +ing skeleton graph S = (U, F) is in Figure 2. +We next compute the depth decomposition of U. The only node that has +no incoming edges is u1, and thus U0 = {u1}. The longest path joining u1 to +u2 is of length 1, and the longest paths from u1 to either u3 or u4 are of lengths +2; hence U1 = {u2} and U2 = {u3, u4}. +Now, for Step 3, there will be two iterations: +1. The first iteration considers the dynamics of the variables associated to +U0 (namely x1, x2) and U1 (namely, x3). We have + + + + + +˙x1 = x2 +˙x2 = −x1 +˙x3 = x2 +2. +(14) +10 + +We observe that the dynamics associated to the nodes in U0 are indeed +linear. Following (7), we set x = (x′, x′′) with x′ := (x1, x2) and x′′ := x3, +p1(x) := x2 +2, and f ′(x′) := (x2, −x1). We obtain that +Lf′p1 = −2x1x2 =: p2 +Lf′p2 = 2(x2 +1 − x2 +2) =: p3 +Lf′p3 = 8x1x2 = −4p2. +The super-linearized system is thus +˙z1 = + + +˙x1 +˙x2 +˙x3 +˙p1 +˙p2 +˙p3 + + += + + +x2 +−x1 +p1 +p2 +p3 +−4p2 + + +=: A1z1. +(15) +2. The second iteration starts with the super-linearized system (15) with +the dynamics of the variables in U2 adjoined. Namely, with + + + + + +˙z1 = A1z1 +˙x4 = x3 + x1x2 +2 +˙x5 = −x5 + x2 +3 + x2 +1x2 +To proceed, we could attempt to super-linearize the vector (x1x2 +2; x2 +3 + +x2 +1x2) at once, or handle each entry consecutively. We choose the latter +option, which deviates slightly from the procedure described in Propo- +sition 2 but requires fewer computations. Also, note that there is some +freedom in how one expresses the nonlinear terms. For example, x1x2 +2 +can also be written as x1p1 or −1 +2x2p2, given the observables introduced +in the first iteration. +We start by setting p4 := x1x2 +2 and f ′(z1) := A1z1. By computation, we +obtain that +Lf′p4 = x3 +2 − 2x2 +1x2 =: p5 +Lf′p5 = −7x1x2 +2 + 2x3 +1 = −7p4 + 2x3 +1 =: p6 +Lf′p6 = −7p5 + 6x2 +1x2 =: p7 +Lf′p7 = −7p6 + 12x1x2 +2 − 6x3 +1 += −7p6 + 12p4 − 3(p6 + 7p4) = −10p6 − 9p4. +11 + +Next, we set p8 := x2 +3 + x2 +1x2 and +Lf′p8 = 2x3p1 + 2x1x2 +2 − x3 +1 += 2x3p1 − 1 +2(p6 + 3p4) =: p9 +Lf′p9 = 2p2 +1 + 2x3p2 − 1 +2(p7 + 3p5) =: p10 +Lf′p10 = 6p1p2 + 2x3p3 + 1 +2(9p4 + 7p6) =: p11 +Lf′p11 = 6p2 +2 + 8p1p3 − 8x3p2 + 1 +2(9p5 + 7p7) =: p12 +Lf′p12 = 20p2p3 − 40p1p2 − 8x3p3 − 1 +2(63p4 + 61p6) =: p13 +Lf′p13 = 20p2 +3 − 120p2 +2 − 48p1p3 + 32x3p2 − 1 +2(63p5 + 61p7) =: p14 +Lf′p14 = −448p2p3 + 224p1p2 + 32x3p3 + 1 +2(549p4 + 547p6) =: p15 +Lf′p15 = 2016p2 +2 − 448p2 +3 + 256p1p3 − 128x3p2 + 1 +2(549p5 + 547p7) =: p16 +Lf′p16 = 7872p2p3 − 1152p1p2 − 128x3p3 − 1 +2(4923p4 + 4921p6) += 1 +2(1485p4 + 1215p6) − 256p11 − 144p13 − 24p15. +We thus obtain the following super-linearization of the original system (5): + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +˙x1 = x2 +˙x2 = −x1 +˙x3 = p1 +˙x4 = x3 + p4 +˙x5 = −x5 + p7 +˙pi = pi+1, for i = 1, 2, 4, 5, 6, 8, · · · , 15 +˙p3 = −4p2 +˙p7 = −10p6 − 9p4 +˙p16 = 1485 +2 p4 + 1215 +2 p6 − 256p11 − 144p13 − 24p15. +4 +Summary and Outlook +We provided in this paper a sufficient condition for a system ˙x(t) = f(x(t)), +with f a polynomial vector field, to be super-linearizable. The condition is +12 + +simply expressed in terms of cycles in what we called the weighted dependency +graph of the system. The proof of the main result is constructive, and we have +sketched an algorithm based on it that produces a super-linearization of vector +fields meeting the sufficient condition. The algorithm was also illustrated on +an example. +The main result of this paper provides a generalized converse of the results +in [8]. Indeed, while the canonical form exhibited there entails that in the +original dynamics, the variables on which the nonlinear terms depend have to +evolve linearly, it is easy to see that this fact does not hold for the system (5). +The gap of course lies in the fact that [8] restricts its scope to systems with +only one visible observable, which precludes the nested super-linearizations +that arise in the inductive step of the proof. +In terms of the vocabulary +introduced in this paper, the results of [8] only deal with skeleton graphs of +depth 1. We will address the converse of the results presented in this paper, +similarly generalize the results of [8], in future work. +References +[1] T. Carleman, “Application de la th´eorie des ´equations int´egrales lin´eaires +aux syst`emes d’´equations diff´erentielles non lin´eaires,” Acta Mathematica, +vol. 59, pp. 63–87, 1932. +[2] B. O. Koopman, “Hamiltonian systems and transformation in Hilbert +space,” Proceedings of the National Academy of Sciences, vol. 17, no. 5, +pp. 315–318, 1931. +[3] K. Kowalski and W.-H. Steeb, Nonlinear dynamical systems and Carle- +man linearization. +World Scientific, 1991. +[4] R. W. Brockett, “Volterra series and geometric control theory,” Automat- +ica, vol. 12, no. 2, pp. 167–176, 1976. +[5] ——, “The early days of geometric nonlinear control,” Automatica, vol. 50, +no. 9, pp. 2203–2224, 2014. +[6] A. Mauroy, Y. Susuki, and I. Mezi´c, Koopman operator in systems and +control. +Springer, 2020. +[7] S. E. Otto and C. W. Rowley, “Koopman operators for estimation and +control of dynamical systems,” Annual Review of Control, Robotics, and +Autonomous Systems, vol. 4, pp. 59–87, 2021. +[8] M.-A. Belabbas, “Canonical forms for polynomial systems with balanced +super-linearizations,” arXiv:2212.12054, 2022. +13 + +[9] S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz, “Koopman +invariant subspaces and finite linear representations of nonlinear dynami- +cal systems for control,” PloS one, vol. 11, no. 2, p. e0150171, 2016. +[10] M.-A. Belabbas, “Visible and hidden observables in super-linearization,” +arXiv:2211.02739, 2022. +14 + diff --git a/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/load_file.txt b/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b1d6388cc504f3f33d72647edacbcf2f9074d30 --- /dev/null +++ b/O9E2T4oBgHgl3EQfrQi-/content/tmp_files/load_file.txt @@ -0,0 +1,390 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf,len=389 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='04048v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='OC] 10 Jan 2023 A Sufficient Condition for the Super-linearization of Polynomial Systems M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Belabbas∗ and Xudong Chen† Abstract We provide in this paper a sufficient condition for a polynomial dynamical system ˙x(t) = f(x(t)) to be super-linearizable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', to be such that all its trajectories are linear projections of the trajectories of a linear dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The condition is expressed in terms of the hereby introduced weighted dependency graph G, whose nodes vi correspond to variables xi and edges vivj have weights ∂fj ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We show that if the product of the edge weights along any cycle in G is a constant, then the system is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The proof is constructive, and we provide an algorithm to obtain super-linearizations and illustrate it on an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 1 Introduction The idea of linearizing system dynamics via embeddings dates back at least to the works of Carleman [1] and Koopman [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' These embeddings are still actively studied a century later, and have found applications in nonlinear control [4,5], and data-driven methods in control [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We derive in this paper a sufficient condition under which a polynomial system can be globally linearized by embedding it into a higher, yet finite- dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In particular, the contribution of this paper is to provide a generalized converse of the result established in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We elaborate on this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' To proceed, we consider the following dynamical system: ˙x = f(x) (1) ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Belabbas is with the Electrical and Computer Engineering Department and the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Email: {belabbas@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' †X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Chen is with the Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Email: xudong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='chen@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Belabbas and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Chen contributed equally to the manuscript in all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 1 where x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This system is said to admit a super-linearization (see Defini- tion 1 below) if there exist m ≥ 0 functions, called observables, which when adjoined to the original system would permit its linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' A typical ex- ample [9] is the following two-dimensional system � ˙x = −x + y2 ˙y = −y (2) Adding the observable w := y2, whose total time derivative is given by ˙w = 2y ˙y = −2y2 = −2w, we obtain the three-dimensional linear system: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ˙x = −x + w ˙y = −y ˙w = −2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (3) Observe that the variables on which the nonlinear part of the dynamics (2) depend (here, the variable y) evolve in a linear, autonomous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', indepen- dent from x) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In a recent paper [8], we showed that if a polynomial system admits a so-called balanced super-linearization with only one visible observable [10], then there exists a linear change of variables under which the nonlinear part of the dynamics depends solely on variables evolving linearly and autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The dynamics resulting from the change of variable are termed the canonical form [8] for the polynomial system (explicitly, the canoni- cal form is given in (6) below), and its existence provides a necessary condition for the super-linearization of that special class of polynomial systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Conversely, we exhibit in this paper a sufficient condition for the super- linearization of general polynomial systems, without any restriction on the number of visible observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In particular, the result of this paper, combined with the ones of [8], provide a necessary and sufficient condition for a class of polynomial systems with a single visible observable to be super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The remainder of the paper is organized as follows: We describe the relevant terminology and notation at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We present the main result in Section 2 and its proof in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The paper ends with a summary and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Terminology and notation used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We let G = (V, E) be a directed graph (possibly with self-loops), with V the node set and E the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We use e = vivj to denote a directed edge of G from node vi to node vj (if vi = vj, then e is a self-loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' A walk is a sequence of nodes w = vi1vi2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' vik such that viℓviℓ+1 is an edge of G for each ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The length of a walk is the number of edges it traverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' A path is a walk which does not visit a node more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We call the depth of G the length of the longest path in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 2 For a dynamical system ˙x(t) = f(x(t)), we denote by etfx0 the solution of the system at time t with initial state x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' For a vector field g : Rn → Rn and a differentiable vector-valued function p : Rn → Rk, we denote the Lie derivative of p along g by Lgp := ∂p ∂xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 2 Statement of the Result We start by defining what it means for system (1) to be super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let m ≥ 0 be an integer, and Π : Rn+m → Rn be the canonical projection onto the first n variables, namely, we have for z ∈ Rn+m that Π(z) = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We reproduce the following definition from [10]: Definition 1 (Super-linearization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The vector field f : Rn → Rn is super- linearizable to the system ˙z = Az + D with A ∈ R(n+m)×(n+m) and D ∈ Rn+m if there exists an injective map p : Rn → Rm so that for all x0 ∈ Rn, the following holds: Π � et(Az+D)z0 � = etfx0 with z0 = (x0, p(x0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (4) We call the functions p : Rn → Rm the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The data of A, D and p is referred to as a super-linearization of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We can express the relation (4) as the following commutative diagram Rn Rn Rn+m Rn+m etf (id, p) et(Az+D) Π We next introduce the following notion: Definition 2 (Weighted dependency graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let f : Rn → Rn be a differen- tiable vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The weighted dependency graph (WDG) G = (V, E, γ) of f is a weighted directed graph (with self-loop) on n nodes v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' For every ordered pair (vi, vj), we define the scalar function: γij(x) := ∂fj(x) ∂xi for 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' There is an edge vivj in G if γij ̸= 0, and its weight is γij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We illustrate the definition on the following example: 3 v3 v4 v5 v2 v1 −1 2x3 2x1x2 x2 1 2x2 1 x2 2 2x1x2 −1 1 Figure 1: The weighted dependency graph of system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Consider the following polynomial system: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙x1 = x2 ˙x2 = −x1 ˙x3 = x2 2 ˙x4 = x3 + x1x2 2 ˙x5 = −x5 + x2 3 + x2 1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (5) Its weighted dependency graph is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Next, for each directed walk w = vi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' vik in G, we let γw := k−1 � j=1 γijij+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In the sequel, we will assume that f is a polynomial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It should be clear that γw(x), for any walk w, is then a polynomial function in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Also, we assume, without loss of generality, that G is weakly connected (otherwise, the original system can be decoupled into sub-systems of lower dimensions and our result, stated below, can be applied to each sub-system independently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The main result of this paper is as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' For a polynomial system ˙x(t) = f(x(t)), let G be the associated weighted dependency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' If γc is a constant for every cycle c of G, then f is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The sufficient condition stated in the Theorem implies that the Jacobian of f is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Note that the weighted dependency graph of system (5), 4 depicted in Figure 1, satisfies the sufficient condition of Theorem 1, and thus system (5) is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' As an illustration of the proof technique used, we will provide toward the end a super-linearization of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 3 Proof of Theorem 1 and an Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1 Proof of the Theorem We start with a simple proposition, dealing with systems where the variables on which the nonlinear part of the dynamics depend evolve linearly and au- tonomously, and show that such systems are super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This result provides a converse of the result of [8], and will be used as a building block to establish the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Suppose that the system ˙x(t) = f(x(t)) takes the following form: � ˙x′(t) = A′x′(t) + D ˙x′′(t) = A′′x′′(t) + g(x′(t)), (6) where x = (x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' x′′), D is a constant vector, and g is a polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' then, sys- tem (6) is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let n′ be the dimension of x′ and d be the degree of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let Pd be the vector space of all polynomials in x′ with real coefficients, whose dimension is r := dim Pd = �n′ + d d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Next, for convenience, we let f ′(x′) := A′x′ + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Since f ′ is affine, Lf′φ ⊆ Pd for any φ ∈ Pd and, hence, Lf′ : Pd → Pd is a linear automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let the minimal polynomial associated with Lf′ be given by sN + αN−1sN−1 + · · · + α0 for some N ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In particular, for any φ ∈ Pd, we have that (LN f′φ) + αN−1(LN−1 f′ φ) + · · · + α0φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Now, define p(x) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 p1(x) p2(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' pN(x) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb := \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 g(x′) Lf′g(x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' LN f′g(x′) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (7) 5 u1 u2 u3 u4 Figure 2: The skeleton graph S of the WDG G of system (5), depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Note that π−1(u1) = {v1, v2}, π−1(u2) = {v3}, π−1(u3) = {v4}, and π−1(u4) = {v5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It then follows that the time derivative of p(x(t)) is d dt \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' pN−1 pN \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 I 0 · · 0 0 0 I · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 0 0 · · 0 I −α0I −α1I · · −αN−2I −αN−1I \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' pN−1 pN \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (8) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We next introduce two notions that are necessary for enabling the recursive use of Proposition 2 in the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The first is the notion of strong component decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Definition 3 (Strong component decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let G be a weakly connected digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The subgraphs Gi = (Vi, Ei), for 1 ≤ i ≤ q, form a strong component decomposition of G if the following items hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The Vi’s partition the vertex set as V = ⊔q i=1Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Each Gi is a subgraph induced by Vi and is strongly connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Any strongly connected subgraph G′ of G is a subgraph of some Gi, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By treating the strongly connected components Gi as single nodes, we obtain the second notion, namely the one of skeleton graph S of G: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let G = (V, E) be a weakly connected digraph, and let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , Gq be the strong component decomposition of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The skeleton graph S = (U, F) is a digraph on q nodes u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , uq, corresponding to G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' There is no self-loop in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' There is an edge uiuj, for ui ̸= uj, only if there exist a node vi′ in Gi and a node vj′ in Gj such that vi′vj′ is an edge in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Further, we denote by π : V → U the map that sends nodes vi′ in Vi to ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 6 (a) u1 u2 u3 (b) u1 u2 (c) (d) Figure 3: Illustration of GS′: (a) A weakly connected digraph G = (V, E), with three strongly connected components highlighted in blue, red, and green, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (b) The skeleton graph S = (U, F) of G, with U = {u1, u2, u3} and F = {u1u2, u1u3, u2u3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (c) A subgraph S′ of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' and (d) The corresponding subgraph GS′ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We illustrate the definition in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' A subgraph S′ = (U′, F ′) of S induces a subgraph of G, obtained by only keeping the nodes of G contained in the strong components represented by nodes of S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' precisely, to S′, we attach the subgraph GS′ of G induced by π−1(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Note that the skeleton graph S is acyclic because otherwise, it will contradict the third item of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let ℓ be the depth of the graph S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' we now introduce a node set decompo- sition, termed the depth decomposition, of S: U = ⊔ℓ m=0Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (9) Starting with U0, we simply let it be the subset of nodes of U without incoming edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Since S is acyclic, U0 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Now to each node uj in U −U0, we assign the set Pj of paths from nodes in U0 to uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It should be clear that Pj is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We define the depth of the node uj, denoted by depth(uj), to be the maximal length of all paths in Pj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', depth(uj) := max{length(w) | w ∈ Pj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The subset Um is then the collection of all nodes in S of depth m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The subsets Um, for 0 ≤ m ≤ ℓ, are all nonempty, pairwise disjoint, and their union is U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' With the preliminaries above, we establish Theorem 1 7 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let G = (V, E, γ) be the weighted dependency graph of the polynomial vector field f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let S = (U, F) be the associated skeleton graph (obtained using Definition 4 and ignoring the weights γ of G), and ℓ be the depth of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Because G is weakly connected by assumption, so is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The proof will be carried out by induction on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Base case ℓ = 0: In this case, since S is weakly connected, it is a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It follows that G is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Next, we claim that all the weights γij for the edges vivj of G are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' To see this, for each edge vivj in G, we let c = vi1vi2 · · · vikvi1 be a cycle in G that contains this edge, with vi1vi2 = vivj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By the hypothesis of Theorem 1, it holds that γc is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We have that γvivjγvi2···vikvi1 = γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Since γc is a constant and since both γvivj and γvi2···vik vi1 are polynomials (over R), it must hold that they are also constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This establishes the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' As a consequence, the vector field f is an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This completes the proof for the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Inductive step: We assume that the statement holds for ℓ ≥ 0 and prove it for (ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let ⊔ℓ+1 m=0Um be the node set decomposition of U introduced in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Consider the subgraph S′ of S induced by the nodes in ⊔ℓ m=0Um, and S′′ the subgraph of S induced by nodes in Uℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It should be clear that S′ is itself an acyclic digraph whose depth is ℓ, and that S′′ is a union of isolated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' To see that the latter statement holds, it suffices to observe that if S′′ has an edge, then it necessarily has nodes with different depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We let x′(t) be the vector with entries taken from x(t) corresponding to nodes in GS′ and x′′(t) be the vector corresponding to GS′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By construction of S′, the dynamics of x′(t) do not depend on x′′(t) and, hence, we can write the said dynamics as ˙x′(t) = f ′(x′(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' On the one hand, by applying the induction hypothesis to each connected component of S′, we have that f ′ is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We set p′ to be the associated observables, on which the super-linearization relies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' On the other hand, the dynamics of x′′(t) may depend on both x′(t) and x′′(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', ˙x′′(t) = f ′′(x′(t), x′′(t)) for f ′′ a polynomial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Since each connected component of GS′′ is strongly connected, every edge in GS′′ belongs to a cycle in GS′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By the hypothesis of Theorem 1 and by the same arguments given in the base case, we then have that all the edge weights in GS′′ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This implies that f ′′(x′, x′′) is affine in x′′ (note that edge weights in GS′′ only take into account differentiation of f ′′ with respect to x′′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', the variables corresponding to nodes GS′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Combining the above, the dynamics can be expressed as � ˙z′(t) = A′z′(t) + D ˙x′′(t) = A′′x′′(t) + g(z′(t)), (10) 8 where, owing to Proposition 2, z′ := (x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' p′), A′ and A′′ are constant matrices, D is a constant vector, and g is a polynomial vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By Proposition 2, system (10) is super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Using arguments similar to the ones of the proof of Theorem 1, one can establish the following fact (with proof omitted): suppose that f is a smooth vector field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' then, the system ˙x = f(x) admits a super-linearization ˙z = Az +D, where z = � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , xn, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , pm � as described below (10), if and only if there exist an integer N > 0 and coefficients ck ∈ R, for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , N−1, such that LN f f = N−1 � k=0 αk Lk ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (11) From that vantage point, the main result of this paper can be restated as follows: if f is a polynomial vector field and if f satisfies the condition of Theorem 1, then f satisfies (11) and is thus super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2 Algorithm for Super-linearization The steps outlined in the proof of Theorem 1 can be formalized as an algorithm, which we will present below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' For ease of presentation, we introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let G be the WDG of a given system ˙x = f(x) and S be the corresponding skeleton graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Let U = ⊔ℓ m=0Um be the depth decomposition, Sm be the subgraph of S induced by Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' With a slight abuse of notation, we will use xm to denote the “sub-vector” of x with entries corresponding to the nodes in GSm, and let fm(x) be defined such that ˙xm(t) = fm(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The algorithm for super-linearization is as follows: Input: A polynomial map f : Rn → Rn for the system ˙x(t) = f(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Step 1: Compute the WDG G of the system and terminate if G does not satisfy the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Step 2: Compute the skeleton graph S = (U, F), its depth ℓ, and the depth decomposition U = ⊔ℓ m=0Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Step 3: Set ℓ′ := 0 and z0 := x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' While ℓ′ < ℓ, repeat: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1: Perform the super-linearization of the following system: � ˙zℓ′(t) = Aℓ′zℓ′(t) + Dℓ′, ˙xℓ′+1(t) = fℓ′+1(x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (12) 9 and obtain the super-linearized dynamics of (12) ˙zℓ′+1(t) = Aℓ′+1zℓ′+1(t) + Dℓ′+1 (13) with observables pℓ′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2: Increase ℓ′ by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Output: The data (Aℓ, Dℓ, pℓ) as a super-linearization of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We elaborate below on a few points of Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1 in the Algorithm: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' When ℓ′ = 0, (12) implies that the dynamics of x0 are necessarily affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' It is indeed the case, and was argued in the proof of Theorem 1 (the base case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In (12), the dynamics of xℓ′+1 depend only on x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , xℓ′+1 and, more- over, linearly in xℓ′+1 as was argued in the proof of Theorem 1 (the inductive step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Note that zℓ′+1 contains the variables x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' , xℓ′+1 and the observables pℓ′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' In order to obtain the super-linearized dynamics (13), one can follow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=', the steps of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The fact that (12) is in the same form as (6) is argued in the second item of this remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' More specifically, the first step is then to determine the degree d of the polynomial vector field fℓ′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Next, upon choosing a basis for Pd, determine the matrix of the linear operator L ¯fℓ′ : Pd → Pd where ¯fℓ′(z) := Aℓ′z + Dℓ′ and compute the minimal polynomial of this matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Finally, introduce the observables p as given in (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' they obey the linear dynamics (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' There exist other ways to obtain a super-linearization of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' we will in fact follow a slightly different approach in the example next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We illustrate the algorithm on the polynomial system given in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Recall that the WDG G of the system is given in Figure 1, and the correspond- ing skeleton graph S = (U, F) is in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We next compute the depth decomposition of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The only node that has no incoming edges is u1, and thus U0 = {u1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The longest path joining u1 to u2 is of length 1, and the longest paths from u1 to either u3 or u4 are of lengths 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' hence U1 = {u2} and U2 = {u3, u4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Now, for Step 3, there will be two iterations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The first iteration considers the dynamics of the variables associated to U0 (namely x1, x2) and U1 (namely, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We have \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ˙x1 = x2 ˙x2 = −x1 ˙x3 = x2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (14) 10 We observe that the dynamics associated to the nodes in U0 are indeed linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Following (7), we set x = (x′, x′′) with x′ := (x1, x2) and x′′ := x3, p1(x) := x2 2, and f ′(x′) := (x2, −x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We obtain that Lf′p1 = −2x1x2 =: p2 Lf′p2 = 2(x2 1 − x2 2) =: p3 Lf′p3 = 8x1x2 = −4p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The super-linearized system is thus ˙z1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ˙x1 ˙x2 ˙x3 ˙p1 ˙p2 ˙p3 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 x2 −x1 p1 p2 p3 −4p2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb =: A1z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' (15) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The second iteration starts with the super-linearized system (15) with the dynamics of the variables in U2 adjoined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Namely, with \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ˙z1 = A1z1 ˙x4 = x3 + x1x2 2 ˙x5 = −x5 + x2 3 + x2 1x2 To proceed, we could attempt to super-linearize the vector (x1x2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' x2 3 + x2 1x2) at once, or handle each entry consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We choose the latter option, which deviates slightly from the procedure described in Propo- sition 2 but requires fewer computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Also, note that there is some freedom in how one expresses the nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' For example, x1x2 2 can also be written as x1p1 or −1 2x2p2, given the observables introduced in the first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We start by setting p4 := x1x2 2 and f ′(z1) := A1z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' By computation, we obtain that Lf′p4 = x3 2 − 2x2 1x2 =: p5 Lf′p5 = −7x1x2 2 + 2x3 1 = −7p4 + 2x3 1 =: p6 Lf′p6 = −7p5 + 6x2 1x2 =: p7 Lf′p7 = −7p6 + 12x1x2 2 − 6x3 1 = −7p6 + 12p4 − 3(p6 + 7p4) = −10p6 − 9p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 11 Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' we set p8 := x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='3 + x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1x2 and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p8 = 2x3p1 + 2x1x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2 − x3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='= 2x3p1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(p6 + 3p4) =: p9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p9 = 2p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='1 + 2x3p2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(p7 + 3p5) =: p10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p10 = 6p1p2 + 2x3p3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(9p4 + 7p6) =: p11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p11 = 6p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2 + 8p1p3 − 8x3p2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(9p5 + 7p7) =: p12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p12 = 20p2p3 − 40p1p2 − 8x3p3 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(63p4 + 61p6) =: p13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p13 = 20p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='3 − 120p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2 − 48p1p3 + 32x3p2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(63p5 + 61p7) =: p14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p14 = −448p2p3 + 224p1p2 + 32x3p3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(549p4 + 547p6) =: p15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p15 = 2016p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2 − 448p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='3 + 256p1p3 − 128x3p2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(549p5 + 547p7) =: p16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='Lf′p16 = 7872p2p3 − 1152p1p2 − 128x3p3 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(4923p4 + 4921p6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content='2(1485p4 + 1215p6) − 256p11 − 144p13 − 24p15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' We thus obtain the following super-linearization of the original system (5): \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙x1 = x2 ˙x2 = −x1 ˙x3 = p1 ˙x4 = x3 + p4 ˙x5 = −x5 + p7 ˙pi = pi+1, for i = 1, 2, 4, 5, 6, 8, · · · , 15 ˙p3 = −4p2 ˙p7 = −10p6 − 9p4 ˙p16 = 1485 2 p4 + 1215 2 p6 − 256p11 − 144p13 − 24p15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' 4 Summary and Outlook We provided in this paper a sufficient condition for a system ˙x(t) = f(x(t)), with f a polynomial vector field, to be super-linearizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The condition is 12 simply expressed in terms of cycles in what we called the weighted dependency graph of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The proof of the main result is constructive, and we have sketched an algorithm based on it that produces a super-linearization of vector fields meeting the sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The algorithm was also illustrated on an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The main result of this paper provides a generalized converse of the results in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' Indeed, while the canonical form exhibited there entails that in the original dynamics, the variables on which the nonlinear terms depend have to evolve linearly, it is easy to see that this fact does not hold for the system (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E2T4oBgHgl3EQfrQi-/content/2301.04048v1.pdf'} +page_content=' The gap of course lies in the fact that [8] restricts its scope to systems with only one visible observable, which precludes the nested super-linearizations that arise in the inductive step of the proof.' metadata={'source': 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Measuring the integrated +information of a quantum +mechanism. Preprints 2023, 1, 0. +https://doi.org/ +Copyright: +© 2023 by the authors. +Licensee MDPI, Basel, Switzerland. +This article is an open access article +distributed +under +the +terms +and +conditions of the Creative Commons +Attribution (CC BY) license (https:// +creativecommons.org/licenses/by/ +4.0/). +Article +Measuring the integrated information of a quantum mechanism +Larissa Albantakis 1,4* +, Robert Prentner 2,4 +and Ian Durham 3,4 +1 +Department of Psychiatry, University of Wisconsin–Madison, Madison, WI 53719, USA +2 +Munich Center for Mathematical Philosophy, Ludwig-Maximilians-University Munich, 80539, Germany; +robert.prentner@amcs.science +3 +Department of Physics, Saint Anselm College, Manchester, NH 03102, USA ; idurham@anselm.edu +4 +Association for Mathematical Consciousness Science, Munich, Germany +* +Correspondence: albantakis@wisc.edu +Abstract: Originally conceived as a theory of consciousness, integrated information theory (IIT) provides +a theoretical framework intended to characterize the compositional causal information that a system, +in its current state, specifies about itself. However, it remains to be determined whether IIT as a theory +of consciousness is compatible with quantum mechanics as a theory of microphysics. Here, we present +an extension of IIT’s latest formalism to evaluate the mechanism integrated information (ϕ) of a system +subset to finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we translate a +recently developed, unique measure of intrinsic information into a density matrix formulation, and extend +the notion of conditional independence to accommodate quantum entanglement. The compositional +nature of the IIT analysis might shed some light on the internal structure of composite quantum states +and operators that cannot be obtained using standard information-theoretical analysis. Finally, our results +should inform theoretical arguments about the link between consciousness, causation, and physics from +the classical to the quantum. +Keywords: causal analysis; causation; quantum information theory; entanglement structure; multivariate +interaction +1. Introduction +Integrated information theory [1–4] stands out as one theory of consciousness that explic- +itly proposes a formal framework for identifying conscious systems. Specifically, IIT provides +requirements about the intrinsic causal structure of a system that supports consciousness, based +on the essential (“phenomenal”) properties of experience. Its formal framework evaluates the +causal powers that a set of interacting physical units exerts on itself in a compositional manner +[1,4–7]. +IIT does not presuppose that consciousness arises at the level of neurons rather than atoms, +molecules, or larger brain areas, but assumes causation to be a central concept for analyzing a +physical system across the hierarchy from the microphysical to the macroscopic [2,8–11]. One +prediction of IIT is that consciousness appears at the level of organization at which the intrinsic +causal powers of a system are maximized [2]. To that end, IIT offers a formal framework for +causal emergence that compares the amount of integrated information of macroscopic causal +models to their underlying microscopic system descriptions [9,10]. Whether causality plays a +fundamental role in physics, in particular in quantum physics, is, however, contested [12,13]. +IIT’s causal framework has been formalized for discrete dynamical systems in the classical +realm [1,4,14–16]. Accordingly, in prior studies [8–10], micro-level systems corresponded to +classical causal networks [17,18], constituted of individual, conditionally independent physical +units, that can (in principle) be manipulated and whose states can be observed. Thus, it remains +to be determined whether IIT is compatible with quantum mechanics [19,20]. +Here, we are interested in the question of whether it is possible to apply or extend the +causal framework of IIT to quantum mechanics, starting with IIT’s measure of mechanism +arXiv:2301.02244v1 [quant-ph] 4 Jan 2023 + +BY2 of 24 +integrated information (ϕ) [4,6]. Several attempts to apply the general principles of IIT to +quantum systems have recently been proposed [21–23]. Of these, the work by Zanardi et al. +[22] comes closest to a direct translation of the previous version of the theory (“IIT 3.0”) [1] into +a quantum-mechanical framework. However, this translation is not unique, does not converge +to the classical formalism for essentially classical state updates, and also does not explicitly +take the philosophical grounding of IIT as a theory of consciousness into account. +Our objective is to accurately transform the various steps of the IIT formalism in its latest +iteration (“IIT 4.0”) [4,6] to be applicable to both classical and quantum systems. As a first step, +here we propose an extension of the IIT formalism to evaluate the integrated information (ϕ) +of a mechanism within a system [6] to quantum mechanisms (e.g. quantum logic gates). To +enable a direct quantitative comparison, quantum integrated information should converge to +the classical formulation if the quantum system under consideration has a classical analog. Our +main contributions, of merit beyond the scope of IIT, are (1) the translation of a newly defined, +unique measure of intrinsic information [6,24] to a quantum density matrix formalism, and (2) +a formulation of the causal constraints specified by a partial quantum state. To that end, we +extend the notion of conditional independence and causal marginalization [18] to accommodate +quantum entanglement. In the results section, we will apply our theoretical developments +to classical computational gates and their quantum analogues (such as the CNOT gate), as +well as quantum states and gates without a classical counterpart. The additional challenges +of evaluating the integrated information of an entire quantum system will be outlined in the +discussion. +While our investigation is based on IIT’s formal framework, it raises questions that apply +to any theory of consciousness and its relation to (micro) physics [21,25,26]. However, we also +want to emphasize that this work is not concerned with the question of whether biological +systems (in particular the brain) should be treated quantum-theoretically or classically. The +question of whether a theory of consciousness is generally applicable across microscopic and +macroscopic scales and thus consistent with our knowledge of micro physics is important +in either case. Our work is also not directly related to the potential role of consciousness in +quantum measurements and the operational collapse of the wave function [27–29], although we +briefly discuss several difficulties in applying IIT’s causal analysis to measurement dynamics. +At the very least, our results should inform theoretical arguments about the link between +consciousness, causation, and physics from the classical to the quantum [30]. Finally, the +compositional nature of the IIT analysis might also shed some light on the internal structure of +composite quantum states and operators that cannot be obtained using standard information- +theoretical analysis. To that end, we provide python code to analyze quantum mechanisms +of two and three qubits, available at https://github.com/Albantakis/QIIT (accessed on 30 +December 2022). +2. Theory +The purpose of IIT’s formal analysis is to evaluate the irreducible causal information that +a system in a particular state specifies about itself. Notably, IIT’s notion of causal information +differs from other information-theoretical measures in multiple ways: it is intrinsic (evaluated +from the perspective of a mechanism within the system), state-dependent (evaluated for +particular states, not state averages), causal (evaluated against all possible counterfactuals of a +system transition [17,18]), and irreducible (evaluated against a partition of the mechanism into +independent parts). Moreover, the IIT analysis is compositional [5]: instead of only analyzing the +system as a whole, or only its elementary components, any system subset counts as a candidate +mechanism that may specify its own irreducible cause and effect within the system. The IIT +analysis thus evaluates the irreducible cause-effect information (ϕ) of every subset of units +within the system [6], which amounts to “unfolding” the system’s cause-effect structure. + +3 of 24 +In the following, we will extend IIT’s ϕ-measure, the integrated information of a mech- +anism, to be applicable to finite-dimensional quantum systems. While the full IIT analysis +assumes a dynamical system of interacting units, mechanism integrated information (ϕ) can +be evaluated in a straightforward manner for any type of input-output logic, such as sets of +logic gates, or whole computational circuits, as well as information channels (see Figure 1 +as an example). For a classical template of our quantum version of mechanism integrated +information (ϕ) we follow Barbosa et al. [6], including minor updates within the most recent +formulation “IIT 4.0” [4], which is briefly reviewed in the following. As a result, the quantum +integrated information of a mechanism as defined below coincides with the classical measure +[4,6] if the quantum system under consideration has a classical analog. +2.1. Classical Systems +In the canonical IIT formalism, a classical physical system S of n interacting units is defined +as a stochastic system S = {S1, S2, . . . , Sn} with finite, discrete state space ΩS = ∏i ΩSi and +current state st ∈ ΩS [6] that evolves according to a transition probability function +TS ≡ p(st+1 | st) = Pr(St+1 = st+1 | St = st), +st, st+1 ∈ ΩS, +(1) +with the additional requirement that S corresponds to a causal network [6]. This implies that +the conditional probabilities p(st+1|st) are well-defined for all possible states +∃ p(st+1|st) ∀ st, st+1 ∈ ΩS, +(2) +with p(st+1|st) = p(st+1|do(st)) [17,18,31,32], where the “do-operator” do(st) indicates that st +is imposed by intervention. Moreover, the individual random variables Si ∈ S are assumed to +be conditionally independent from each other given the preceding state of S, +p(st+1 | st) = +n +∏ +i=1 +p(si,t+1|st), +(3) +which has to be revisited in the quantum case. If S is an open system within a larger universe +U with current state ut ∈ ΩU, variables W = U \ S are treated as fixed background conditions +throughout the causal analysis [1,6]. +A mechanism M ⊆ S is a subset of the system S with current state mt ∈ ΩM. The +intrinsic information that a mechanism M in state mt specifies over a “purview” Zt±1 ⊆ S, is +defined by a difference measure ii(mt, Zt±1), which quantifies how much mt constrains the +state of Zt±1 compared to chance, but also takes its selectivity into account (how much the +mechanism specifies a particular state of Zt±1) [4,6]. The mechanism’s integrated information +ϕ(mt, Z, θ) is then evaluated over the maximal cause and effect states z′ +c/e identified by the +intrinsic information measure. It quantifies how much the mechanism mt constrains z′ +c/e as one +mechanism, compared to a partition θ +θ = {(M(1), Z(1)), (M(2), Z(2)), . . . , (M(k), Z(k))}, +(4) +of the mechanism and purview into k independent parts [1,6]. Below we will define all relevant +quantities for computing the mechanism integrated information ϕ(mt) following Barbosa et al. +[6] with minor updates from [4]. Figure 1 outlines the steps of IIT’s causal analysis for a simple +example system, a COPY-XOR gate. +2.1.1. Cause and effect repertoires +How the state of a mechanism M = m constrains the possible states of a purview Z +is captured by a product probability distribution π(Z|m), which can be computed from the + +4 of 24 +Figure 1. Outline of the IIT analysis applied to a classical COPY-XOR gate. (a) The COPY-XOR gate and its +(deterministic) transition probability function TS depicted by a probability matrix. Unit C is a copy of the +input bit A, and D corresponds to an XOR function of both input bits (A, B). For input state AB = 10 (also +denoted 10AB) the COPY-XOR gate outputs CD = 11 (denoted 11CD). (b) Based on TS, we can identify +the intrinsic effect of a mechanism M in its current state m over a purview Z as the effect state z′e with +maximal intrinsic effect information iie. For m = 10AB and Z = CD, the intrinsic effect is z′e = 11CD. (c) +Next, we assess the irreducibility of the intrinsic effect by computing the integrated information ϕe(m, Z) +over the minimum partition (MIP). (d) To identify the maximally irreducible effect of a mechanism m, +we compare ϕe(m, Z) across all possible effect purviews Z. Here, the maximally irreducible effect of +m = 10AB is z∗e = 11CD, because it specifies a maximum of ϕe and is the largest purview that does so +(see text for details). (e) For a given system, we identify all maximally irreducible causes and effects. +Given the input state AB = 10, the classical IIT analysis identifies two irreducible effects, the first-order +mechanism 1A specifies the effect 1C, and the second-order mechanism 10AB specifies the effect 11CD. +Given the output state CD = 11, the IIT analysis identifies three irreducible mechanisms, including the +mechanism 1D with purview 10AB or 01AB (which are tied). Both intrinsic information (ii) and integrated +information (ϕ) are quantified in “ibit” units (see text below). + +(a) COPY-XOR +Ts=p(st+1st),VSt,St+1E2s +next +state +1 +A= 1 +C = 1 +D +A B +00 +1 +. +9 +0 +D = 1 +current +10 +0 +0 +0 +1 +B = 0 +state +01 +0 +0 +1 +0 +11 +0 +1 +0 +(b) Intrinsic effect of m = 10AB on Z = CD +(c) Irreducibility Pe(m= 10AB,z=11cD) +next +next +state +state +C +0 +c +0 +D +0 +0 +D +0 +A B +A B +ii.= 2.0 +00 +1 +0 +0 +00 +1 +0 +0 +Pe=1.0 +current +10 +0 +0.25 +current +10 +0 +1.9 +0 +0 +state +0 +state +01 +0 +0 +0 +01 +0 +0 +1 +0 +11 +0 +10 +0 +1 1 +0 +1 +0 + 1 +MIE +>z(m=10AB,Z=CD)=11cD +(d) Maximally irreducible effect of m = 10AB +(e) Set of max. irreducible causes and effects +Pe(10AB,Z=CD)=1ibit Iz*=11cD +Effects: +Pe(10AB)= 1ibit Iz* =11cD +Pe(10AB,Z=D)=1ibit +Iz* = 1D +Pe(1A)= 1ibit +I z* = 1c +Pe(10AB,Z=C)=0 ibit +Causes: +Pc(11cD) = 1ibit Iz* = 10AB +→z*(10AB) =11cD +Pc(1c)=1ibit +I z* = 1A +→e(10AB) = 1 ibit +c(1D) = 0.5 ibit Iz* = [10,01}AB5 of 24 +system’s transition probability function (Eqn. 1) [1,6,18]. Specifically, πc(Z|m) = π(Zt−1|mt) +is the “cause repertoire” of m over Z, and πe(Z|m) = π(Zt+1|mt) is the “effect repertoire”. +Without lack of generality, in what follows we will focus on effects of mt on purviews Z = Zt+1 +and omit update indices (t − 1, t, t + 1) unless necessary. +To capture the constraints on Z that are due to the mechanism in its state (M = m) and +nothing else, it is important to remove any contributions to the repertoire from outside the +mechanism. This is done by “causally marginalizing” all variables in X = S \ M [1,6,18]. When +evaluating the constraints of m onto a single unit Zi ∈ Z, causal marginalization amounts to +imposing a uniform distribution as p(Xt). The effect repertoire of a single unit Zi ∈ Z is thus +defined as +πe(Zi | m) = |ΩX|−1 ∑ +xt∈ΩX +p(Zi,t+1 | mt, xt). +(5) +In the general case of an effect repertoire over a set Z of |Z| units (where |Z| denotes the +cardinality of the set of units Z), each Zi ∈ Z must receive independent inputs from units in +X = S \ M to discount correlations from units in X with divergent outputs to multiple units in Z +(see Figure 2). Formally, this amounts to using product probabilities π(Z|m) instead of standard +conditional probabilities p(Z|m) (again imposing a uniform interventional distribution). The +effect repertoire over a set Z of |Z| units Zi is thus defined as the product of the effect repertoires +over individual units +πe(Z | m) = +|Z| +� +i=1 +πe(Zi | m), +(6) +where � is the Kronecker product of the probability distributions. As in [4], we define the +unconstrained effect repertoire as the marginal distribution +πe(Z; M) = |ΩM|−1 ∑ +m∈ΩM +πe(Z | m). +(7) +The cause repertoire πc(Z|m) is obtained using Bayes’ rule over the product distributions of +the corresponding effect repertoire (for details see [4,6]). The unconstrained cause repertoire +πc(Z) is simply the uniform distribution over the states of Z. +2.1.2. Intrinsic difference (ID) +The classical version of mechanism integrated information (ϕ) evaluates the difference +between two probability distributions P = [p1, ..., pN] and Q = [q1, ..., qN] based on a newly +developed information measure, the “intrinsic difference” (ID) [6,24]. The ID measure is +uniquely defined based on three desired properties: causality, specificity, and intrinsicality, +which align with the postulates of IIT [4,6,24]. Specifically, +ID(P, Q) = max +α +� +pα log +� pα +qα +�� +, +(8) +where α denotes a particular state in the distribution. +Formally, the ID is related to the Kullback-Leibler Divergence (KLD) or “relative entropy” +measure, +KLD(P, Q) = ∑ +α +pα log +� pα +qα +� +. +(9) + +6 of 24 +Figure 2. Causal marginalization. Let us assume we want to identify the effect of the input bit B = 0 (or +0B) on the output CD in the COPY-XOR system of Figure 1. Intuitively, by itself, 0B does not have an effect +on C, as it does not input into C. It also has no effect on D, because, by itself, it specifies no information +about the output state of the XOR D. However, simply marginalizing the input A (averaging over all +possible input states of A, while maintaining the common inputs from A to C and D) would result in +a “spurious” correlation between the output bits that is not due to B, but instead due to the common +inputs from A. Capturing the fact that 0B by itself has no effect on CD requires causal marginalization +(independent marginal inputs to each unit in the effect purview). +While the KLD can be viewed as an average of the point-wise mutual information log +� pα +qα +� +across states, the ID is instead defined based on the state that maximizes the difference between +distributions (specificity property). For fully selective distributions (there is one state with +probability one), the ID thus coincides with the KLD and is additive. Otherwise, the ID is +subadditive and decreases with indeterminism (intrinsicality property). As argued in [6], this +allows the ID to capture the information specified by a mechanism within a particular system. +From the perspective of a mechanism the system has to be taken as is (intrinsic perspective), +while the KLD evaluates information from the perspective of a channel designer with the +possibility to perform error correction (extrinsic perspective) [24]. To highlight this difference, +the unit assigned to the ID measure is labeled an “ibit” or “intrinsic bit”. Logarithms are +evaluated with base 2 throughout. Formally, the “ibit” corresponds to a point-wise information +value measured in bits weighted by a probability. +2.1.3. Identifying intrinsic causes and effects +Based on the intrinsic difference (8), the intrinsic effect information that the mechanism +M = m specifies over a purview Z can be quantified by comparing its effect repertoire πe(Z|m) +to chance, that is, to the unconstrained effect repertoire πe(Z; M) (7), +iie(m, Z) = ID(πe(Z|m), πe(Z; M)) +(10) +The specific state z′ +e ∈ ΩZ over which (10) is maximized corresponds to the intrinsic effect +of the mechanism M = m on the purview Z, + +What is the effect of O on CD? +Marginalizeoverthe stateof A +Causally marginalizeA +(0,1]A +[0,1]A +[0,1]A +c +(0,1)A +C +(0,1)A +[0,1]A +O b +D +[0,1]B +Ob +D +{0,1)B +D +1 +p(CD|0B) +1 +p(CD) +元e(CDIOB) +πe(CD;B) +元e（C)?元e(DIOB) +0.5 +0.5 +0.5 +0.5 +0.25 +0.25 +0.25 +0.25 +00 +10 +TO +11 +00 +1 +0 +1 +00 +1 +0 +11 +0 +1 +0 +11 +0 +1 +1 +0 +1 +0 +0 +D(p(CDIOB) II p(CD) ) > 0 +D(e(CDI0)Ie(CD;B))=0 +Causal marginalization is required to discount extraneous correlations7 of 24 +z′ +e(m, Z) = argmax +z∈ΩZ +� +πe(Z|m) log +� πe(Z|m) +πe(Z; M) +�� +. +(11) +The intrinsic cause z′ +c(m, Z) is defined in the same way based on the respective cause repertoires. +(Note that the definition of the intrinsic cause information iic and, consequently, also the +integrated cause information ϕc, has been updated in [4] compared to [6]. However, this +update of the classical formulation is of no consequence in the quantum case and is thus not +further discussed herein.) +2.1.4. Disintegrating partitions +The integrated effect information ϕe(m, Z, θ) quantifies how much the mechanism m +specifies the intrinsic effect z′ +e(m, Z) as one mechanism and is assessed by comparing the effect +probability π(z′ +e | m) to a partitioned effect probability πθ +e (z′ +e | m) in which certain connections +from M to Z are severed (causally marginalized). +Barbosa et al. [6] (see also [4,18]) define the set of possible partitions θ ∈ Θ(M, Z) as +Θ(M, Z) = +� +{(M(i), Z(i))}k +i=1 +�����k ∈ {2, 3, 4, . . .}, M(i) ∈ P(M), Z(i) ∈ P(Z), +� +M(i) = M, +� +Z(i) = Z, Z(i) ∩ Z(j) = M(i) ∩ M(j) = ∅ ∀ i ̸= j, M(i) = M =⇒ Z(i) = ∅ +� +. +(12) +In words, for each θ ∈ Θ(M, Z) it holds that {M(i)} is a partition of M and {Z(i)} is a partition +of Z (as indicated in Eqn. (4)), but the empty set may also be used as a part (P denotes the +powerset). However, if the whole mechanism is one part (M(i) = M), then it must be cut away +from the entire purview. This definition guarantees that any θ ∈ Θ(M, Z) is a “disintegrating +partition” of {M, Z}: it either “cuts” the mechanism into at least two independent parts if +|M| > 1, or it severs all connections between M and Z, which is always the case if |M| = 1, +where again |M| denotes the cardinality of the set of units M. +Given a partition θ ∈ Θ(M, Z) constituted of k parts (see Eq. (12)), we can define the +partitioned effect repertoire +πθ +e (Z | m) = +k +� +i=1 +πe(Z(i) | m(i)), +(13) +with π(∅|m(i)) = π(∅) = 1. In the case of m(i) = ∅, πe(Z(i)|∅) corresponds to the fully +partitioned effect repertoire +πe(Z | ∅) = +|Z| +� +i=1 ∑ +st∈ΩS +p(Zi,t+1 | st)|ΩS|−1. +(14) +2.1.5. Mechanism integrated information +In all, the general form of ϕe(m, Z, θ) corresponds to that of the intrinsic difference ID (8), +albeit over the specific effect state z′ +e +ϕe(m, Z, θ) = ϕe(m, z′ +e, θ) = πe(z′ +e | m) log +� πe(z′ +e | m) +πθe (z′e | m) +� +. +(15) + +8 of 24 +Quantifying the integrated effect information of a mechanism mt within a system S, +moreover requires an optimization across all possible partitions θ ∈ Θ to identify the minimum +partition (MIP) +θ′ = argmin +θ∈Θ(M,Z) +ϕe(m, Z, θ) +max +T ′ +S +ϕe(m, Z, θ). +(16) +The normalization factor maxT ′ +S ϕe(m, Z, θ) ensures that the minimum partition is evaluated +against its maximum possible value across all possible system T ′ +S of the same dimensions as the +original system. It was introduced in [4] and shown to correspond to the number of possible +pairwise interactions affected by the partition. +The integrated effect information of a mechanism over a particular purview Z then +corresponds to ϕe(m, Z) = ϕe(m, Z, θ′) (which is not normalized, see [4]). Within system S, +ϕe(m) is then defined as the integrated effect information of m evaluated across all possible +purviews Z ⊆ S with ϕe(m) = max +Z +ϕe(m, Z). +The effect purview +Z∗ +e (m) = argmax +Z⊆S +ϕe(m, Z), +(17) +in state +z∗ +e (m) = argmax +{z′e|Z⊆S} +ϕ(m, Z = z′ +e) = argmax +{z′e|Z⊆S} +� +πe(z′ +e | m) log +� πe(z′ +e | m) +πθ′ +e (z′e | m) +�� +(18) +corresponds to the maximally irreducible intrinsic effect of M = m within S. +To summarize, +ϕe(m) = ϕ(m, z∗ +e ) = max +Z⊆S +� +πe(z′ +e | m) log +� πe(z′ +e | m) +πθ′ +e (z′e | m) +�� +, +(19) +with θ′ as in (16) and analogously for ϕc(m). +Finally, the set of all irreducible causes and effects {z∗ +c/e : m ⊆ s, ϕc/e(m) > 0} within a +system S in state s forms the basis of the system’s state-dependent cause-effect structure. +(While the value ϕe(m) is unique, there may be multiple purviews Z∗ +e , or multiple states +z∗ +e within a purview Z∗ +e that maximize ϕe(m) [4,6,33,34]. As outlined in IIT 4.0 [4], such ties in +z∗ +e are resolved according to the “maximum existence principle” at the system level by selecting +the z∗ +e that maximizes the amount of structured information Φ within the system. Here, we +apply the simplified criterion that larger purviews are selected in case of ties across purviews +with different numbers of units |Ze|, as larger purviews typically allow for larger Φ values. +Any remaining ties are reported in the examples below.) +2.2. Quantum Systems +Our objective is to define a quantum version of IIT’s mechanism integrated information +ϕ(m) that is applicable to composite quantum systems and coincides with the classical measure +[4,6] if there is a classical analog to the quantum system. To that end, we start with a composite +quantum system Q in state ρ = ∑s|ψs⟩⟨ψs|, which can be pure or mixed and is described by its +density matrix [22,23]. +Q consists of n units H1, . . . , Hn, which are each described by a finite dimensional Hilbert +space such that HQ = �n +i=1 Hi. Without lack of generality [16], we will focus on systems +constituted of n qubits. The system’s time evolution is defined by a completely positive +(trace-preserving) linear map T = {Tα} [35], as + +9 of 24 +ρt+1 = T (ρt) = ∑ +α +TαρtT† +α. +(20) +We will mainly consider unitary transformations (U) +ρt+1 = UρtU†, +(21) +where U†U = 1, which means that U is reversible and the inverse of U corresponds to its +adjoint (U−1 = U†). However, we will also address quantum measurements F = {Fα} with +∑α F† +α Fα = I, where the probability of obtaining the result α is given by Pr(α) = tr(F† +α Fαρt) +in the discussion section. If Q is an open system with environment E, such that the joint +system evolves under a unitary transformation, we can evaluate the subsystem Q by treating +the environment E in its current state et as a fixed background condition (but see Section 4.3 +below). +A mechanism M ⊆ Q is a subset of Q with current state m = ρM +t += trM′(ρt) within the +corresponding Hilbert space HM = � +i∈M Hi, where M′ = Q \ M and trM′ denotes the trace +over the Hilbert space HM′. +The quantum integrated information of a mechanism M should quantify how much the +state ρM +t +constrains the state of a purview, a system subset Zt±1 ⊆ Q, before or after an update +T of the system, compared to a partition θ of the mechanism and purview into k independent +parts (Eqn. (4)). As above, we will omit the update indices (t − 1, t, t + 1) unless necessary and +focus on effects. +2.2.1. Quantum cause and effect repertoires +To translate the cause and effect repertoires into a density matrix description, we first treat +the special case of a single purview node Z = Zi with |Z| = 1, for which πe(Z|m) = p(Zt+1|mt) +in the classical case. Replacing the probability distributions with the corresponding density +matrices, we obtain +πe(Zi|m) = ρZi|m +t+1 = trZ′ +i +� +T (ρM ⊗ ρM′ +mm) +� +, +(22) +where ’ denotes the complement of a set in Q and ρM′ +mm is the maximally mixed state of M′ = +Q \ M (see also [22,23]). +Next, we consider the case of purviews comprised of multiple units (|Z| > 1). In the +classical case, units in M′ may induce correlations between units in Z, as shown in Figure 2 +by example of the COPY-XOR gate. The quantum equivalent of a classical COPY-XOR gate +is the CNOT gate (Figure 3). For classical inputs, the CNOT behaves identically to the COPY- +XOR gate and thus the same considerations apply. This means that, also in quantum systems, +extraneous correlations should be discounted when evaluating the causal constraints of a +system subset M, since they do not correspond to constraints due to the mechanism M itself. +In the following, we will use ρZ|m +t+1 to denote trZ′ +� +T (ρM ⊗ ρM′ +mm) +� +, while πe(Z|m) corresponds +to the final effect repertoire, after discounting extraneous correlations. +In the quantum case, units in Z may be correlated due to entanglement, which means +quantum systems may violate the conditional independence assumption imposed for classical +systems (Eqn. 3). (Note that incomplete knowledge or a coarse-grained temporal scale can +lead to a violation of conditional independence in a classical system, but those “instantaneous +interactions” are not considered intrinsic to the system and are thus ignored in IIT’s causal +analysis [10]). Simply inserting Eqn. (22) into Eqn. (6) would inadvertently destroy correlations +in Z that are due to entanglement (either preserved or produced during the transformation +T ). In order to correctly capture correlations due to entanglement and discount extraneous + +10 of 24 +correlations due to correlated “noise” from units in M′, the entanglement structure of ρZ|m +t+1 +must be taken into account. +The multipartite entanglement structure of an n-qubit pure state |ψ⟩ can be identified +through partial traces. Following [36], we define a partition Pr(V) = {V(1), . . . , V(r)} with +r = |Pr| ≤ n, � V(i) = V and V(i) � V(j) = ∅ if i ̸= j. +Definition 1. An n-qubit pure state |ψ⟩ is Pr-separable iff it can be written as |ψ⟩ = �r +i=1 +���ψ(i)� +. +In the general case that ρZ|m +t+1 is a mixed state, it has to be decomposed into a convex +mixture of pure states to identify its entanglement structure. +Definition 2. An n-qubit mixed state ρ is Pr-separable iff it can be decomposed into a convex mixture +ρ = ∑s ps|ψs⟩⟨ψs|, with ps ≥ 0, ∀s and ∑s ps = 1, such that every |ψs⟩ in the mixture is a Pr- +separable pure state |ψs⟩ = �r +i=1 +���ψ(i) +s +� +under the same partition Pr. +Note that Definition (2) differs from that in [36], as we require the same partition Pr for +all |ψs⟩ in the mixture. +Definition 3. Out of the set of partitions {Pr}ρ = {Pr|ρ is Pr-separable}, we define the maximal +partition P∗(ρ) as the one with the maximal number of parts r∗ = maxPr r and r∗ = |P∗| ≤ n. +Definition 4. Given the maximal partition P∗ of ρZ|m +t+1 , we can define the quantum effect repertoire of +mechanism m over purview Z as +πe(Z | m) = +r∗ +� +i=1 +πe(Z(i) | m) = +r∗ +� +i=1 +ρZ(i)|m +t+1 +. +(23) +The product in (23) is thus taken over the reduced density matrices of all subsets Z(i) ⊆ Z +that are entangled within themselves but not entangled with the other qubits in Z. Note that P∗ +is a simple set partition, and should not be confused with the disintegrating partitions Θ(M, Z) +(12) used to evaluate the integrated information ϕ(m, Z, θ). Identifying the entanglement +structure for multipartite mixed states remains an area of active research [37–39]. For 2-qubit +mixed states, separability can be determined using the Peres-Horodecki criterion of the positive +partial transform [40,41]. In general, however, this criterion is only a necessary condition for +separability [41] and may thus miss certain complex forms of entanglement [42]. +Several implications follow from the definition of the effect repertoire (23): +1. +If ρZ|m +t+1 corresponds to a pure state, the purview qubits are fully determined by the +mechanism qubits. Thus, ρZ|m +t+1 is not influenced by qubits outside of m. It follows that +πe(Z|m) = ρZ|m +t+1 if the latter is pure. This is analogous to the classical case, where +πe(Z|m) = p(Zt+1|mt) if p(Zt+1|mt) is deterministic. +2. +Conceptually, entangled subsets are treated as indivisible units in the effect repertoire. If +a purview is fully entangled then πe(Z|m) = ρZ|m +t+1 . +3. +Extraneous classical correlations are successfully discounted, which means they will not +contribute to the integrated information of a mechanism (Figure 3). +The cause repertoire of a mechanism in state m over a purview Z also requires causal +marginalization (independent noise applied to conditionally independent subsets) to isolate +the causal constraints of m over Z. In the classical case, the cause repertoire is obtained by + +11 of 24 +applying Bayes’ rule to the effect product probabilities. The quantum case is more complex as +the entanglement structure of ρM might need to be taken into account. +If T is a unitary transformation (21), the cause repertoire for any subset m(i) ∈ P∗(ρM) +that is, itself, mutually entangled (e.g. the subset could consist of an entangled pair of qubits) +but is not entangled with units of other subsets (e.g. other qubits) can be obtained by applying +the adjoint operator T † +πc(Z | m(i)) = ρZ|m(i) +t−1 += trZ′ +� +T †(ρM(i) ⊗ ρM′(i) +mm ) +� +. +(24) +Definition 5. Given the maximal partition P∗ of ρM, we can define the quantum cause repertoire of +mechanism m over purview Z as +πc(Z | m) = +∏r∗ +i=1 πc(Z | m(i)) +tr +� +∏r∗ +i=1 πc(Z | m(i)) +�. +(25) +Note that the product here is over parts of ρM, not of ρZ|m +t−1 . This introduces an asymmetry +in the formulation of cause and effect repertoires, as in the classical case [1,18]. This asymmetry +is a direct implication of treating non-entangled subsets as “physical” causal units, rather than +abstract statistical variables. Causal units are conditionally independent in the present given +the past, but not vice versa. This means that in the effect repertoire, purview subsets that +are not entangled with other units are conditionally independent given the mechanism and +independent noise from outside the mechanism (due to causal marginalization). By contrast, +the cause repertoire is inferred from the conditionally independent mechanism subsets, but +is not itself conditionally independent. The set of effects specified by a quantum state ρt +undergoing a unitary transformation (U) may thus differ from the set of causes specified by +ρt+1 = UρtU† (Figure 3). (The assumption of conditional independence, paired with causal +marginalization, distinguishes IIT’s causal analysis from standard information-theoretical +analyses of information flow [18,32].) +As pointed out in [23], the quantum IIT formalism proposed by Zanardi et al. [22] does +not include causal marginalization (which was formulated in terms of “virtual units” in [1]). +We will show below that causal marginalization (23, 25) is necessary to isolate the causes and +effects of system subsets also in the quantum case—an observation that should be of relevance +to the causal analysis of quantum systems beyond IIT. +2.2.2. Quantum intrinsic information (QID) +Our goal is to define a quantum version of the intrinsic difference measure, which coincides +with the classical measure (8) [6] in the classical case. In quantum information theory, the +classical definition of the KLD (9), or relative entropy, is extended from probability distributions +to density matrices based on the von Neumann entropy. The quantum relative entropy of the +density matrix ρ with respect to another density matrix σ is then defined as: +S(ρ||σ) = Trρ log ρ − Trρ log σ, +(26) +which coincides with the classical case if ρσ = σρ. Unitary operations, including a change +of basis, leave S(ρ||σ) invariant [35]. Specifically, if ρ and σ are expressed as orthonormal +decompositions ρ = ∑i pi|i⟩⟨i| and σ = ∑j qj|j⟩⟨j|, we can write [43] +S(ρ||σ) = ∑ +i +pi +� +log(pi) − ∑ +j +Pij log(qj) +� +, +(27) + +12 of 24 +where Pij = ⟨i|j⟩⟨j|i⟩. In this formulation, a quantum version of the intrinsic difference measure +can be defined as +QID(ρ||σ) = max +i +pi +� +log(pi) − ∑ +j +Pij log(qj) +� +, +(28) +analogous to the classical measure. As for the relative entropy, QID(ρ||σ) coincides with the +classical case if ρσ = σρ, because in that case Pij = δij. Moreover, QID(ρ||σ) = S(ρ||σ) if ρ is +pure, as in the classical case for fully selective distributions. Otherwise, the QID is subadditive, +as desired [24]. +Zanardi et al. [22] proposed the trace distance as a measure of the cause/effect information +based on its simplicity and widespread use in quantum-information theory. The trace distance +quantifies the maximal difference in probability between two quantum states across all possible +POVM measures [43], which is a useful quantity from the perspective of an experimenter. +By contrast, QID is a measure of the intrinsic information of a quantum mechanism. Its +value is maximized over the eigenvectors {|i⟩} of ρ (28). If ρ is pure, there is only one non- +zero eigenvalue and the state identified by the QID measure is simply ρ. If ρ is mixed, the +eigenvalue pi that maximizes equation (28) may be degenerate. In that case the QID specifies +the eigenspace spanned by the set of eigenvectors for which the difference between ρ and σ is +maximal. Otherwise, the QID specifies the eigenvector of ρ with the optimal eigenvalue. +2.2.3. Identifying intrinsic causes and effects +Equipped with the quantum intrinsic difference (QID) measure (28), the intrinsic effect +information that the quantum mechanism M = m specifies over a purview Z can be quantified +as +iie(m, Z) = QID(πe(Z|m), πe(Z)), +(29) +where πe(Z) = πc(Z) = ρZ +mm is the maximally mixed state in the quantum case. +Following from equation (28), with ρ = πe(Z | m) = ∑i pi|i⟩⟨i| as the effect repertoire +and σ = πe(Z) = ∑j qj|j⟩⟨j| = ρZ +mm as the unconstrained effect repertoire, the intrinsic effect of +mechanism m on purview Z is +z′ +e(m, Z) = argmax +i∈HZ +pi +� +log pi − ∑ +j +Pij log(qj) +� += argmax +i∈HZ +pi +� +log pi − log |HZ|−1� +, +(30) +where |HZ| denotes the cardinality of HZ. The intrinsic effect z′ +e(m, Z) is thus simply the +eigenvector |i⟩ of πe(Z|m) with the maximal eigenvalue. If the maximal eigenvalue of ρ = +πe(Z | m) is degenerate, z∗ +e (m) corresponds to the subspace of HZ∗e spanned by the set of +eigenvectors belonging to the maximal eigenvalue (and the same for the intrinsic cause z′ +c(m, Z) +evaluated over πc(Z|m)). +Note that, in the case that πe(Z|m) is a mixed quantum state (corresponding to a probabil- +ity distribution with multiple possible effect states in the classical case), this means that the +intrinsic effect z′ +e(m, Z) differs from ρ = πe(Z|m) = ∑i pi|i⟩⟨i|. + +13 of 24 +2.2.4. Disintegrating partitions +As in the classical case, the quantum integrated information ϕ(m, Z, θ) is evaluated by com- +paring the effect repertoire πe(Z|m) to a partitioned effect repertoire πθ +e (Z|m) (and analogously +for ϕc(m, Z, θ)). +The set of possible partitions θ ∈ Θ(M, Z) is the same as for the classical case (Eqn. 12). +Likewise, the partitioned effect repertoire is defined as in (13), as a product over the parts in +the partition. In the quantum case, πe(Z(i)|∅) corresponds to the maximally mixed state ρZ(i) +mm. +The partitioned cause repertoire is defined in the same way. +Note that the disintegrating partition θ ∈ Θ(M, Z) (12) here is applied on top of P∗ (Defi- +nition 3). Partitioning may thus affect entanglement within the repertoire. Conceptually, any +entanglement in πe(Z | m) that is destroyed by the partition θ will count towards ϕe(m, Z, θ). +Ultimately, however, ϕe(m, Z) is again evaluated over θ′ (16), the minimum information parti- +tion (MIP). This means that everything else being equal, partitions that affect entanglement +less are more likely to correspond to the MIP. +2.2.5. Quantum mechanism integrated information +Having identified the specific effect state z′ +e as an eigenstate |i⟩ of ρ = πe(Z|m), the +integrated effect information ϕ(m, Z, θ) is evaluated as the QID(ρ||σ) over that eigenstate, such +that +ϕ(m, Z, θ) = ϕ(m, z′ +e, θ) = pi +� +log pi − ∑ +j +Pij log(pθ +j ) +� +, +(31) +where σ = πθ +e (Z|m) = ∑j pθ +j |j⟩⟨j| is now the partitioned effect repertoire. +As above, quantifying the integrated effect information ϕe(m) of a mechanism m within a +quantum system Q requires a search over all possible partitions θ ∈ Θ(M, Z) to identify the +MIP, and a search across all possible purviews Z ⊆ Q, such that +ϕe(m) = max +Z⊆Q ϕe(m, Z) = max +Z⊆Q ϕ(m, Z, θ′), +(32) +as in (19), with θ′ as in (16), and analogously for ϕc(m). +The maximally irreducible effect purview Z∗ +e (m) +Z∗ +e (m) = argmax +Z⊆Q +ϕe(m, Z) +(33) +again corresponds to the subset of Q upon which the mechanism M = m has the maximally +irreducible intrinsic effect z∗ +e , which corresponds to the eigenstate of ρ = πe(Z∗|m) that +maximizes Eq. (30), or the eigenspace spanned by a set of eigenvectors corresponding to a +degenerate maximal eigenvalue. +As in the classical case, Z∗ +e is not necessarily unique and we again choose the larger +purview in case of a tie between purviews of different sizes (see above). Any remaining ties are +reported in the examples below. +2.2.6. The intrinsic structure of a quantum system +Standard approaches for studying the causal or informational properties of a system +typically assume either a reductionist perspective (focused on individual units) or holistic +perspective (describing the system as a whole). As the units in a quantum system can be +entangled, focusing on individual units is ill-suited at the quantum level. However, a purely +holistic description of a quantum system will still miss differences in the internal structure of a + +14 of 24 +quantum state (see below the comparison between the maximally entangled GHZ-type and +W-type states [44]). +In IIT, causation is neither reductionist nor holistic but compositional: the IIT analysis +considers the intrinsic causes and effects of every subset within a system and quantifies their +irreducibility as ϕc/e(m) [5]. As a result, it can elucidate the internal structure of composite +quantum states and operators, as we will show in the next section. +We note that typically, the IIT analysis assumes a current system state st and identifies its +compositional causes at t − 1 and effects at t + 1. A subset m ⊆ s with an irreducible cause and +effect forms a “causal distinction” within the system s, where ϕ(m) = min(ϕc(m), ϕe(m)) is +the integrated (cause-effect) information of m. +According to IIT, the phenomenal experience of a physical system S in state s is identical +to its cause-effect structure, composed of a system’s causal distinctions and their relations +[45]. Unfolding the full cause-effect structure requires assessing the integrated (cause-effect) +information ϕ(m) of every subset of units m ⊆ s. +For ease of demonstration, in the following, we will instead evaluate examples of system +transitions from state t to t + 1 and identify the intrinsic effects of the system in state st and the +intrinsic causes of the system in state st+1 (see also [18]). +3. Results +For a direct comparison between classical and quantum systems, we will focus our +attention on computational quantum systems (see [46] for an overview and comparison to +classical systems), constituted of a finite number of quantum units with a finite-dimensional +Hilbert space, evolving in discrete updates according to unitary transformations, expressed in +the computational (or “classical”) basis unless stated otherwise. +To compute classical IIT quantities, we made use of the openly available PyPhi python +toolbox, developed by the Tononi lab [15,16], using the “iit-4.0” feature branch with standard +IIT 4.0 settings. To compute quantum IIT results, we implemented a QIIT toolbox (https: +//github.com/Albantakis/QIIT, accessed 30 December 2022), applicable to unitary quantum +mechanisms of two and three qubits. +3.1. CNOT +3.1.1. Classical case +As a first example, we will evaluate the “controlled-NOT” (CNOT) gate. Classically, the +CNOT gate corresponds to a reversible XOR gate, with a COPY operation performed on the +first input bit (A) and an XOR operation comparing the two input bits A and B as the second +output (Figure 1). For instance, the input state AB = (1, 0) leads to the output CD = (1, 1). In +what follows, we will abbreviate the states of system subsets (mechanisms and purviews) by +the state plus a set subscript, for example, 10AB for AB = (1, 0). +Given the input state AB = (1, 0), the IIT analysis identifies two irreducible mechanisms, +one first-order and one second-order mechanism. The mechanism 1A specifies the effect +purview 1C with ϕ = 1 ibit; the second-order mechanism 10AB specifies the effect purview +11CD also with ϕ = 1 ibit (while there is a tie with the effect 1D, we choose the larger purview +as described above). Notably, 0B by itself (with A replaced by independent noise) does not +specify any information about the next state of CD (Figure 2). While this conclusion should +be straightforward, it relies on the use of product probabilities instead of simple conditional +probabilities (6). The latter would mistakenly count the correlation between C and D as an +effect of B, although it is actually due to the common input A. +By contrast to 0B on the effect side, 1D on the cause side specifies irreducible cause +information about the previous state of AB in addition to 1C and 11CD, albeit only ϕc(1D) = 0.5 + +15 of 24 +ibit due to the remaining uncertainty about the state of AB (note the quantitative difference +between the ID measure (8) and the KLD (9), which would return a value of 1 bit). +3.1.2. Quantum case +For a CNOT gate with the input state ρAB = |10⟩⟨10| (or |10⟩AB), we obtain the same +results as for a COPY-XOR gate with input state AB = (1, 0) using the formalism outlined +above (Figure 3a). With essentially classical inputs, the CNOT gate thus reproduces the intrinsic +causal structure of the classical COPY-XOR gate. +To that end, it was necessary to discount the spurious correlation between qubits A and B +through product distributions (23). This demonstrates that standard conditional probabilities +are insufficient to identify the causes and effects of system subsets also in the quantum case. +Note that for the CNOT gate the role of the “control” (COPY) and the “target” (XOR) qubit +changes depending on the input state, which is not true for the COPY-XOR gate. For an input +state in the Hadamard basis, e.g. |−+⟩AB, information seems to flow from B to C, not A to D as +Figure 3. CNOT gate. The CNOT operator is shown in the top box. (a) For a pure input state in the +classical basis, we obtain the same results as in the classical case (Fig. 1). (b) For a pure input state in the +Hadamard basis, the role of the “control” (here B) and “target” (here A) is reversed compared to (a) (as +indicated in the circuit diagram). (c) The CNOT is often used to produce a “Bell state” of two maximally +entangled qubits. In this exclusively quantum scenario, only the second order mechanisms |+0⟩AB and +��B+� +CD specify an effect or cause, respectively. None of the subsets has any cause or effect information +(ϕ = 0 ibit). (d) Conversely, given the input state |0+⟩AB all second order mechanisms are fully reducible +(ϕ = 0 ibit) and only the first order mechanisms specify causes and effects. + +/ 1.0 +0.0 +0.0 +0.0 +CNOT = +0.0 +1.0 +0.0 +0.0 +0.0 +0.0 +0.0 +1.0 +0.0 +0.0 +1.0 +0.0 / +(a) Classical basis +Effects: +Causes: +Classical equivalent: +Pe(/10)AB) = 1 ibit |z* =[11)cD +c(/11)cp)= 1 ibit +I z* = [10)AB +[1] A +[1)c +(Pe(|1)A)= 1 ibit |z*=|1)c +c(/1)c) = 1 ibit +2* = [1)A +c(/1)p) = 0.5 ibit +Iz* = span(|10),[01)) +[0)B +— [1)D +(b) Hadamard basis +Effects: +Causes: +Classicalequivalent: +e(I- +)AB) = 1 ibit | z* = [- +)cD +c(/- +)cD) = 1 ibit +I 2* = [- +)AB +I-)A +一 - +c(1-)c) = 0.5 ibit +Iz* = span(I- +),I+ -) AB +(Pe(I+)B) = 1 ibit +I z* = |+)D +c(I+)p) = 1 ibit +Iz* = |+)B +I+)B +I+)D +(c) Bell state +Effects: +Causes: +Classical equivalent: +I+)A +Pe(lI+0)AB) = 2 ibit Iz*=[B+)CD +Pc(/B+)cD) = 2 ibit +Iz* = |+0)AB +None +IB+)=(I00)+[11)cD +10)B +(d) No interaction +Effects: +Causes: +Classical equivalent: +[0 A +[0)c +e(l0)A) = 1 ibit +1 2* = |0)c +c(10)c) = 1 ibit +12* = [0)A +Pe(I+)B) = 1 ibit +I z* = |+)D +Pc(/+)p) = 1 ibit +Iz* = |+)B +I+)B +I+)p16 of 24 +for a classical input. Accordingly, the quantum IIT analysis now identifies |+⟩B and |−+⟩AB +as irreducible mechanisms with ϕ = 1 ibit, while |−⟩A by itself does not specify any effect +information (Figure 3b). Yet, |+⟩C does specify irreducible cause information about AB. +In quantum systems, the CNOT is often used to produce the maximally entangled Bell +state |B+⟩ = +1 +√ +2(|00⟩ + |11⟩). CD = |B+⟩ results from the input state AB = |+0⟩, a transition +for which there is no classical circuit equivalent [47]. In this case, the quantum IIT analysis +identifies only the second order mechanisms (constituted of two qubits) |+0⟩AB and |B+⟩CD +with ϕ = 2 ibit each. Individual qubits specify no cause or effect information (Figure 3c). An +analogous result obtains for the Bell state as the input to the CNOT gate. +Finally, with AB = |0+⟩ as the input, there appears to be no interaction between qubits +and the quantum IIT analysis only identifies first order mechanisms on the cause and effect +side (Figure 3d). +3.1.3. Mixed states and extensions to larger systems +The purpose of the IIT analysis is to evaluate the cause-effect power of a system in its +current state. Evaluating statistical ensembles is conceptually not in line with the theory. +Accordingly, the classical IIT analysis always assumes a particular (fully determined) state for +the mechanism m. However, in quantum mechanics, mixed states not only describe statistical +ensembles, but also subsets of entangled pure states. +If we apply an even mixture ρAB = 0.5 ∗ (|00⟩⟨00| + |11⟩⟨11|) to the CNOT gate, we obtain +ρCD = 0.5 ∗ (|00⟩⟨00| + |10⟩⟨10|) as a result. In this case, only the second order mechanism +m = ρAB has an irreducible effect with ϕe = 1.0 ibit over z∗ = |0⟩D. There is no effect on C, +as C by itself is undetermined (maximally mixed). In turn, only |0⟩D specifies an irreducible +cause with ϕ(|0D⟩) = 0.5 ibit over purview Z∗ = AB with z∗ corresponding to the subspace +spanned by |00⟩AB and |11⟩AB (Figure 4a). Note the difference to the causal analysis of the Bell +state |B+⟩ = +1 +√ +2(|00⟩ + |11⟩) above, where |+0⟩AB and |B+⟩CD both specified second order +mechanisms with ϕ = 2 ibit each. +Figure 4. Mixed states and entanglement with the environment. (a) IIT analysis of the CNOT gate with a +mixed input state ρAB = 0.5 ∗ (|00⟩⟨00| + |11⟩⟨11|). (b) It is possible to describe the mixed state as a pure +state entangled with the environment. Analyzing such an extended system for the case in (a), the cause +and effect of the subsystem are preserved in the larger system (gray), but we obtain additional causes and +effects that span all three qubits (black). |GHZ⟩′ denotes a maximally entangled superposition of states +|001⟩ and |110⟩. + +(a) CNOT mixed state +0.5 * (/0)01 + /1)X1D)c +Effects: +Pe(pAB) = 1 ibit +I z* = [0)D +0.5 * (/00)(00| + |11X11)AB +Causes: +10D +Pc(/1)p) = 0.5 ibit +I z* = span(l00),[11)AE +(b) I CNOT +Effects: +[B+) = (I00) + [11))DE +βe(IGHZ)ABc) = 2 ibit Iz* = [B+)AB +Pe(pBC) = 1 ibit +I z* =0>F +IGHZ)=(I000) +[111)ABC +0.5*(/0)0| +|1)(1)E +Causes: +0.5 *(/00)(00| + /11)11))BC +(Pc(|B+0)DEF) = 1 ibit Iz* =|GHZ)ABC +HBC +e(p4) = 0.28 ibit +1 z* = |0)A +Pe(pB) = 0.28 ibit +I z* = [0)B +Pe(p) = 0.28 ibit + z* = [0)c18 of 24 +4. Discussion +Our goal in this study was to extend the mathematical formalism of IIT from discrete, +classical dynamical systems to finite-dimensional quantum systems, starting with IIT’s mech- +anism integrated information ϕ(m) [6]. To that end, we translated IIT’s intrinsic difference +measure [6,24] into a density matrix formalism, and extended the notion of conditional inde- +pendence and causal marginalization [18] to allow for quantum entanglement. Our results +demonstrate that it is possible to extend the applicability of IIT’s formal framework to finite- +dimensional quantum systems evolving according to unitary transformations, such that the +quantum formulation converges to the classical formulation for essentially classical state up- +dates (as demonstrated by the example of the CNOT gate, Figures 1 and 3). In the following, +we will compare our work to previous attempts of applying IIT to quantum systems [21–23], +discuss several difficulties in applying IIT’s causal analysis to measurement dynamics, and +highlight several limitations and implications of our QIIT formalism. +4.1. Comparison with previous approaches +Potential extensions of IIT to quantum systems have been explored in [21–23,28,29]. Of +these, only Zanardi et al. [22] aimed for a direct translation of the IIT formalism (specifically, +“IIT 3.0” [1]) from a classical into a quantum-mechanical framework. As demonstrated by +Kleiner and Tull [23], the quantum IIT formalism proposed in [22] captures the higher-level +mathematical structure of the canonical framework (IIT 3.0). However, it does not converge +to the classical IIT framework and thus does not allow for a quantitative comparison across +quantum and classical systems. Among other differences, Zanardi et al. omitted the causal +marginalization of variables outside the cause or effect repertoires and across partitions. As we +have shown above (Figure 1 and 3), causal marginalization is necessary to identify the causal +constraints specific to a subset of variables within the system also in the quantum case. Paired +with the conditional independence assumption, this also implies that the IIT formalism does +not obey time-reversal symmetry, even when applied to unitary transformations (see also [5] +for classical reversible systems). +Compared to [22], we have, moreover, incorporated several updates of the IIT formalism +from “IIT 3.0” [1] to “IIT 4.0” [4,6]. These include an updated partitioning scheme [6,18], as well +as a novel measure of intrinsic information based on the intrinsic difference (ID) introduced +in [24]. While Zanardi et al. [22] used the trace distance to quantify ϕ, we have developed a +quantum version of the novel intrinsic information measure, starting from the quantum relative +entropy between two density matrices. In combination with the implementation of causal +marginalization in quantum systems, the QIIT formalism proposed above thus converges to +the classical version for essentially classical state updates. +While [22,23] are mainly concerned with the mathematical framework of IIT, [28] and +[29] apply the notion of integrated information within the context of a consciousness-induced +collapse model of quantum mechanics. To that end, Chalmers and McQueen [29] utilize +the QIIT framework proposed in [22]. Kremnitzer and Ranchin [28] present an independent +quantum integrated information measure based on the quantum relative entropy. However, +their measure applies to the quantum state itself and does not take the dynamics of the +quantum system into account. Our work has a different focus. IIT does not require a role for +consciousness in the collapse of the wave function (but see section 4.2 below). +Finally, Tegmark [21] leans on the general principles of IIT’s approach to understand and +explain consciousness in physical systems and addresses the so-called “quantum factorization +problem” [50] using generalized measures of information integration. While we regard the +quantum factorization problem as a serious issue, it is beyond the scope of this work. Our +assumed starting point is a particular density matrix that undergoes a particular unitary +transformation (21). While the QID measure (28) is basis independent, a system’s cause-effect + +19 of 24 +structure and the mechanism integrated information values ϕ(m) of its subsets m ⊆ Q will +typically change under an additional unitary transformation. +4.2. Measurement dynamics +The dynamics of a quantum measurement can be described by a quantum operator +F = {Fα} with ∑α F† +α Fα = I. While the output of a unitary transformation is a density matrix +corresponding to a pure or mixed quantum state, the outcome of a measurement is probabilistic +with Pr(α) = tr(F† +α Fαρt) for measurement outcome α [43]. +The IIT analysis evaluates the potential effects and potential causes of a mechanism in a +state. From the perspective of the quantum state ρt being measured, the measurement outcome +is still unknown. The effect repertoire of the quantum state ρt directly before the measurement +(23) could thus be computed from +ρt+1 = ∑ +α +FαρtF† +α, +(34) +following equation (20). The density matrix ρt+1 then corresponds to a mixed state, that is, a +probability distribution of possible measurement outcomes. +Measurement dynamics become problematic, if we want to evaluate the quantum state +directly after the measurement. Here, the cause repertoire has to be computed from the perspec- +tive of the quantum state post measurement ρα +t+1, corresponding to a particular measurement +outcome α +ρα +t+1 = +FαρtF† +α +tr(F†α Fαρ). +(35) +Since measurements are not unitary transformations, the adjoint operator T † is not the +same as the inverse T −1. For this reason, we cannot use equation (24) to compute the cause +repertoire of ρα +t+1 (note that the same holds for prior proposals [22,23]). +In the classical case, the cause repertoire of an irreversible mechanism can be computed +using Bayes’ Rule [4,6]. However, in the quantum case, all information about the basis of the +original quantum state before the measurement is lost, which means that there are infinitely +many possible past states. While different past states should still be more or less likely, we do +not know of any available method for obtaining a probability distribution of possible causes in +this case. +That said, the amount of cause information specified by a post-measurement state ρα +t+1 +depends on the way the measurement dynamics are conceptualized, and thus on the specific +quantum theory applied. While ρα +t+1 specifies (almost) no cause information under spontaneous +collapse theories, the case may be quite different for deterministic hidden variable theories. No, +or very low, cause information at the quantum level would imply that quantum systems are +poor substrates for consciousness and may offer room for macro level descriptions to reach +maximal values of integrated information, as predicted by IIT. +Finally, the technical difficulties introduced by probabilistic measurement dynamics would +naturally be avoided by so-called “no-collapse” models of quantum mechanics, such as the +Many-Worlds Interpretation. However, theories that rely only on a density matrix encoding +the state of the universe and a unitary transformation determining its time-evolution [21] face +a different issue when it comes to identifying conscious entities through causal, informational, +or computational means. If applied at the fundamental level, any entities obtained would cor- +respond to subsets of the universal density matrix, never subsets within individual “branches” +only (see for example Fig. 3c). While the QIIT measures (and other quantities) could formally +be applied within a branch, there is no principled justification for doing so from the perspective + +20 of 24 +of a fundamental theory of consciousness (note that the notion of decoherence cannot resolve +this issue). +4.3. Formal considerations and limitations +Formally, the restriction to unitary transformations eliminated differences between the +unconstrained cause and effect repertoire that commonly arise in the classical formulation. +Nevertheless, due to the assumption of conditional independence on the effect side, but not the +cause side, cause repertoires are formally distinct from effect repertoires even under unitary +transformations. +The quantum formulation also provides justification for treating all variables outside the +candidate system under consideration as fixed background conditions, which is motivated by +IIT’s intrinsicality postulate [1,18]: by the no-communication theorem [43], any unitary trans- +formation on a system will leave the density matrix of its environment unchanged. However, +not all subsets of unitary transforms are unitary. Future work should explore the implications +of assuming fixed background conditions in such cases. +The IIT formalism for classical systems starts from a transition probability matrix (TPM) +which corresponds to a complete set of transition probabilities (from every possible system state +to every possible system state) (1). This has led some to critize IIT on conceptual grounds, as it +seems to imply that subjective experience would depend not only on the actual states a system +inhabits in the course of its dynamical evolution, but also on hypothetical counterfactuals that +may never happen [51]. In the QIIT formalism, the role of the classical transition probability +matrix (TPM) is assumed by the unitary transform (21) applied to the quantum state. Just as +evolution operators in quantum mechanics essentially are TPMs, in IIT, the TPM simply serves +as a complete description of the system’s dynamics. +In this work, we have focused on mechanism integrated information ϕ [6]. In principle, +it should be possible to formally extend our QIIT formalism to incorporate the full “IIT 4.0” +framework, including the system integrated information (ϕs) [52], a full characterization of the +system’s cause-effect structure comprised of causal distinctions and causal relations [45], and +the amount of structure information (Φ) specified by a system. +Nevertheless, there are several conceptual issues that need to be resolved before the QIIT +formalism can be applied to identify conscious systems, which have to comply with all of IIT’s +requirements for being a substrate of consciousness (IIT’s “postulates”) [1]. For example, it is +unclear whether mixed states should count as permissible states for evaluating the system’s +integrated information. While only specific sets of units, not ensembles, qualify as substrates, +a particular set of units may still be in a mixed state if it is entangled with the environment +(Fig. 4). Yet, IIT’s information postulate requires systems and mechanisms to have specific +cause-effect power. It thus remains to be determined whether mixed states can comply with +IIT’s information postulate. +Recurrent quantum systems are another issue. In the classical formulation, recurrent +connections between system units are required for positive system integrated information [1,52]. +Physical units (e.g., neurons, transistors,...) are thus assumed to be dynamically persistent +variables with at least two possible states. However, it is less obvious whether qubits, or qudits +more generally, may indeed be treated as variables that maintain a causal identity across their +state updates. +4.4. From micro to macro? +Current empirical evidence suggests that consciousness and its contents are correlated +with the dynamics and activity of neurons in some parts of the cerebral cortex [53]. While our +experiences seem to unfold over macroscopic spatial and temporal scales, the brain can, in +principle, be described at a multitude of levels, for example, as a network comprised of a few + +21 of 24 +interacting brain regions, or a microphysical quantum system. Why is it then that the contents +of our experiences correlate with neural activity in certain regions of the cortex rather than +their underlying microphysical processes [9,21]? +IIT offers a single, general principle for identifying conscious systems: a substrate of +consciousness must correspond to a set of units that forms a maximum of intrinsic cause-effect +power over grains of units, updates, and states [2,9,10]. However, it remains to be determined +whether IIT’s propositions are compatible with our current best knowledge about micro physics +[19,20]. +The QIIT formalism presented above allows for a quantitative comparison between clas- +sical and quantum systems. Squaring IIT (as well as any other causal, computational, or +information-based theory of consciousness) with our current knowledge of micro physics, +moreover, requires a method for obtaining macroscopic causal models from microscopic dynam- +ics. This could be achieved by a "black-boxing" of quantum circuits into suitable macro-units +[10], or a quantitative framework that formalizes the emergence of well-defined probability +distributions [54]. +To identify the maximally irreducible description of a system across a hierarchy of spatio- +temporal scales, we have to compare micro- and macro-level descriptions of the same system. +While it is always possible to implement the global function performed by a classical system +with a quantum circuit [43], these systems will typically not have the same causal structure (the +CNOT gate described in Fig. 3 is exceptional in that way). One reason is that quantum gates +have to be reversible, and thus require so-called “ancilla qubits” to implement convergent logic +gates such as AND-gates, or NOR-gates. These ancilla qubits cannot simply be ignored in the +IIT analysis, as this would introduce an observer-dependent, extrinsic perspective. They also +cannot typically be treated as a fixed background conditions. Understanding whether and how +irreversible logic functions might emerge from reversible quantum circuits is thus an important +subject for future investigations. +As is, QIIT and its classical counterpart are only partially overlapping in their domains +of applicability. While QIIT is in principle more fundamental as an extension of IIT’s classical +causal framework to quantum systems, it is currently limited to reversible, unitary transforma- +tions, and thus cannot directly be applied to irreversible processes. +Overall, we see it as a positive development that the updated IIT 4.0 formalism for +computing the mechanism integrated information [6] is readily applicable within a quantum +mechanical framework. Our work revealed several conceptual issues regarding theories of +consciousness as they relate to fundamental physics. Regardless, the theoretical framework for +identifying causes and effects of subsets of units within a quantum system should be of interest +within the field of quantum information theory and quantum causal models more generally. +Author Contributions: “Conceptualization, L.A., R.P., and I.D.; methodology, L.A. and I.D.; software, +L.A.; validation, L.A., R.P., and I.D.; formal analysis, L.A.; investigation, L.A.; writing—original draft +preparation, L.A.; writing—review and editing, R.P. and I.D; visualization, L.A.. All authors have read +and agreed to the published version of the manuscript.” +Funding: L.A. acknowledges the support of a grant from the Templeton World Charity Foundation +(TWCF-2020-20526). +Acknowledgments: The authors thank the Association for Mathematical Consciousness Science (AMCS) +for its institutional support. L.A. thanks William Marshall, Alireza Zaeemzadeh, and Giulio Tononi for +helpful discussions and comments on earlier drafts of this article, Will Mayner for maintaining PyPhi +and his advice on implementing the QIIT toolbox, and Matteo Grasso for his help with figures. R.P. +acknowledges the support of the Munich Center for Mathematical Philosophy. + +22 of 24 +Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of +the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the +decision to publish the results. +References +1. +Oizumi, M.; Albantakis, L.; Tononi, G. From the Phenomenology to the Mechanisms of Con- +sciousness: Integrated Information Theory 3.0. +PLoS Computational Biology 2014, 10, e1003588. +https://doi.org/10.1371/journal.pcbi.1003588. +2. +Tononi, G.; Boly, M.; Massimini, M.; Koch, C. Integrated information theory: from consciousness to +its physical substrate. 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Entropy 2020, 22, 568. + diff --git a/QdE0T4oBgHgl3EQfUACA/content/tmp_files/load_file.txt b/QdE0T4oBgHgl3EQfUACA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d942d06e531d63cee211f8826c02c5759173b108 --- /dev/null +++ b/QdE0T4oBgHgl3EQfUACA/content/tmp_files/load_file.txt @@ -0,0 +1,1311 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf,len=1310 +page_content='Citation: Albantakis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Prentner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Durham, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Measuring the integrated information of a quantum mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Preprints 2023, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='org/ Copyright: © 2023 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Licensee MDPI, Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='org/licenses/by/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Article Measuring the integrated information of a quantum mechanism Larissa Albantakis 1,4* , Robert Prentner 2,4 and Ian Durham 3,4 1 Department of Psychiatry, University of Wisconsin–Madison, Madison, WI 53719, USA 2 Munich Center for Mathematical Philosophy, Ludwig-Maximilians-University Munich, 80539, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='prentner@amcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='science 3 Department of Physics, Saint Anselm College, Manchester, NH 03102, USA ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' idurham@anselm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='edu 4 Association for Mathematical Consciousness Science, Munich, Germany Correspondence: albantakis@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='edu Abstract: Originally conceived as a theory of consciousness, integrated information theory (IIT) provides a theoretical framework intended to characterize the compositional causal information that a system, in its current state, specifies about itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, it remains to be determined whether IIT as a theory of consciousness is compatible with quantum mechanics as a theory of microphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Here, we present an extension of IIT’s latest formalism to evaluate the mechanism integrated information (ϕ) of a system subset to finite-dimensional quantum systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=', quantum logic gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, we translate a recently developed, unique measure of intrinsic information into a density matrix formulation, and extend the notion of conditional independence to accommodate quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The compositional nature of the IIT analysis might shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information-theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Finally, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Keywords: causal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' causation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' quantum information theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' entanglement structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' multivariate interaction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Introduction Integrated information theory [1–4] stands out as one theory of consciousness that explic- itly proposes a formal framework for identifying conscious systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Specifically, IIT provides requirements about the intrinsic causal structure of a system that supports consciousness, based on the essential (“phenomenal”) properties of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Its formal framework evaluates the causal powers that a set of interacting physical units exerts on itself in a compositional manner [1,4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' IIT does not presuppose that consciousness arises at the level of neurons rather than atoms, molecules, or larger brain areas, but assumes causation to be a central concept for analyzing a physical system across the hierarchy from the microphysical to the macroscopic [2,8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' One prediction of IIT is that consciousness appears at the level of organization at which the intrinsic causal powers of a system are maximized [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, IIT offers a formal framework for causal emergence that compares the amount of integrated information of macroscopic causal models to their underlying microscopic system descriptions [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Whether causality plays a fundamental role in physics, in particular in quantum physics, is, however, contested [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' IIT’s causal framework has been formalized for discrete dynamical systems in the classical realm [1,4,14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Accordingly, in prior studies [8–10], micro-level systems corresponded to classical causal networks [17,18], constituted of individual, conditionally independent physical units, that can (in principle) be manipulated and whose states can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Thus, it remains to be determined whether IIT is compatible with quantum mechanics [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Here, we are interested in the question of whether it is possible to apply or extend the causal framework of IIT to quantum mechanics, starting with IIT’s measure of mechanism arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='02244v1 [quant-ph] 4 Jan 2023 BY2 of 24 integrated information (ϕ) [4,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Several attempts to apply the general principles of IIT to quantum systems have recently been proposed [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Of these, the work by Zanardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [22] comes closest to a direct translation of the previous version of the theory (“IIT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0”) [1] into a quantum-mechanical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, this translation is not unique, does not converge to the classical formalism for essentially classical state updates, and also does not explicitly take the philosophical grounding of IIT as a theory of consciousness into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Our objective is to accurately transform the various steps of the IIT formalism in its latest iteration (“IIT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0”) [4,6] to be applicable to both classical and quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As a first step, here we propose an extension of the IIT formalism to evaluate the integrated information (ϕ) of a mechanism within a system [6] to quantum mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' quantum logic gates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To enable a direct quantitative comparison, quantum integrated information should converge to the classical formulation if the quantum system under consideration has a classical analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Our main contributions, of merit beyond the scope of IIT, are (1) the translation of a newly defined, unique measure of intrinsic information [6,24] to a quantum density matrix formalism, and (2) a formulation of the causal constraints specified by a partial quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, we extend the notion of conditional independence and causal marginalization [18] to accommodate quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the results section, we will apply our theoretical developments to classical computational gates and their quantum analogues (such as the CNOT gate), as well as quantum states and gates without a classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The additional challenges of evaluating the integrated information of an entire quantum system will be outlined in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' While our investigation is based on IIT’s formal framework, it raises questions that apply to any theory of consciousness and its relation to (micro) physics [21,25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, we also want to emphasize that this work is not concerned with the question of whether biological systems (in particular the brain) should be treated quantum-theoretically or classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The question of whether a theory of consciousness is generally applicable across microscopic and macroscopic scales and thus consistent with our knowledge of micro physics is important in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Our work is also not directly related to the potential role of consciousness in quantum measurements and the operational collapse of the wave function [27–29], although we briefly discuss several difficulties in applying IIT’s causal analysis to measurement dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' At the very least, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Finally, the compositional nature of the IIT analysis might also shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information- theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, we provide python code to analyze quantum mechanisms of two and three qubits, available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='com/Albantakis/QIIT (accessed on 30 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Theory The purpose of IIT’s formal analysis is to evaluate the irreducible causal information that a system in a particular state specifies about itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Notably, IIT’s notion of causal information differs from other information-theoretical measures in multiple ways: it is intrinsic (evaluated from the perspective of a mechanism within the system), state-dependent (evaluated for particular states, not state averages), causal (evaluated against all possible counterfactuals of a system transition [17,18]), and irreducible (evaluated against a partition of the mechanism into independent parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Moreover, the IIT analysis is compositional [5]: instead of only analyzing the system as a whole, or only its elementary components, any system subset counts as a candidate mechanism that may specify its own irreducible cause and effect within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The IIT analysis thus evaluates the irreducible cause-effect information (ϕ) of every subset of units within the system [6], which amounts to “unfolding” the system’s cause-effect structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3 of 24 In the following, we will extend IIT’s ϕ-measure, the integrated information of a mech- anism, to be applicable to finite-dimensional quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' While the full IIT analysis assumes a dynamical system of interacting units, mechanism integrated information (ϕ) can be evaluated in a straightforward manner for any type of input-output logic, such as sets of logic gates, or whole computational circuits, as well as information channels (see Figure 1 as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For a classical template of our quantum version of mechanism integrated information (ϕ) we follow Barbosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [6], including minor updates within the most recent formulation “IIT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0” [4], which is briefly reviewed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As a result, the quantum integrated information of a mechanism as defined below coincides with the classical measure [4,6] if the quantum system under consideration has a classical analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Classical Systems In the canonical IIT formalism, a classical physical system S of n interacting units is defined as a stochastic system S = {S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' , Sn} with finite, discrete state space ΩS = ∏i ΩSi and current state st ∈ ΩS [6] that evolves according to a transition probability function TS ≡ p(st+1 | st) = Pr(St+1 = st+1 | St = st), st, st+1 ∈ ΩS, (1) with the additional requirement that S corresponds to a causal network [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This implies that the conditional probabilities p(st+1|st) are well-defined for all possible states ∃ p(st+1|st) ∀ st, st+1 ∈ ΩS, (2) with p(st+1|st) = p(st+1|do(st)) [17,18,31,32], where the “do-operator” do(st) indicates that st is imposed by intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Moreover, the individual random variables Si ∈ S are assumed to be conditionally independent from each other given the preceding state of S, p(st+1 | st) = n ∏ i=1 p(si,t+1|st), (3) which has to be revisited in the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If S is an open system within a larger universe U with current state ut ∈ ΩU, variables W = U \\ S are treated as fixed background conditions throughout the causal analysis [1,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' A mechanism M ⊆ S is a subset of the system S with current state mt ∈ ΩM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The intrinsic information that a mechanism M in state mt specifies over a “purview” Zt±1 ⊆ S, is defined by a difference measure ii(mt, Zt±1), which quantifies how much mt constrains the state of Zt±1 compared to chance, but also takes its selectivity into account (how much the mechanism specifies a particular state of Zt±1) [4,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The mechanism’s integrated information ϕ(mt, Z, θ) is then evaluated over the maximal cause and effect states z′ c/e identified by the intrinsic information measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' It quantifies how much the mechanism mt constrains z′ c/e as one mechanism, compared to a partition θ θ = {(M(1), Z(1)), (M(2), Z(2)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' , (M(k), Z(k))}, (4) of the mechanism and purview into k independent parts [1,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Below we will define all relevant quantities for computing the mechanism integrated information ϕ(mt) following Barbosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [6] with minor updates from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Figure 1 outlines the steps of IIT’s causal analysis for a simple example system, a COPY-XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Cause and effect repertoires How the state of a mechanism M = m constrains the possible states of a purview Z is captured by a product probability distribution π(Z|m), which can be computed from the 4 of 24 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Outline of the IIT analysis applied to a classical COPY-XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (a) The COPY-XOR gate and its (deterministic) transition probability function TS depicted by a probability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Unit C is a copy of the input bit A, and D corresponds to an XOR function of both input bits (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For input state AB = 10 (also denoted 10AB) the COPY-XOR gate outputs CD = 11 (denoted 11CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (b) Based on TS, we can identify the intrinsic effect of a mechanism M in its current state m over a purview Z as the effect state z′e with maximal intrinsic effect information iie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For m = 10AB and Z = CD, the intrinsic effect is z′e = 11CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (c) Next, we assess the irreducibility of the intrinsic effect by computing the integrated information ϕe(m, Z) over the minimum partition (MIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (d) To identify the maximally irreducible effect of a mechanism m, we compare ϕe(m, Z) across all possible effect purviews Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Here, the maximally irreducible effect of m = 10AB is z∗e = 11CD, because it specifies a maximum of ϕe and is the largest purview that does so (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (e) For a given system, we identify all maximally irreducible causes and effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given the input state AB = 10, the classical IIT analysis identifies two irreducible effects, the first-order mechanism 1A specifies the effect 1C, and the second-order mechanism 10AB specifies the effect 11CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given the output state CD = 11, the IIT analysis identifies three irreducible mechanisms, including the mechanism 1D with purview 10AB or 01AB (which are tied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Both intrinsic information (ii) and integrated information (ϕ) are quantified in “ibit” units (see text below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (a) COPY-XOR Ts=p(st+1st),VSt,St+1E2s next state 1 A= 1 C = 1 D A B 00 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 9 0 D = 1 current 10 0 0 0 1 B = 0 state 01 0 0 1 0 11 0 1 0 (b) Intrinsic effect of m = 10AB on Z = CD (c) Irreducibility Pe(m= 10AB,z=11cD) next next state state C 0 c 0 D 0 0 D 0 A B A B ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 00 1 0 0 00 1 0 0 Pe=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 current 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='25 current 10 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='9 0 0 state 0 state 01 0 0 0 01 0 0 1 0 11 0 10 0 1 1 0 1 0 1 MIE >z(m=10AB,Z=CD)=11cD (d) Maximally irreducible effect of m = 10AB (e) Set of max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' irreducible causes and effects Pe(10AB,Z=CD)=1ibit Iz*=11cD Effects: Pe(10AB)= 1ibit Iz* =11cD Pe(10AB,Z=D)=1ibit Iz* = 1D Pe(1A)= 1ibit I z* = 1c Pe(10AB,Z=C)=0 ibit Causes: Pc(11cD) = 1ibit Iz* = 10AB →z*(10AB) =11cD Pc(1c)=1ibit I z* = 1A →e(10AB) = 1 ibit c(1D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ibit Iz* = [10,01}AB5 of 24 system’s transition probability function (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 1) [1,6,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Specifically, πc(Z|m) = π(Zt−1|mt) is the “cause repertoire” of m over Z, and πe(Z|m) = π(Zt+1|mt) is the “effect repertoire”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Without lack of generality, in what follows we will focus on effects of mt on purviews Z = Zt+1 and omit update indices (t − 1, t, t + 1) unless necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To capture the constraints on Z that are due to the mechanism in its state (M = m) and nothing else, it is important to remove any contributions to the repertoire from outside the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This is done by “causally marginalizing” all variables in X = S \\ M [1,6,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' When evaluating the constraints of m onto a single unit Zi ∈ Z, causal marginalization amounts to imposing a uniform distribution as p(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The effect repertoire of a single unit Zi ∈ Z is thus defined as πe(Zi | m) = |ΩX|−1 ∑ xt∈ΩX p(Zi,t+1 | mt, xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (5) In the general case of an effect repertoire over a set Z of |Z| units (where |Z| denotes the cardinality of the set of units Z), each Zi ∈ Z must receive independent inputs from units in X = S \\ M to discount correlations from units in X with divergent outputs to multiple units in Z (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Formally, this amounts to using product probabilities π(Z|m) instead of standard conditional probabilities p(Z|m) (again imposing a uniform interventional distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The effect repertoire over a set Z of |Z| units Zi is thus defined as the product of the effect repertoires over individual units πe(Z | m) = |Z| � i=1 πe(Zi | m), (6) where � is the Kronecker product of the probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As in [4], we define the unconstrained effect repertoire as the marginal distribution πe(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' M) = |ΩM|−1 ∑ m∈ΩM πe(Z | m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (7) The cause repertoire πc(Z|m) is obtained using Bayes’ rule over the product distributions of the corresponding effect repertoire (for details see [4,6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The unconstrained cause repertoire πc(Z) is simply the uniform distribution over the states of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Intrinsic difference (ID) The classical version of mechanism integrated information (ϕ) evaluates the difference between two probability distributions P = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=', pN] and Q = [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=', qN] based on a newly developed information measure, the “intrinsic difference” (ID) [6,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The ID measure is uniquely defined based on three desired properties: causality, specificity, and intrinsicality, which align with the postulates of IIT [4,6,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Specifically, ID(P, Q) = max α � pα log � pα qα �� , (8) where α denotes a particular state in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Formally, the ID is related to the Kullback-Leibler Divergence (KLD) or “relative entropy” measure, KLD(P, Q) = ∑ α pα log � pα qα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (9) 6 of 24 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Causal marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Let us assume we want to identify the effect of the input bit B = 0 (or 0B) on the output CD in the COPY-XOR system of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Intuitively, by itself, 0B does not have an effect on C, as it does not input into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' It also has no effect on D, because, by itself, it specifies no information about the output state of the XOR D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, simply marginalizing the input A (averaging over all possible input states of A, while maintaining the common inputs from A to C and D) would result in a “spurious” correlation between the output bits that is not due to B, but instead due to the common inputs from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Capturing the fact that 0B by itself has no effect on CD requires causal marginalization (independent marginal inputs to each unit in the effect purview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' While the KLD can be viewed as an average of the point-wise mutual information log � pα qα � across states, the ID is instead defined based on the state that maximizes the difference between distributions (specificity property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For fully selective distributions (there is one state with probability one), the ID thus coincides with the KLD and is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Otherwise, the ID is subadditive and decreases with indeterminism (intrinsicality property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As argued in [6], this allows the ID to capture the information specified by a mechanism within a particular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' From the perspective of a mechanism the system has to be taken as is (intrinsic perspective), while the KLD evaluates information from the perspective of a channel designer with the possibility to perform error correction (extrinsic perspective) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To highlight this difference, the unit assigned to the ID measure is labeled an “ibit” or “intrinsic bit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Logarithms are evaluated with base 2 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Formally, the “ibit” corresponds to a point-wise information value measured in bits weighted by a probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Identifying intrinsic causes and effects Based on the intrinsic difference (8), the intrinsic effect information that the mechanism M = m specifies over a purview Z can be quantified by comparing its effect repertoire πe(Z|m) to chance, that is, to the unconstrained effect repertoire πe(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' M) (7), iie(m, Z) = ID(πe(Z|m), πe(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' M)) (10) The specific state z′ e ∈ ΩZ over which (10) is maximized corresponds to the intrinsic effect of the mechanism M = m on the purview Z, What is the effect of O on CD?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Marginalizeoverthe stateof A Causally marginalizeA (0,1]A [0,1]A [0,1]A c (0,1)A C (0,1)A [0,1]A O b D [0,1]B Ob D {0,1)B D 1 p(CD|0B) 1 p(CD) 元e(CDIOB) πe(CD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='B) 元e（C)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='元e(DIOB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='25 00 10 TO 11 00 1 0 1 00 1 0 11 0 1 0 11 0 1 1 0 1 0 0 D(p(CDIOB) II p(CD) ) > 0 D(e(CDI0)Ie(CD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='B))=0 Causal marginalization is required to discount extraneous correlations7 of 24 z′ e(m, Z) = argmax z∈ΩZ � πe(Z|m) log � πe(Z|m) πe(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' M) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (11) The intrinsic cause z′ c(m, Z) is defined in the same way based on the respective cause repertoires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (Note that the definition of the intrinsic cause information iic and, consequently, also the integrated cause information ϕc, has been updated in [4] compared to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, this update of the classical formulation is of no consequence in the quantum case and is thus not further discussed herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Disintegrating partitions The integrated effect information ϕe(m, Z, θ) quantifies how much the mechanism m specifies the intrinsic effect z′ e(m, Z) as one mechanism and is assessed by comparing the effect probability π(z′ e | m) to a partitioned effect probability πθ e (z′ e | m) in which certain connections from M to Z are severed (causally marginalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Barbosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [6] (see also [4,18]) define the set of possible partitions θ ∈ Θ(M, Z) as Θ(M, Z) = � {(M(i), Z(i))}k i=1 �����k ∈ {2, 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' }, M(i) ∈ P(M), Z(i) ∈ P(Z), � M(i) = M, � Z(i) = Z, Z(i) ∩ Z(j) = M(i) ∩ M(j) = ∅ ∀ i ̸= j, M(i) = M =⇒ Z(i) = ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (12) In words, for each θ ∈ Θ(M, Z) it holds that {M(i)} is a partition of M and {Z(i)} is a partition of Z (as indicated in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (4)), but the empty set may also be used as a part (P denotes the powerset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, if the whole mechanism is one part (M(i) = M), then it must be cut away from the entire purview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This definition guarantees that any θ ∈ Θ(M, Z) is a “disintegrating partition” of {M, Z}: it either “cuts” the mechanism into at least two independent parts if |M| > 1, or it severs all connections between M and Z, which is always the case if |M| = 1, where again |M| denotes the cardinality of the set of units M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given a partition θ ∈ Θ(M, Z) constituted of k parts (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (12)), we can define the partitioned effect repertoire πθ e (Z | m) = k � i=1 πe(Z(i) | m(i)), (13) with π(∅|m(i)) = π(∅) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the case of m(i) = ∅, πe(Z(i)|∅) corresponds to the fully partitioned effect repertoire πe(Z | ∅) = |Z| � i=1 ∑ st∈ΩS p(Zi,t+1 | st)|ΩS|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Mechanism integrated information In all, the general form of ϕe(m, Z, θ) corresponds to that of the intrinsic difference ID (8), albeit over the specific effect state z′ e ϕe(m, Z, θ) = ϕe(m, z′ e, θ) = πe(z′ e | m) log � πe(z′ e | m) πθe (z′e | m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (15) 8 of 24 Quantifying the integrated effect information of a mechanism mt within a system S, moreover requires an optimization across all possible partitions θ ∈ Θ to identify the minimum partition (MIP) θ′ = argmin θ∈Θ(M,Z) ϕe(m, Z, θ) max T ′ S ϕe(m, Z, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (16) The normalization factor maxT ′ S ϕe(m, Z, θ) ensures that the minimum partition is evaluated against its maximum possible value across all possible system T ′ S of the same dimensions as the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' It was introduced in [4] and shown to correspond to the number of possible pairwise interactions affected by the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The integrated effect information of a mechanism over a particular purview Z then corresponds to ϕe(m, Z) = ϕe(m, Z, θ′) (which is not normalized, see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Within system S, ϕe(m) is then defined as the integrated effect information of m evaluated across all possible purviews Z ⊆ S with ϕe(m) = max Z ϕe(m, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The effect purview Z∗ e (m) = argmax Z⊆S ϕe(m, Z), (17) in state z∗ e (m) = argmax {z′e|Z⊆S} ϕ(m, Z = z′ e) = argmax {z′e|Z⊆S} � πe(z′ e | m) log � πe(z′ e | m) πθ′ e (z′e | m) �� (18) corresponds to the maximally irreducible intrinsic effect of M = m within S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To summarize, ϕe(m) = ϕ(m, z∗ e ) = max Z⊆S � πe(z′ e | m) log � πe(z′ e | m) πθ′ e (z′e | m) �� , (19) with θ′ as in (16) and analogously for ϕc(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Finally, the set of all irreducible causes and effects {z∗ c/e : m ⊆ s, ϕc/e(m) > 0} within a system S in state s forms the basis of the system’s state-dependent cause-effect structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (While the value ϕe(m) is unique, there may be multiple purviews Z∗ e , or multiple states z∗ e within a purview Z∗ e that maximize ϕe(m) [4,6,33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As outlined in IIT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 [4], such ties in z∗ e are resolved according to the “maximum existence principle” at the system level by selecting the z∗ e that maximizes the amount of structured information Φ within the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Here, we apply the simplified criterion that larger purviews are selected in case of ties across purviews with different numbers of units |Ze|, as larger purviews typically allow for larger Φ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Any remaining ties are reported in the examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Quantum Systems Our objective is to define a quantum version of IIT’s mechanism integrated information ϕ(m) that is applicable to composite quantum systems and coincides with the classical measure [4,6] if there is a classical analog to the quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, we start with a composite quantum system Q in state ρ = ∑s|ψs⟩⟨ψs|, which can be pure or mixed and is described by its density matrix [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Q consists of n units H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' , Hn, which are each described by a finite dimensional Hilbert space such that HQ = �n i=1 Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Without lack of generality [16], we will focus on systems constituted of n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The system’s time evolution is defined by a completely positive (trace-preserving) linear map T = {Tα} [35], as 9 of 24 ρt+1 = T (ρt) = ∑ α TαρtT† α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (20) We will mainly consider unitary transformations (U) ρt+1 = UρtU†, (21) where U†U = 1, which means that U is reversible and the inverse of U corresponds to its adjoint (U−1 = U†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, we will also address quantum measurements F = {Fα} with ∑α F† α Fα = I, where the probability of obtaining the result α is given by Pr(α) = tr(F† α Fαρt) in the discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If Q is an open system with environment E, such that the joint system evolves under a unitary transformation, we can evaluate the subsystem Q by treating the environment E in its current state et as a fixed background condition (but see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' A mechanism M ⊆ Q is a subset of Q with current state m = ρM t = trM′(ρt) within the corresponding Hilbert space HM = � i∈M Hi, where M′ = Q \\ M and trM′ denotes the trace over the Hilbert space HM′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The quantum integrated information of a mechanism M should quantify how much the state ρM t constrains the state of a purview, a system subset Zt±1 ⊆ Q, before or after an update T of the system, compared to a partition θ of the mechanism and purview into k independent parts (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As above, we will omit the update indices (t − 1, t, t + 1) unless necessary and focus on effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Quantum cause and effect repertoires To translate the cause and effect repertoires into a density matrix description, we first treat the special case of a single purview node Z = Zi with |Z| = 1, for which πe(Z|m) = p(Zt+1|mt) in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Replacing the probability distributions with the corresponding density matrices, we obtain πe(Zi|m) = ρZi|m t+1 = trZ′ i � T (ρM ⊗ ρM′ mm) � , (22) where ’ denotes the complement of a set in Q and ρM′ mm is the maximally mixed state of M′ = Q \\ M (see also [22,23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Next, we consider the case of purviews comprised of multiple units (|Z| > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the classical case, units in M′ may induce correlations between units in Z, as shown in Figure 2 by example of the COPY-XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The quantum equivalent of a classical COPY-XOR gate is the CNOT gate (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For classical inputs, the CNOT behaves identically to the COPY- XOR gate and thus the same considerations apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This means that, also in quantum systems, extraneous correlations should be discounted when evaluating the causal constraints of a system subset M, since they do not correspond to constraints due to the mechanism M itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the following, we will use ρZ|m t+1 to denote trZ′ � T (ρM ⊗ ρM′ mm) � , while πe(Z|m) corresponds to the final effect repertoire, after discounting extraneous correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the quantum case, units in Z may be correlated due to entanglement, which means quantum systems may violate the conditional independence assumption imposed for classical systems (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (Note that incomplete knowledge or a coarse-grained temporal scale can lead to a violation of conditional independence in a classical system, but those “instantaneous interactions” are not considered intrinsic to the system and are thus ignored in IIT’s causal analysis [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Simply inserting Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (22) into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (6) would inadvertently destroy correlations in Z that are due to entanglement (either preserved or produced during the transformation T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In order to correctly capture correlations due to entanglement and discount extraneous 10 of 24 correlations due to correlated “noise” from units in M′, the entanglement structure of ρZ|m t+1 must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The multipartite entanglement structure of an n-qubit pure state |ψ⟩ can be identified through partial traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Following [36], we define a partition Pr(V) = {V(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' , V(r)} with r = |Pr| ≤ n, � V(i) = V and V(i) � V(j) = ∅ if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' An n-qubit pure state |ψ⟩ is Pr-separable iff it can be written as |ψ⟩ = �r i=1 ���ψ(i)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the general case that ρZ|m t+1 is a mixed state, it has to be decomposed into a convex mixture of pure states to identify its entanglement structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' An n-qubit mixed state ρ is Pr-separable iff it can be decomposed into a convex mixture ρ = ∑s ps|ψs⟩⟨ψs|, with ps ≥ 0, ∀s and ∑s ps = 1, such that every |ψs⟩ in the mixture is a Pr- separable pure state |ψs⟩ = �r i=1 ���ψ(i) s � under the same partition Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note that Definition (2) differs from that in [36], as we require the same partition Pr for all |ψs⟩ in the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Out of the set of partitions {Pr}ρ = {Pr|ρ is Pr-separable}, we define the maximal partition P∗(ρ) as the one with the maximal number of parts r∗ = maxPr r and r∗ = |P∗| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given the maximal partition P∗ of ρZ|m t+1 , we can define the quantum effect repertoire of mechanism m over purview Z as πe(Z | m) = r∗ � i=1 πe(Z(i) | m) = r∗ � i=1 ρZ(i)|m t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (23) The product in (23) is thus taken over the reduced density matrices of all subsets Z(i) ⊆ Z that are entangled within themselves but not entangled with the other qubits in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note that P∗ is a simple set partition, and should not be confused with the disintegrating partitions Θ(M, Z) (12) used to evaluate the integrated information ϕ(m, Z, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Identifying the entanglement structure for multipartite mixed states remains an area of active research [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For 2-qubit mixed states, separability can be determined using the Peres-Horodecki criterion of the positive partial transform [40,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In general, however, this criterion is only a necessary condition for separability [41] and may thus miss certain complex forms of entanglement [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Several implications follow from the definition of the effect repertoire (23): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If ρZ|m t+1 corresponds to a pure state, the purview qubits are fully determined by the mechanism qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Thus, ρZ|m t+1 is not influenced by qubits outside of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' It follows that πe(Z|m) = ρZ|m t+1 if the latter is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This is analogous to the classical case, where πe(Z|m) = p(Zt+1|mt) if p(Zt+1|mt) is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Conceptually, entangled subsets are treated as indivisible units in the effect repertoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If a purview is fully entangled then πe(Z|m) = ρZ|m t+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Extraneous classical correlations are successfully discounted, which means they will not contribute to the integrated information of a mechanism (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The cause repertoire of a mechanism in state m over a purview Z also requires causal marginalization (independent noise applied to conditionally independent subsets) to isolate the causal constraints of m over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the classical case, the cause repertoire is obtained by 11 of 24 applying Bayes’ rule to the effect product probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The quantum case is more complex as the entanglement structure of ρM might need to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If T is a unitary transformation (21), the cause repertoire for any subset m(i) ∈ P∗(ρM) that is, itself, mutually entangled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' the subset could consist of an entangled pair of qubits) but is not entangled with units of other subsets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' other qubits) can be obtained by applying the adjoint operator T † πc(Z | m(i)) = ρZ|m(i) t−1 = trZ′ � T †(ρM(i) ⊗ ρM′(i) mm ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (24) Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given the maximal partition P∗ of ρM, we can define the quantum cause repertoire of mechanism m over purview Z as πc(Z | m) = ∏r∗ i=1 πc(Z | m(i)) tr � ∏r∗ i=1 πc(Z | m(i)) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (25) Note that the product here is over parts of ρM, not of ρZ|m t−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This introduces an asymmetry in the formulation of cause and effect repertoires, as in the classical case [1,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This asymmetry is a direct implication of treating non-entangled subsets as “physical” causal units, rather than abstract statistical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Causal units are conditionally independent in the present given the past, but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This means that in the effect repertoire, purview subsets that are not entangled with other units are conditionally independent given the mechanism and independent noise from outside the mechanism (due to causal marginalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' By contrast, the cause repertoire is inferred from the conditionally independent mechanism subsets, but is not itself conditionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The set of effects specified by a quantum state ρt undergoing a unitary transformation (U) may thus differ from the set of causes specified by ρt+1 = UρtU† (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (The assumption of conditional independence, paired with causal marginalization, distinguishes IIT’s causal analysis from standard information-theoretical analyses of information flow [18,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=') As pointed out in [23], the quantum IIT formalism proposed by Zanardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [22] does not include causal marginalization (which was formulated in terms of “virtual units” in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' We will show below that causal marginalization (23, 25) is necessary to isolate the causes and effects of system subsets also in the quantum case—an observation that should be of relevance to the causal analysis of quantum systems beyond IIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Quantum intrinsic information (QID) Our goal is to define a quantum version of the intrinsic difference measure, which coincides with the classical measure (8) [6] in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In quantum information theory, the classical definition of the KLD (9), or relative entropy, is extended from probability distributions to density matrices based on the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The quantum relative entropy of the density matrix ρ with respect to another density matrix σ is then defined as: S(ρ||σ) = Trρ log ρ − Trρ log σ, (26) which coincides with the classical case if ρσ = σρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Unitary operations, including a change of basis, leave S(ρ||σ) invariant [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Specifically, if ρ and σ are expressed as orthonormal decompositions ρ = ∑i pi|i⟩⟨i| and σ = ∑j qj|j⟩⟨j|, we can write [43] S(ρ||σ) = ∑ i pi � log(pi) − ∑ j Pij log(qj) � , (27) 12 of 24 where Pij = ⟨i|j⟩⟨j|i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In this formulation, a quantum version of the intrinsic difference measure can be defined as QID(ρ||σ) = max i pi � log(pi) − ∑ j Pij log(qj) � , (28) analogous to the classical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As for the relative entropy, QID(ρ||σ) coincides with the classical case if ρσ = σρ, because in that case Pij = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Moreover, QID(ρ||σ) = S(ρ||σ) if ρ is pure, as in the classical case for fully selective distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Otherwise, the QID is subadditive, as desired [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Zanardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' [22] proposed the trace distance as a measure of the cause/effect information based on its simplicity and widespread use in quantum-information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The trace distance quantifies the maximal difference in probability between two quantum states across all possible POVM measures [43], which is a useful quantity from the perspective of an experimenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' By contrast, QID is a measure of the intrinsic information of a quantum mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Its value is maximized over the eigenvectors {|i⟩} of ρ (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If ρ is pure, there is only one non- zero eigenvalue and the state identified by the QID measure is simply ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If ρ is mixed, the eigenvalue pi that maximizes equation (28) may be degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In that case the QID specifies the eigenspace spanned by the set of eigenvectors for which the difference between ρ and σ is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Otherwise, the QID specifies the eigenvector of ρ with the optimal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Identifying intrinsic causes and effects Equipped with the quantum intrinsic difference (QID) measure (28), the intrinsic effect information that the quantum mechanism M = m specifies over a purview Z can be quantified as iie(m, Z) = QID(πe(Z|m), πe(Z)), (29) where πe(Z) = πc(Z) = ρZ mm is the maximally mixed state in the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Following from equation (28), with ρ = πe(Z | m) = ∑i pi|i⟩⟨i| as the effect repertoire and σ = πe(Z) = ∑j qj|j⟩⟨j| = ρZ mm as the unconstrained effect repertoire, the intrinsic effect of mechanism m on purview Z is z′ e(m, Z) = argmax i∈HZ pi � log pi − ∑ j Pij log(qj) � = argmax i∈HZ pi � log pi − log |HZ|−1� , (30) where |HZ| denotes the cardinality of HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The intrinsic effect z′ e(m, Z) is thus simply the eigenvector |i⟩ of πe(Z|m) with the maximal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If the maximal eigenvalue of ρ = πe(Z | m) is degenerate, z∗ e (m) corresponds to the subspace of HZ∗e spanned by the set of eigenvectors belonging to the maximal eigenvalue (and the same for the intrinsic cause z′ c(m, Z) evaluated over πc(Z|m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note that, in the case that πe(Z|m) is a mixed quantum state (corresponding to a probabil- ity distribution with multiple possible effect states in the classical case), this means that the intrinsic effect z′ e(m, Z) differs from ρ = πe(Z|m) = ∑i pi|i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 13 of 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Disintegrating partitions As in the classical case, the quantum integrated information ϕ(m, Z, θ) is evaluated by com- paring the effect repertoire πe(Z|m) to a partitioned effect repertoire πθ e (Z|m) (and analogously for ϕc(m, Z, θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The set of possible partitions θ ∈ Θ(M, Z) is the same as for the classical case (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Likewise, the partitioned effect repertoire is defined as in (13), as a product over the parts in the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In the quantum case, πe(Z(i)|∅) corresponds to the maximally mixed state ρZ(i) mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The partitioned cause repertoire is defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note that the disintegrating partition θ ∈ Θ(M, Z) (12) here is applied on top of P∗ (Defi- nition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Partitioning may thus affect entanglement within the repertoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Conceptually, any entanglement in πe(Z | m) that is destroyed by the partition θ will count towards ϕe(m, Z, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Ultimately, however, ϕe(m, Z) is again evaluated over θ′ (16), the minimum information parti- tion (MIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This means that everything else being equal, partitions that affect entanglement less are more likely to correspond to the MIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Quantum mechanism integrated information Having identified the specific effect state z′ e as an eigenstate |i⟩ of ρ = πe(Z|m), the integrated effect information ϕ(m, Z, θ) is evaluated as the QID(ρ||σ) over that eigenstate, such that ϕ(m, Z, θ) = ϕ(m, z′ e, θ) = pi � log pi − ∑ j Pij log(pθ j ) � , (31) where σ = πθ e (Z|m) = ∑j pθ j |j⟩⟨j| is now the partitioned effect repertoire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As above, quantifying the integrated effect information ϕe(m) of a mechanism m within a quantum system Q requires a search over all possible partitions θ ∈ Θ(M, Z) to identify the MIP, and a search across all possible purviews Z ⊆ Q, such that ϕe(m) = max Z⊆Q ϕe(m, Z) = max Z⊆Q ϕ(m, Z, θ′), (32) as in (19), with θ′ as in (16), and analogously for ϕc(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The maximally irreducible effect purview Z∗ e (m) Z∗ e (m) = argmax Z⊆Q ϕe(m, Z) (33) again corresponds to the subset of Q upon which the mechanism M = m has the maximally irreducible intrinsic effect z∗ e , which corresponds to the eigenstate of ρ = πe(Z∗|m) that maximizes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (30), or the eigenspace spanned by a set of eigenvectors corresponding to a degenerate maximal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As in the classical case, Z∗ e is not necessarily unique and we again choose the larger purview in case of a tie between purviews of different sizes (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Any remaining ties are reported in the examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The intrinsic structure of a quantum system Standard approaches for studying the causal or informational properties of a system typically assume either a reductionist perspective (focused on individual units) or holistic perspective (describing the system as a whole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As the units in a quantum system can be entangled, focusing on individual units is ill-suited at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, a purely holistic description of a quantum system will still miss differences in the internal structure of a 14 of 24 quantum state (see below the comparison between the maximally entangled GHZ-type and W-type states [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In IIT, causation is neither reductionist nor holistic but compositional: the IIT analysis considers the intrinsic causes and effects of every subset within a system and quantifies their irreducibility as ϕc/e(m) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' As a result, it can elucidate the internal structure of composite quantum states and operators, as we will show in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' We note that typically, the IIT analysis assumes a current system state st and identifies its compositional causes at t − 1 and effects at t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' A subset m ⊆ s with an irreducible cause and effect forms a “causal distinction” within the system s, where ϕ(m) = min(ϕc(m), ϕe(m)) is the integrated (cause-effect) information of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' According to IIT, the phenomenal experience of a physical system S in state s is identical to its cause-effect structure, composed of a system’s causal distinctions and their relations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Unfolding the full cause-effect structure requires assessing the integrated (cause-effect) information ϕ(m) of every subset of units m ⊆ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For ease of demonstration, in the following, we will instead evaluate examples of system transitions from state t to t + 1 and identify the intrinsic effects of the system in state st and the intrinsic causes of the system in state st+1 (see also [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Results For a direct comparison between classical and quantum systems, we will focus our attention on computational quantum systems (see [46] for an overview and comparison to classical systems), constituted of a finite number of quantum units with a finite-dimensional Hilbert space, evolving in discrete updates according to unitary transformations, expressed in the computational (or “classical”) basis unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To compute classical IIT quantities, we made use of the openly available PyPhi python toolbox, developed by the Tononi lab [15,16], using the “iit-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0” feature branch with standard IIT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To compute quantum IIT results, we implemented a QIIT toolbox (https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='com/Albantakis/QIIT, accessed 30 December 2022), applicable to unitary quantum mechanisms of two and three qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' CNOT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Classical case As a first example, we will evaluate the “controlled-NOT” (CNOT) gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Classically, the CNOT gate corresponds to a reversible XOR gate, with a COPY operation performed on the first input bit (A) and an XOR operation comparing the two input bits A and B as the second output (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For instance, the input state AB = (1, 0) leads to the output CD = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In what follows, we will abbreviate the states of system subsets (mechanisms and purviews) by the state plus a set subscript, for example, 10AB for AB = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Given the input state AB = (1, 0), the IIT analysis identifies two irreducible mechanisms, one first-order and one second-order mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The mechanism 1A specifies the effect purview 1C with ϕ = 1 ibit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' the second-order mechanism 10AB specifies the effect purview 11CD also with ϕ = 1 ibit (while there is a tie with the effect 1D, we choose the larger purview as described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Notably, 0B by itself (with A replaced by independent noise) does not specify any information about the next state of CD (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' While this conclusion should be straightforward, it relies on the use of product probabilities instead of simple conditional probabilities (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The latter would mistakenly count the correlation between C and D as an effect of B, although it is actually due to the common input A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' By contrast to 0B on the effect side, 1D on the cause side specifies irreducible cause information about the previous state of AB in addition to 1C and 11CD, albeit only ϕc(1D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 15 of 24 ibit due to the remaining uncertainty about the state of AB (note the quantitative difference between the ID measure (8) and the KLD (9), which would return a value of 1 bit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Quantum case For a CNOT gate with the input state ρAB = |10⟩⟨10| (or |10⟩AB), we obtain the same results as for a COPY-XOR gate with input state AB = (1, 0) using the formalism outlined above (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' With essentially classical inputs, the CNOT gate thus reproduces the intrinsic causal structure of the classical COPY-XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' To that end, it was necessary to discount the spurious correlation between qubits A and B through product distributions (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' This demonstrates that standard conditional probabilities are insufficient to identify the causes and effects of system subsets also in the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note that for the CNOT gate the role of the “control” (COPY) and the “target” (XOR) qubit changes depending on the input state, which is not true for the COPY-XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' For an input state in the Hadamard basis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' |−+⟩AB, information seems to flow from B to C, not A to D as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' CNOT gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' The CNOT operator is shown in the top box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (a) For a pure input state in the classical basis, we obtain the same results as in the classical case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (b) For a pure input state in the Hadamard basis, the role of the “control” (here B) and “target” (here A) is reversed compared to (a) (as indicated in the circuit diagram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (c) The CNOT is often used to produce a “Bell state” of two maximally entangled qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In this exclusively quantum scenario, only the second order mechanisms |+0⟩AB and ��B+� CD specify an effect or cause, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' None of the subsets has any cause or effect information (ϕ = 0 ibit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (d) Conversely, given the input state |0+⟩AB all second order mechanisms are fully reducible (ϕ = 0 ibit) and only the first order mechanisms specify causes and effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 CNOT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 / (a) Classical basis Effects: Causes: Classical equivalent: Pe(/10)AB) = 1 ibit |z* =[11)cD c(/11)cp)= 1 ibit I z* = [10)AB [1] A [1)c (Pe(|1)A)= 1 ibit |z*=|1)c c(/1)c) = 1 ibit 2* = [1)A c(/1)p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ibit Iz* = span(|10),[01)) [0)B — [1)D (b) Hadamard basis Effects: Causes: Classicalequivalent: e(I- +)AB) = 1 ibit | z* = [- +)cD c(/- +)cD) = 1 ibit I 2* = [- +)AB I-)A 一 - c(1-)c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ibit Iz* = span(I- +),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='I+ -) AB (Pe(I+)B) = 1 ibit I z* = |+)D c(I+)p) = 1 ibit Iz* = |+)B I+)B I+)D (c) Bell state Effects: Causes: Classical equivalent: I+)A Pe(lI+0)AB) = 2 ibit Iz*=[B+)CD Pc(/B+)cD) = 2 ibit Iz* = |+0)AB None IB+)=(I00)+[11)cD 10)B (d) No interaction Effects: Causes: Classical equivalent: [0 A [0)c e(l0)A) = 1 ibit 1 2* = |0)c c(10)c) = 1 ibit 12* = [0)A Pe(I+)B) = 1 ibit I z* = |+)D Pc(/+)p) = 1 ibit Iz* = |+)B I+)B I+)p16 of 24 for a classical input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Accordingly, the quantum IIT analysis now identifies |+⟩B and |−+⟩AB as irreducible mechanisms with ϕ = 1 ibit, while |−⟩A by itself does not specify any effect information (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Yet, |+⟩C does specify irreducible cause information about AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In quantum systems, the CNOT is often used to produce the maximally entangled Bell state |B+⟩ = 1 √ 2(|00⟩ + |11⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' CD = |B+⟩ results from the input state AB = |+0⟩, a transition for which there is no classical circuit equivalent [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In this case, the quantum IIT analysis identifies only the second order mechanisms (constituted of two qubits) |+0⟩AB and |B+⟩CD with ϕ = 2 ibit each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Individual qubits specify no cause or effect information (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' An analogous result obtains for the Bell state as the input to the CNOT gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Finally, with AB = |0+⟩ as the input, there appears to be no interaction between qubits and the quantum IIT analysis only identifies first order mechanisms on the cause and effect side (Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Mixed states and extensions to larger systems The purpose of the IIT analysis is to evaluate the cause-effect power of a system in its current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Evaluating statistical ensembles is conceptually not in line with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Accordingly, the classical IIT analysis always assumes a particular (fully determined) state for the mechanism m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' However, in quantum mechanics, mixed states not only describe statistical ensembles, but also subsets of entangled pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' If we apply an even mixture ρAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ∗ (|00⟩⟨00| + |11⟩⟨11|) to the CNOT gate, we obtain ρCD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ∗ (|00⟩⟨00| + |10⟩⟨10|) as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In this case, only the second order mechanism m = ρAB has an irreducible effect with ϕe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='0 ibit over z∗ = |0⟩D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' There is no effect on C, as C by itself is undetermined (maximally mixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' In turn, only |0⟩D specifies an irreducible cause with ϕ(|0D⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ibit over purview Z∗ = AB with z∗ corresponding to the subspace spanned by |00⟩AB and |11⟩AB (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Note the difference to the causal analysis of the Bell state |B+⟩ = 1 √ 2(|00⟩ + |11⟩) above, where |+0⟩AB and |B+⟩CD both specified second order mechanisms with ϕ = 2 ibit each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Mixed states and entanglement with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (a) IIT analysis of the CNOT gate with a mixed input state ρAB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ∗ (|00⟩⟨00| + |11⟩⟨11|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (b) It is possible to describe the mixed state as a pure state entangled with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' Analyzing such an extended system for the case in (a), the cause and effect of the subsystem are preserved in the larger system (gray), but we obtain additional causes and effects that span all three qubits (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' |GHZ⟩′ denotes a maximally entangled superposition of states |001⟩ and |110⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content=' (a) CNOT mixed state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 * (/0)01 + /1)X1D)c Effects: Pe(pAB) = 1 ibit I z* = [0)D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 * (/00)(00| + |11X11)AB Causes: 10D Pc(/1)p) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 ibit I z* = span(l00),[11)AE (b) I CNOT Effects: [B+) = (I00) + [11))DE βe(IGHZ)ABc) = 2 ibit Iz* = [B+)AB Pe(pBC) = 1 ibit I z* =0>F IGHZ)=(I000) +[111)ABC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5*(/0)0| +|1)(1)E Causes: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfUACA/content/2301.02244v1.pdf'} +page_content='5 *(/00)(00| + /11)11))BC (Pc(|B+0)DEF) = 1 ibit Iz* =|GHZ)ABC H j, +(12) +where Zi := I⊗(i−1) +2 +⊗ Z ⊗ I⊗(n−i) +2 +(similarly for Xi, Yi). The Hamiltonian +H := +� +(u,v)∈E +� +i,j∈[n] +[Cuv]ij +� +k∈[n] +P (u) +ik +⊗ P (v) +jk +(13) +defines our quantum relaxation of the objective f over O(n)m. The notation A(v) denotes the operator A acting only +on the Hilbert space of vertex v, and we overload this notation to indicate either the n-qubit operator or mn-qubit +operator acting trivially on the remaining vertices. When the context is clear we typically omit writing the trivial +support. +For optimization over SO(n)m, we consider instead the (n − 1)-qubit Pauli operators +�Pij := Π0PijΠT +0 , +(14) +Π0 = +1 +√ +2 +Ä +⟨+| ⊗ I⊗(n−1) +2 ++ ⟨−| ⊗ Z⊗(n−1)ä +, +(15) + +7 +FIG. 1. A cartoon description of the quantum and classical encodings of an LNCG problem, followed by classical and quantum +rounding. (Left) The description of the problem that we consider, which is described by a graph ([m], E) and n × n matrices +Cuv for each edge (u, v). We wish to assign elements of O(n) or SO(n) to each vertex such that the quadratic form of Eq. (11) +is maximized. (Center top) The description of the standard classical relaxation of the LNCG problem as an mn × mn PSD +matrix M ⪰ 0, which is optimized using a semidefinite program. (Right top) The classical rounding procedure, which returns a +collection of orthogonal matrices from M. (Center bottom) A description of our quantum formulation of the LNCG problem as +a two-body interacting Hamiltonian. On each vertex we place a d-dimensional Hilbert space, and the Hamiltonian corresponds +to interaction terms Huv on the edges (L(H) is the set of linear operators on a Hilbert space H). The classical solution of the +LNCG problem lies in a subset of the full Hilbert space containing separable Gaussian states. (Right bottom) Our proposed +quantum rounding protocols. One protocol requires knowledge of the two-body reduced density matrices across edges, while +the other uses the one-body reduced density matrices on each vertex. +where Π0 : H2n → H2n−1 represents the projection onto the even-parity subspace of H2n. The construction of the +relaxed Hamiltonian for SO(n) is then analogous to Eq. (13): +�H := +� +u,v∈E +� +i,j∈[n] +[Cuv]ij +� +k∈[n] +�P (u) +ik +⊗ �P (v) +jk , +(16) +where now the relaxed quantum problem is defined over m(n − 1) qubits. +These Hamiltonians serve as relaxations to Problem (2) in the following sense. +First, we show that for every +R ∈ O(n), there is an n-qubit state |φ(R)⟩ which is the maximum eigenstate of +F(R) = +� +i,j∈[n] +RijPij. +(17) +In particular, F(R) is a free-fermion Hamiltonian, so |φ(R)⟩ is a fermionic Gaussian state. +If R ∈ SO(n), then +furthermore |φ(R)⟩ is an even-parity state, i.e., ⟨φ(R)|Z⊗n|φ(R)⟩ = 1, so it is only supported on a subspace of +dimension 2n−1 (the image of Π0). +This correspondence establishes a reformulation of the classical optimization +problem as a constrained Hamiltonian problem: +max +R∈Gm f(R) = +max +|ψ⟩=� +v∈[m] |φ(Rv)⟩ +Rv∈G +⟨ψ|H|ψ⟩. +(18) +Dropping these constraints on |ψ⟩ implies the inequalities +max +R∈O(n)m f(R) ≤ +max +ρ∈D(H⊗m +2n ) +tr(Hρ), +(19) +max +R∈SO(n)m f(R) ≤ +max +ρ∈D(H⊗m +2n−1) +tr( �Hρ), +(20) +where D(H) denotes the set of density operators on a Hilbert space H. This establishes the quantum Hamiltonian +relaxation. + +8 +Algorithm 1: conv G-based rounding of edge marginals +Data: Quantum state ρ ∈ D(H⊗m +d +) over a graph of m vertices, each with local Hilbert space of dimension d = 2n if +G = O(n), or d = 2n−1 if G = SO(n) +Result: Orthogonal matrices on each vertex, R1, . . . , Rm ∈ G +M ← Imn ; +for u ̸= v ∈ [m] do +for (i, j) ∈ [n]2 do +if G = O(n) then +[Muv]ij ← 1 +n tr(Γ(u,v) +ij +ρ) ; +else if G = SO(n) then +[Muv]ij ← 1 +n tr(�Γ(u,v) +ij +ρ) ; +end +end +end +for v ∈ [m] do +Rv ← arg minY ∈G ∥Y − M1v∥F ; +end +Algorithm 2: Rounding vertex marginals +Data: Quantum state ρ ∈ D(H⊗m +d +) over a graph of m vertices, each with local Hilbert space of dimension d = 2n if +G = O(n), or d = 2n−1 if G = SO(n) +Result: Orthogonal matrices on each vertex, R1, . . . , Rm ∈ G +for v ∈ [m] do +Qv ← 0 ∈ Rn×n ; +for (i, j) ∈ [n]2 do +if G = O(n) then +[Qv]ij ← tr(P (v) +ij ρ) ; +else if G = SO(n) then +[Qv]ij ← tr( �P (v) +ij ρ) ; +end +end +end +for v ∈ [m] do +Rv ← arg minY ∈G ∥Y − Qv∥F ; +end +B. +Quantum rounding +In order to recover orthogonal matrices from a relaxed quantum solution ρ, we propose two rounding procedures, +summarized in Algorithms 1 and 2. These rounding procedures operate on local (i.e., single- or two-vertex observables) +expectation values of ρ stored in classical memory, which can be efficiently estimated, e.g., by partial state tomography. +Algorithm 1 is inspired by constructing a quantum analogue of the PSD variable appearing in semidefinite relax- +ations to Problem (2). Consider the mn × mn matrix of expectation values +M := +� +���� +In +T12 +· · · T1m +T21 +In +· · · T2m +... +... +... +... +Tm1 Tm2 · · · +In +� +���� , +(21) + +9 +where the off-diagonal blocks are defined as +Tuv := 1 +n +� +�� +tr(Γ(u,v) +11 +ρ) · · · tr(Γ(u,v) +1n +ρ) +... +... +... +tr(Γ(u,v) +n1 +ρ) · · · tr(Γ(u,v) +nn +ρ) +� +�� = T T +vu, +(22) +Γ(u,v) +ij +:= +� +k∈[n] +P (u) +ik +⊗ P (v) +jk +(23) +when G = O(n), and we replace the operators Pij with �Pij when G = SO(n). We show that M satisfies the following +properties for all states ρ: +M ⪰ 0, +(24) +Muv ∈ conv G +∀u, v ∈ [m], +(25) +where conv G is the convex hull of G. Thus when G = SO(n), M obeys the same constraints as the conv SO(n)- +based semidefinite relaxation proposed by Saunderson et al. [21]. However, whereas the classical representation of the +conv SO(n) constraints requires at least matrices of size 2n−1 × 2n−1 for each edge, our quantum state automatically +satisfies these constraints (using only n − 1 qubits per vertex). +Algorithm 2 uses the single-vertex information tr(P (v) +ij ρ) of ρ, as opposed to the two-vertex information tr(Γ(u,v) +ij +ρ). +We consider this rounding procedure due to the fact that, if ρ is a pure Gaussian state satisfying the constraint of +Eq. (18), then the matrix of expectation values +Qv := +� +�� +tr(P (v) +11 ρ) · · · tr(P (v) +1n ρ) +... +... +... +tr(P (v) +n1 ρ) · · · tr(P (v) +nn ρ) +� +�� +(26) +lies in O(n). On the other hand, for arbitrary density matrices we have the relaxation Qv ∈ conv O(n), and again +when we replace P (v) +ij +with �P (v) +ij +then Qv ∈ conv SO(n). +Both rounding procedures use the standard projection of the matrices X ∈ conv G (e.g., the matrices Tuv or Qv +measured from the quantum state) to some R ∈ G by finding the nearest (special) orthogonal matrix according to +Frobenius-norm distance: +R = arg min +Y ∈G +∥X − Y ∥F . +(27) +This can be solved efficiently as a classical postprocessing step, essentially by computing the singular value decom- +position of X = UΣV T. When G = O(n), the solution is R = UV T. When G = SO(n), we instead use the so-called +special singular value decomposition of X = U �Σ�V T, where �Σ = ΣJ and �V = V J, with J being the diagonal matrix +J = +ïIn−1 +0 +0 +det(UV T) +ò +, +(28) +assuming that the singular values σi(X) are in descending order, σ1(X) ≥ · · · ≥ σn(X). Then the solution to Eq. (27) +is R = U �V T ∈ SO(n). +V. +QUANTUM FORMALISM FOR OPTIMIZATION OVER ORTHOGONAL MATRICES +Our key insight into encoding orthogonal matrices into quantum states comes from the construction of the orthogonal +group from a Clifford algebra. We review this mathematical construction in Appendix A and only discuss the main +aspects here. The Clifford algebra Cl(n) is a 2n-dimensional real vector space equipped with an inner product and +multiplication operation satisfying the anticommutation relation +eiej + ejei = −2δij11, +(29) +where e1, . . . , en is an orthonormal basis for Rn and 11 is the multiplicative identity of the algebra. The orthogonal +group is then realized through a quadratic map Q : Cl(n) → Rn×n and the identification of a subgroup Pin(n) ⊂ Cl(n) + +10 +such that Q(Pin(n)) = O(n). Notably, the elements of Pin(n) have unit norm (with respect to the inner product +on Cl(n)). The special orthogonal group, meanwhile, is constructed by considering only the even-parity elements of +Cl(n), denoted by Cl0(n). The group Spin(n) = Pin(n) ∩ Cl0(n) then yields Q(Spin(n)) = SO(n). +Because the Clifford algebra Cl(n) is a 2n-dimensional vector space, we observe that it can be identified with a +Hilbert space of n qubits.5 In this section we explore this connection in detail, showing how to represent orthogonal +matrices as quantum states and how the mapping Q acts as a linear functional on those states. +A. +Qubit representation of the Clifford algebra +First we describe the canonical isomorphism between Cl(n) and H2n := (R2)⊗n as Hilbert spaces. We denote the +standard basis of Cl(n) by {eI := ei1 · · · eik | I = {i1, . . . , ik} ⊆ [n]}. By convention we assume that the elements +of I are ordered as i1 < · · · < ik. +Each basis element eI maps onto to a computational basis state |b⟩, where +b = b1 · · · bn ∈ {0, 1}n, via the correspondence +eI ≡ +� +i∈[n] +|bi⟩, +where bi = +® +1 +if i ∈ I, +0 +otherwise. +(30) +The inner products on both spaces coincide since this associates one orthonormal basis to another. This correspondence +also naturally equates the grade |I| of the Clifford algebra with the Hamming weight |b| of the qubits. The notion of +parity, |I|mod2 = |b|mod2, is therefore preserved, so Cl0(n) corresponds to the subspace of H2n with even Hamming +weight. +To represent the multiplication of algebra elements in this Hilbert space, we use the fact that left- and right- +multiplication are linear automorphisms on Cl(n), which are denoted by +λx(y) = xy, +ρx(y) = yx. +(31) +The action of the algebra can therefore be represented on H2n as linear operators. We shall use the matrix represen- +tation provided in Ref. [24], as it precisely coincides with the n-qubit computational basis described above. Because +of linearity, it suffices to specify left- and right-multiplication by the generators ei, which are the operators +λi ≡ Z⊗(i−1) ⊗ (−iY ) ⊗ I⊗(n−i) +2 +, +(32) +ρi ≡ I⊗(i−1) +2 +⊗ (−iY ) ⊗ Z⊗(n−i). +(33) +It will also be useful to write down the parity automorphism α(eI) = (−1)|I|eI under this matrix representation. As +the notion of parity is equivalent between Cl(n) and H2n, α is simply the n-qubit parity operator, +α ≡ Z⊗n. +(34) +It will also be useful to represent the subspace Cl0(n) explicitly as an (n − 1)-qubit Hilbert space. This is achieved +by the projection from Cl(n) to Cl0(n), expressed in Ref. [24] as the 2n−1 × 2n matrix +Π0 := +1 +√ +2 +Ä +⟨+| ⊗ I⊗(n−1) +2 ++ ⟨−| ⊗ Z⊗(n−1)ä +. +(35) +It is straightforward to check that Π0|b⟩ = 0 if |b| mod 2 = 1, and that its image is a 2n−1-dimensional Hilbert space. +B. +The quadratic mapping as quantum expectation values +The quadratic map Q : Cl(n) → Rn×n is defined as +Q(x)(v) := πRn(α(x)vx) +∀x ∈ Cl(n), v ∈ Rn, +(36) +where πRn is the projector from Cl(n) to Rn and the conjugation operation is eI = (−1)|I|eik · · · ei1. +This map +associates Clifford algebra elements with orthogonal matrices via the relations Q(Pin(n)) = O(n) and Q(Spin(n)) = +5 In fact, n rebits suffice since Cl(n) is a real vector space, but to keep the presentation straightforward we will not make such a distinction. + +11 +SO(n) (see Appendix A for a review of the construction). In the standard basis of Rn, the linear map Q(x) : Rn → Rn +has the matrix elements +[Q(x)]ij = ⟨ei, Q(x)(ej)⟩ += ⟨ei, α(x)ejx⟩. +(37) +Using the linear maps λi, ρj of left- and right-multiplication by ei, ej, as well as the conjugation identity ⟨x, yz⟩ = +⟨xz, y⟩ in the Clifford algebra, these matrix elements of Q(x) can be rearranged as +[Q(x)]ij = ⟨ei, α(x)ejx⟩ += ⟨eix, α(x)ej⟩ += ⟨λi(x), ρj(α(x))⟩ += ⟨x, λ† +i(ρj(α(x)))⟩. +(38) +We now transfer this expression to the quantum representation developed above. First, define the following n-qubit +Pauli operators as the composition of the linear maps appearing in Eq. (38): +Pij := λ† +iρjα = +� +� +� +� +� +−I⊗(i−1) +2 +⊗ X ⊗ Z⊗(j−i−1) ⊗ X ⊗ I⊗(n−j) +2 +i < j, +I⊗(i−1) +2 +⊗ Z ⊗ I⊗(n−i) +2 +i = j, +−I⊗(j−1) +2 +⊗ Y ⊗ Z⊗(i−j−1) ⊗ Y ⊗ I⊗(n−i) +2 +i > j, +(39) +where the expressions in terms of Pauli matrices follow from Eqs. (32) to (34). Then we may rewrite Eq. (38) as +[Q(x)]ij = ⟨x|Pij|x⟩, +(40) +where |x⟩ ∈ H2n is the quantum state identified with x ∈ Cl(n). Hence, the matrix elements of Q(x) ∈ Rn×n possess +the interpretation as expectation values of a collection of n2 Pauli observables {Pij}i,j∈[n]. Furthermore, recall that +Q(x) ∈ O(n) if and only if x ∈ Pin(n), and Q(x) ∈ SO(n) if and only if x ∈ Spin(n). Because Spin(n) = Pin(n)∩Cl0(n), +one can work in the even-parity sector directly by projecting the operators as +�Pij := Π0PijΠT +0 . +(41) +These are (n − 1)-qubit Pauli operators, and we provide explicit expressions in Appendix C. When necessary, we may +specify another map �Q : Cl0(n) → Rn×n, +[ �Q(x)]ij := ⟨x| �Pij|x⟩, +(42) +for which �Q(Spin(n)) = SO(n). +In general, these double covers are only a subset of the unit sphere in Hd (d = 2n or 2n−1), so not all quantum +states mapped by Q yield orthogonal matrices. In Section V C we characterize the elements of Pin(n) and Spin(n) as +a class of well-studied quantum states, namely, pure fermionic Gaussian states. +C. +Fermionic representation of the construction +1. +Notation +First we establish some notation. A system of n fermionic modes, described by the creation operators a† +1, . . . , a† +n, +can be equivalently represented by the 2n Majorana operators +γi = ai + a† +i, +(43) +�γi = −i(ai − a† +i), +(44) +for all i ∈ [n]. These operators form a representation for the Clifford algebra Cl(2n), as they satisfy6 +γiγj + γjγi = �γi�γj + �γj�γi = 2δij11, +(45) +γi�γj + �γjγi = 0. +(46) +6 Note that we adopt the physicist’s convention here, which takes the generators to be Hermitian, as opposed to Eq. (A1) wherein they +square to −11. + +12 +The Jordan–Wigner mapping allows us to identify this fermionic system with an n-qubit system via the relations +γi = Z⊗(i−1) ⊗ X ⊗ I⊗(n−i) +2 +, +(47) +�γi = Z⊗(i−1) ⊗ Y ⊗ I⊗(n−i) +2 +. +(48) +We will work with the two representations interchangeably. +A central tool for describing noninteracting fermions is the Bogoliubov transformation γ �→ Oγ, where O ∈ O(2n) +and +γ := ��γ1 · · · �γn γ1 · · · γn +�T . +(49) +This transformation is achieved by fermionic Gaussian unitaries, which are equivalent to matchgate circuits on qubits +under the Jordan–Wigner mapping [53–55]. In particular, we will make use of a subgroup of such unitaries corre- +sponding to O(n) × O(n) ⊂ O(2n). For any U, V ∈ O(n), let U(U,V ) be the fermionic Gaussian unitary with the +adjoint action +U(U,V )�γiU† +(U,V ) = +� +j∈[n] +Uij�γj, +(50) +U(U,V )γiU† +(U,V ) = +� +j∈[n] +Vijγj. +(51) +In contrast to arbitrary O(2n) transformations, these unitaries do not mix between the γ- and �γ-type Majorana +operators. +2. +Linear optimization as free-fermion models +Applying the representation of Majorana operators under the Jordan–Wigner transformation, Eqs. (47) and (48), +to the Clifford algebra automorphisms, Eqs. (32) to (34), we see that λ† +i = i�γi and ρjα = γj. Therefore the Pauli +operators Pij defining the quadratic map Q are equivalent to fermionic one-body operators, +Pij = i�γiγj. +(52) +Consider now a linear objective function ℓ(X) := ⟨C, X⟩ for some fixed C ∈ Rn×n, which we wish to optimize over +O(n): +max +X∈O(n) ℓ(X) = +max +X∈O(n)⟨C, X⟩. +(53) +Because we require X ∈ O(n), it is equivalent to search over all x ∈ Pin(n) through Q: +max +X∈O(n)⟨C, X⟩ = +max +x∈Pin(n)⟨C, Q(x)⟩. +(54) +Writing out the matrix elements explicitly, we see that the objective takes the form +ℓ(X) = +� +i,j∈[n] +Cij[Q(x)]ij += +� +i,j∈[n] +Cij⟨x|Pij|x⟩ += ⟨x|F(C)|x⟩, +(55) +where we have defined the noninteracting fermionic Hamiltonian +F(C) := +� +i,j∈[n] +CijPij = i +� +i,j∈[n] +Cij�γiγj. +(56) +The linear optimization problem is therefore equivalent to solving a free-fermion model, +max +X∈O(n)⟨C, X⟩ = +max +x∈Pin(n)⟨x|F(C)|x⟩, +(57) + +13 +the eigenvectors of which are fermionic Gaussian states. As such, this problem can be solved efficiently by a classical +algorithm. In fact, the known classical algorithm for solving the optimization problem is exactly the same as that +used for diagonalizing F(C). +We now review the standard method to diagonalize F(C). Consider the singular value decomposition of C = UΣV T, +which is computable in time O(n3). This decomposition immediately reveals the diagonal form of the Hamiltonian: +F(C) = i +� +i,j∈[n] +[UΣV T]ij�γiγj += i +� +k∈[n] +σk(C) +Ñ +� +i∈[n] +[U T]ki�γi +éÑ +� +j∈[n] +[V T]kjγj +é += U† +(U,V ) +Ñ +� +k∈[n] +σk(C)i�γkγk +é +U(U,V ). +(58) +Because i�γkγk = Zk, it follows that the eigenvectors of F(C) are the fermionic Gaussian states +|φb⟩ = U† +(U,V )|b⟩, +b ∈ {0, 1}n, +(59) +with eigenvalues +Eb = +� +k∈[n] +(−1)bkσk(C). +(60) +The maximum energy is E0n = tr Σ since all singular values are nonnegative. The corresponding eigenstate |φ0n⟩ +is the maximizer of Eq. (57), so it corresponds to an element φ0n ∈ Pin(n). It is straightforward to see this by +recognizing that [Q(φ0n)]ij = ⟨φ0n|i�γiγj|φ0n⟩ = [UV T]ij. The fact that Q(φ0n) ∈ O(n) if and only if φ0n ∈ Pin(n) +concludes the argument. +Indeed, the standard classical algorithm [56] for solving Eq. (53) uses precisely the same decomposition. From the +cyclic property of the trace and the fact that O(n) is a group, we have +max +X∈O(n)⟨UΣV T, X⟩ = +max +X′∈O(n)⟨Σ, X′⟩, +(61) +where we have employed the change of variables X′ := U TXV . +Again, because Σ has only nonnegative entries, +⟨Σ, X′⟩ achieves its maximum, tr Σ, when X′ = In. This implies that the optimal solution is X = UV T. Note that +this problem is equivalent to minimizing the Frobenius-norm distance, since +arg min +X∈O(n) +∥C − X∥2 +F = arg min +X∈O(n) +�∥C∥2 +F + ∥X∥2 +F − 2⟨C, X⟩� += arg max +X∈O(n) +⟨C, X⟩. +(62) +Now suppose we wish to optimize ℓ over SO(n). In this setting, one instead computes X = U �V T from the special +singular value decomposition of C = U �Σ�V T. This ensures that det(X) = 1 while maximizing ℓ(X), as only the +smallest singular value σn(C) has its sign potentially flipped to guarantee the positive determinant constraint. This +sign flip also has a direct analogue within the free-fermion perspective. Recall that the determinant of Q(x) ∈ O(n) +is given by the parity of x ∈ Pin(n), or equivalently the parity of the state |x⟩ in the computational basis. Note +also that all fermionic states are eigenstates of the parity operator. To optimize over SO(n), we therefore seek the +maximal eigenstate |φb⟩ of F(C) which has even parity. If ⟨φ0n|Z⊗n|φ0n⟩ = 1 then we are done. On the other hand, if +⟨φ0n|Z⊗n|φ0n⟩ = −1 then we need to flip only a single bit in 0n to reach an even-parity state. The smallest change in +energy by such a flip is achieved from changing the occupation of the mode corresponding to the smallest singular value +of C. The resulting eigenstate |φ0n−11⟩ is then the even-parity state with the largest energy, E0n−11 = tr Σ − 2σn(C). +Finally, we point out that all elements of Pin(n) are free-fermion states. To see this, observe that C is arbitrary. +We can therefore construct the family of Hamiltonians {F(C) | C ∈ O(n)}. Clearly, the maximum ⟨C, X⟩ = n +within this family is achieved when X = C, each of which corresponds to a fermionic Gaussian state |φ⟩ satisfying +F(C)|φ⟩ = n|φ⟩ and Q(φ) = C. We note that this argument generalizes the mathematical one presented in Ref. [24], +which only considered the eigenvectors lying in Spin(n). + +14 +3. +Mixed states and the convex hull +First we review descriptions of the convex hull of orthogonal and rotation matrices, the latter of which was charac- +terized by Saunderson et al. [24]. The convex hull of O(n) is the set of all matrices with operator norm bounded by +1, +conv O(n) = �X ∈ Rn×n | σ1(X) ≤ 1�. +(63) +On the other hand, the convex hull of SO(n) has a more complicated description in terms of special singular values: +conv SO(n) = +� +� +�X ∈ Rn×n +����� +� +i∈[n]\I +�σi(X) − +� +i∈I +�σi(X) ≤ n − 2 +∀I ⊆ [n], |I| odd +� +� +�. +(64) +Saunderson et al. [24] establish that this convex body is a spectrahedron, the feasible region of a semidefinite program. +The representation that we will be interested in is called a PSD lift: +conv SO(n) = +� +� +� +� +� +� +�� +⟨ �P11, ρ⟩ · · · +⟨ �P1n, ρ⟩ +... +... +... +⟨ �Pn1, ρ⟩ · · · ⟨ �Pnn, ρ⟩ +� +�� +����� ρ ⪰ 0, tr ρ = 1 +� +� +� +� +� +, +(65) +where the 2n−1 × 2n−1 matrices �Pij are defined in Eq. (41).7 +Recall that the density operators on a Hilbert space H form the convex hull of its pure states: +D(H) := conv{|ψ⟩⟨ψ| | |ψ⟩ ∈ H, ⟨ψ|ψ⟩ = 1} = {ρ ∈ L(H) | ρ ⪰ 0, tr ρ = 1}. +(66) +From Eq. (65) one immediately recognizes that the PSD lift of conv SO(n) corresponds to D(H2n−1), where we +recognize that H2n−1 ∼= Cl0(n). Furthermore, the projection of the lift is achieved through the convexification of the +map Q : Cl(n) → Rn×n, where the fact that Q is quadratic in Cl(n) translates to being linear in D(Cl(n)). Specifically, +by a slight abuse of notation we shall extend the definition of Q to act on density operators as +Q(ρ) = +� +µ +pµQ(xµ), +where ρ = +� +µ +pµ|xµ⟩⟨xµ|. +(67) +Then Eq. (65) is the statement that Q(D(Cl0(n))) = conv SO(n). +In Appendix B 1 we show that this statement straightforwardly generalizes for Q(D(Cl(n))) = conv O(n). We prove +this using the fermionic representation developed in Section V C 2, and furthermore use these techniques to provide +an alternative derivation for the PSD lift of conv SO(n). The core of our argument is showing that the singular-value +conditions of Eqs. (63) and (64) translate into bounds on the largest eigenvalue of corresponding n-qubit observables: +σi(X) = tr(i�γiγiρ) ≤ 1, +(68) +� +i∈[n]\I +�σi(X) − +� +i∈I +�σi(X) = tr +� +�ρ0 +Ñ +� +i∈[n]\I +i�γiγi − +� +i∈I +i�γiγi +é� +� ≤ n − 2, +(69) +where ρ ∈ D(Cl(n)) and ρ0 ∈ D(Cl0(n)). The physical interpretation here is that not all pure quantum states map +onto to orthogonal or rotation matrices (which is clear from the fact that fermionic Gaussian states are only a subset +of quantum states). However, all density operators do map onto to their convex hulls, and the distinction between +conv O(n) and conv SO(n) can be automatically specified by restricting the support of ρ to the even-parity subspace. +VI. +QUANTUM RELAXATION FOR THE QUADRATIC PROBLEM +We now arrive at the primary problem of interest in this work, the little noncommutative Grothendieck problem +over the (special) orthogonal group. While the linear problem of Eq. (53) can be solved classically in polynomial +7 Technically, Saunderson et al. [24] use the definition �Pij = −Π0λiρjΠT +0 because they employ the standard adjoint representation, which +differs from our use of the twisted adjoint representation which includes the parity automorphism α. However since α(x) = x for all +x ∈ Cl0(n), both definitions of �Pij coincide. + +15 +time, quadratic programs are considerably more difficult. Here, we use the quantum formalism of the Pin and Spin +groups developed above to construct a quantum relaxation of this problem. Then in Section VII we describe rounding +procedures to recover a collection of orthogonal matrices from the quantum solution to this relaxation. +Recall the description of the input to Problem (2). Let (V, E) be a graph, and associate to each edge (u, v) ∈ E a +matrix Cuv ∈ Rn×n. We label the vertices as V = [m]. We wish to maximize the objective +f(R1, . . . , Rm) = +� +(u,v)∈E +⟨Cuv, RuRT +v ⟩. +(70) +over (R1, . . . , Rm) ∈ O(n)m. First, expand this expression in terms of matrix elements: +� +(u,v)∈E +⟨Cuv, RuRT +v ⟩ = +� +(u,v)∈E +� +i,j∈[n] +[Cuv]ij +� +k∈[n] +[Ru]ik[RT +v ]kj. +(71) +From the quadratic mapping Q : Cl(n) → Rn×n, we know that for each R ∈ G there exists some φ ∈ Pin(n) such that +Rij = ⟨φ|Pij|φ⟩. Hence we can express the matrix product as +[Ru]ik[RT +v ]kj = ⟨φu|Pik|φu⟩⟨φv|Pjk|φv⟩ += ⟨φu ⊗ φv|Pik ⊗ Pjk|φu ⊗ φv⟩, +(72) +which is now the expectation value of a 2n-qubit Pauli operator with respect to a product state of two Gaussian states +|φu⟩, |φv⟩. To extend this over the entire graph, we define a Hilbert space of m registers of n qubits each. For each +edge (u, v) ∈ E we introduce the Hamiltonian terms +Huv := +� +i,j∈[n] +[Cuv]ijΓ(u,v) +ij +, +(73) +where +Γ(u,v) +ij +:= +Ñ +� +k∈[n] +P (u) +ik +⊗ P (v) +jk +é +� +w∈V \{u,v} +I(w) +2n . +(74) +To simplify notation, we shall omit the trivial support � +w∈V \{u,v} I(w) +2n when the context is clear. +The problem is now reformulated as optimizing the mn-qubit Hamiltonian +H := +� +(u,v)∈E +Huv = +� +(u,v)∈E +� +i,j∈[n] +[Cuv]ij +� +k∈[n] +P (u) +ik +⊗ P (v) +jk . +(75) +The exact LNCG problem over O(n) then corresponds to +max +R∈Gm f(R) = +max +|ψ⟩∈H⊗m +2n +⟨ψ|H|ψ⟩ +subject to +� +� +� +� +� +⟨ψ|ψ⟩ = 1, +|ψ⟩ = � +v∈[m] |φv⟩, +|φv⟩ = U(Rv,In)|0n⟩, +Rv ∈ O(n) ∀v ∈ [m]. +(76) +The hardness of this problem is therefore related to finding the optimal separable state for local Hamiltonians, which +is NP-hard in general [57–59]. Dropping these constraints on the state provides a relaxation of the problem, since +max +|ψ⟩∈H⊗m +2n , +⟨ψ|ψ⟩=1 +⟨ψ|H|ψ⟩ ≥ max +R∈Gm f(R). +(77) +We point out here that the Hamiltonian terms Huv can be interpreted as two-body fermionic interactions. Note that +there is an important distinction between two-body fermionic operators (Clifford-algebra products of four Majorana +operators) and two-body qudit operators (tensor products of two qudit Pauli operators). Recall that Pij = i�γiγj +is one-body in the fermionic sense. While the operators P (u) +ik +⊗ P (v) +jk +appear to mix both notions, here they in fact +coincide. To see this, we consider a global algebra of Majorana operators {γi+(v−1)n, �γi+(v−1)n | i ∈ [n], v ∈ [m]} + +16 +acting on a Hilbert space of mn fermionic modes. While it is not true that the local single-mode Majorana operators +map onto the global single-mode operators, i.e., +γ(v) +i +� +w∈V \{v} +I(w) +2n ̸= γi+(v−1)n, +(78) +the local two-mode Majorana operators in fact do correspond to global two-mode operators: +�γ(v) +i +γ(v) +j +� +w∈V \{v} +I(w) +2n = �γi+(v−1)nγj+(v−1)n. +(79) +Thus, taking the tensor product of two local two-mode Majorana operators on different vertices is equivalent to taking +the product of two global two-mode Majorana operators: +�γ(u) +i +γ(u) +j +⊗ �γ(v) +k γ(v) +l +� +w∈V \{u,v} +I(w) +2n = �γi+(u−1)nγj+(u−1)n�γk+(v−1)nγl+(v−1)n. +(80) +Therefore Eq. (75) can be equivalently expressed as a Hamiltonian with two-body fermionic interactions. +Finally, when we wish to optimize over (R1, . . . , Rm) ∈ SO(n)m, it is straightforward to see that we can simply +replace the terms Pij with �Pij. Defining +�Γ(u,v) +ij +:= +Ñ +� +k∈[n] +�P (u) +ik +⊗ �P (v) +jk +é +� +w∈V \{u,v} +I(w) +2n−1, +(81) +�Huv := +� +i,j∈[n] +[Cuv]ij�Γ(u,v) +ij +, +(82) +the quantum relaxation for the SO(n) problem is given by the m(n − 1)-qubit Hamiltonian +�H := +� +(u,v)∈E +�Huv. +(83) +VII. +ROUNDING ALGORITHMS +Optimizing the energy of a local Hamiltonian is a well-studied problem, both from the perspective of quantum +and classical algorithms. In this section we will assume that such an algorithm has been used to produce the state +ρ ∈ D(H⊗m +d +) which (approximately) maximizes the energy tr(Hρ). We wish to round this state into the feasible space, +namely the set of product states of Gaussian states. We do so by rounding the expectation values of ρ appropriately, +such that we return some valid approximation R1, . . . , Rm ∈ G. In this section we propose two approaches to perform +this quantum rounding. +The first uses insight from the fact that our quantum relaxation is equivalent to a classical semidefinite relaxation +with additional constraints based on the convex hull of the orthogonal group. +This is approach is particularly +advantageous when optimizing over SO(n), as conv SO(n) has a matrix representation exponential in n (its PSD +lift). To build the semidefinite variable from the quantum state, we require measurements of the expectation values +of the two-vertex operators Γ(u,v) +ij += � +k∈[n] P (u) +ik +⊗ P (v) +jk +for each pair of vertices (u, v). +We refer this procedure +as conv SO(n)-based rounding.8 Our second rounding protocol uses the expectation values of P (v) +ij +of each vertex v +directly. In this case, rather than expectation values of two-vertex operators as before, we only require the information +of single-vertex marginals ρv := tr¬v(ρ). Therefore we call this approach vertex-marginal rounding. +If ρ is produced by a deterministic classical algorithm, then the relevant expectation values can be exactly computed +(to machine precision). However if the state is produced by a randomized algorithm, or is otherwise prepared by a +quantum computer, then we can only estimate the expectation values to within statistical error by some form of +sampling. +In the quantum setting, this can be achieved either by partial state tomography [60, 61] or a more +sophisticated measurement protocol [62].9 +See Appendix H for further comments on this quantum measurement +aspect. +The rounding algorithms then operate entirely as classical postprocessing after estimating the necessary +expectation values. +8 This rounding can also be applied to the optimization problem over O(n) as well, but we are particularly interested in the conv SO(n) +constraints due to their exponentially large classical representation. +9 For the present discussion we do not consider the effects of finite sampling, although we expect that rounding is fairly robust to such +errors since it will always return a solution in the feasible space. + +17 +A. +Approximation ratio for rounding the classical SDP +Before describing our quantum rounding protocols, we first review classical relaxations and rounding procedures +for Problem (2). The standard semidefinite relaxation can be expressed as the SDP +max +M∈Rmn×mn⟨C, M⟩ +subject to +® +M ⪰ 0, +Mvv = In +∀v ∈ [m], +(84) +where C ∈ Rmn×mn is the matrix with n × n blocks Cuv. If an additional nonconvex constraint rank(M) = n is +imposed, then the solution would be exact: +M = RRT = +� +���� +In +R1RT +2 +· · · R1RT +m +R2RT +1 +In +· · · R2RT +m +... +... +... +... +RmRT +1 RmRT +2 · · · +In +� +���� . +(85) +Problem (84) is there a relaxation of the original problem. However, the solution M ∈ Rmn×mn is still PSD, so it can +be decomposed as M = XXT, where +X = +� +�� +X1 +... +Xm +� +�� , +Xv ∈ Rn×mn. +(86) +The rounding algorithm of Bandeira et al. [16] then computes, for each v ∈ [m], +Ov = P(XvZ) := arg min +Y ∈O(n) +∥Y − XvZ∥F , +(87) +where Z is an mn × n Gaussian random matrix whose entries are drawn i.i.d. from N(0, 1/n). When optimizing over +G = O(n), this rounded solution guarantees (in expectation) an approximation ratio of +α2 +O(n) = E +� +� 1 +n +� +i∈[n] +σi(Z1) +� +� +2 +, +(88) +where Z1 is a random n × n matrix with i.i.d. entries from N(0, 1/n). +In Appendix E we extend the argument used to obtain this result for the optimization problem over G = SO(n), +and we show a corresponding approximation ratio of +α2 +SO(n) = E +� +� 1 +n +� +i∈[n−1] +σi(Z1) +� +� +2 +(89) +where the only change to the rounding algorithm is that we project to the nearest SO(n) element via �P, which is +defined as +�P(X) := arg min +Y ∈SO(n) +∥Y − X∥F . +(90) +Note that singular values are nonnegative, and in particular we show that E[σn(Z1)] > 0 for all finite n. Hence it +follows that α2 +SO(n) < α2 +O(n), which provides evidence for the claim that solving for rotations is generally a more +difficult problem (see Ref. [23, Section 4.3] for a brief discussion). For small values of n, the numerical values of these +approximation ratios are (computed using Mathematica): +α2 +O(2) ≈ 0.6564, +α2 +SO(2) ≈ 0.3927, +(91) +α2 +O(3) ≈ 0.6704, +α2 +SO(3) ≈ 0.5476, +(92) +α2 +O(4) ≈ 0.6795, +α2 +SO(4) ≈ 0.6096. +(93) + +18 +In Appendix E we provide an integral expression for αSO(n) which can be evaluated for arbitrary n. +For the problem over SO(n), Saunderson et al. [21] propose augmenting this SDP by adding the constraints that +each block of M lies in conv SO(n): +max +M∈Rmn×mn⟨C, M⟩ +subject to +� +� +� +� +� +M ⪰ 0, +Mvv = In +∀v ∈ [m], +Muv ∈ conv SO(n) +∀u, v ∈ [m]. +(94) +Although they do not prove approximation guarantees for this enhanced SDP, they first show that, if one reintroduces +the rank constraint on M, then the convex constraint Muv ∈ conv SO(n) in fact suffices to guarantee the much +stronger condition Muv ∈ SO(n). Then, when dropping the rank constraint (but leaving the conv SO(n) constraint) +they show that the relaxed problem is still exact over certain types of graphs, such as tree graphs. Finally, they provide +numerical evidence that even when the relaxation is not exact, it returns substantially more accurate approximations +than the standard SDP (84). +B. +Quantum Gram matrix +Analogous to the classical SDP solution M, we can form a matrix M ∈ Rmn×mn from the expectation values of ρ +as +M := +� +���� +In +T12 +· · · T1m +T21 +In +· · · T2m +... +... +... +... +Tm1 Tm2 · · · +In +� +���� , +(95) +where +Tuv := 1 +n +� +�� +tr(Γ(u,v) +11 +ρ) · · · tr(Γ(u,v) +1n +ρ) +... +... +... +tr(Γ(u,v) +n1 +ρ) · · · tr(Γ(u,v) +nn +ρ) +� +�� +(96) +and Tvu = T T +uv for u < v. Just as ⟨C, M⟩ gives the relaxed objective value (up to rescaling and constant shifts), +here we have that ⟨C, M⟩ = 2 +n tr(Hρ) + tr(C). In Appendix B we show that for any quantum state, M satisfies the +following properties: +� +� +� +� +� +M ⪰ 0, +Mvv = In +∀v ∈ [m], +Muv ∈ conv O(n) +∀u, v ∈ [m]. +(97) +Furthermore, we show that when ρ is supported only on the even subspace of each single-vertex Hilbert space (or +equivalently, if we replace Γ(u,v) +ij +with �Γ(u,v) +ij +in Eq. (96)), then +1 +n +� +�� +tr(�Γ(u,v) +11 +ρ) · · · tr(�Γ(u,v) +1n +ρ) +... +... +... +tr(�Γ(u,v) +n1 +ρ) · · · tr(�Γ(u,v) +nn +ρ) +� +�� ∈ conv SO(n) +∀ρ ∈ D(H⊗m +2n−1). +(98) +Therefore when optimizing the relaxed Hamiltonian �H for the SO(n) setting, we are guaranteed to automatically +satisfy the conv SO(n) constraints. +C. +conv SO(n)-based rounding +Given the construction of the M from quantum expectation values, we proceed to round the Gram matrix as in +the classical SDP with conv SO(n) constraints [21]. This consists of computing the matrices +Rv = �P(M1v), +(99) + +19 +where the projection to SO(n) can be efficiently computed from the special singular value decomposition, i.e., +�P(X) = U �V T +(100) +(recall Eq. (28)). Our choice of rounding using the first n × mn “row” of M amounts to fixing R1 = In. We note that +the same rounding procedure can naturally be applied to the O(n) setting as well, replacing �P with P. +D. +Vertex-marginal rounding +The single-vertex marginals are obtained by tracing out the qudits associated to all but one vertex v ∈ [m], +ρv = tr¬v(ρ). +(101) +As ρv ∈ D(H⊗m +d +), from Section V C 3 we have that Q(ρv) ∈ conv G, where we linearly extend the definition of Q to +Q(ρv) := +� +�� +tr(P (v) +11 ρ) · · · tr(P (v) +1n ρ) +... +... +... +tr(P (v) +n1 ρ) · · · tr(P (v) +nn ρ) +� +�� . +(102) +The rounding scheme we propose here then projects Q(ρv) to G using either P or �P: +Rv = arg min +Y ∈G +∥Y − Q(ρv)∥F . +(103) +We point out that the relaxed Hamiltonian only has two-vertex terms which we seek to maximize. In Appendix F +we show that H commutes with U⊗m +(In,V ) for all V ∈ O(n), which we further show implies that H may possess +eigenstates whose single-vertex marginals obey Q(σv) = 0. This indicates that there may exist eigenstates of H whose +single-vertex marginals yield no information, despite the fact that their two-vertex marginals are nontrivial. In our +numerical studies, we observe that breaking this symmetry resolves this issue. We accomplish this by including small +perturbative one-body terms which correspond to the trace of Q(σv): +H1 = +� +v∈[m] +� +i∈[n] +P (v) +ii , +(104) +Note that this trace quantity is importantly invariant with respect to the choice of basis for Rn. We then augment +the objective Hamiltonian with H1, defining +H′(ζ) := H + ζH1 +(105) +where ζ > 0 is a small regularizing parameter. +While this one-body perturbation does not correspond to any +terms in the original quadratic objective function, any arbitrarily small ζ > 0 suffices to break the O(n) symmetry. +Furthermore, the rounding procedure always guarantees that the solution is projected back into the feasible space +Gm. When G = SO(n) we define �H′(ζ) analogously. +VIII. +NUMERICAL EXPERIMENTS +To explore the potential of our quantum relaxation and rounding procedures, we performed numerical experiments +on randomly generated instances of the group synchronization problem. Because the Hilbert-space dimension grows +exponentially in both m and n, our classical simulations here are limited to small problem sizes. However, optimizing +over rotations in R3 (requiring only two qubits per vertex) is highly relevant to many practical applications, so +here we focus on the problem of SO(3) group synchronization. For example, this problem appears in the context of +cryo-EM as described in Section III A. To model the problem, we generated random instances by selecting random +three-regular graphs ([m], E), uniformly randomly sampling m rotations g1, . . . , gm ∈ SO(n), and then constructing +Cuv = gugT +v + σWuv for each (u, v) ∈ E, where the Gaussian noise matrix Wuv ∈ Rn×n has i.i.d. elements drawn from +N(0, 1) and σ ≥ 0 represents the strength (standard deviation) of this noise. +While the classical conv SO(n)-based SDP is not guaranteed to find the optimal solution, the problems studied +here were selected for such that this enhanced SDP in fact does solve the exact problem. We verify this property + +20 +4 +6 +8 +10 +Number of vertices m +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Approximation ratio +Quantum (CR) +Quantum (VR) +Classical SDP +10 +3.3 +10 +2.8 +10 +2.3 +10 +1.8 +10 +1.3 +10 +0.8 +10 +0.3 +Noise strength +0.2 +0.4 +0.6 +0.8 +1.0 +Approximation ratio +Quantum (CR) +Quantum (VR) +Classical SDP +FIG. 2. +Approximation ratios for solutions obtained from rounding the maximum eigenvector of the relaxed Hamiltonian +� +H. Violin plots show the distribution of approximation ratios over 50 randomly generated instance, and with the median +being indicated by the center marker. CR refers to rounding according to the conv SO(n)-based scheme (Section VII C), while +VR denotes the vertex-marginal rounding scheme (Section VII D). The classical SDP solution was rounded by the standard +randomized algorithm [16], and we report the best solution over 1000 rounding trials. (Left) Varying the number of vertices +m in the graph (random 3-regular graphs). Note that the number of qubits required here is 2m. (Right) Varying the noise +strength parameter σ which defines the problem via Cuv = gugT +v + σWuv. +by confirming that rank(M) = n before rounding on each problem instance. In this way we are able to calculate an +approximation ratio for the other methods (as it is not clear how to solve for the globally optimal solution in general, +even with an exponential-time classical algorithm). The methods compared here include our quantum relaxation with +conv SO(n)-based rounding (denoted CR), vertex-marginal rounding (VR), and the classical SDP (without conv SO(n) +constraints but using the �P projection to guarantee that the rounded solutions are elements of SO(n)). When using +the vertex-rounding method, we employ �H′(ζ) as the objective Hamiltonian with ζ = 10−6. +A. +Exact eigenvectors +First, we consider the solution obtained by rounding the maximum eigenvector of �H. Although the hardness of +preparing such a state is equivalent that of the ground-state problem, this nonetheless provides us with a benchmark +for the ultimate approximation quality of our quantum relaxation. In Figure 2 we plot the approximation ratio of the +rounded quantum states and compare to that of the classical SDP on the same problem instances. Each violin plot +was constructed from the results of 50 random instances. +The results here demonstrate that, while the approximation quality of the classical SDP quickly falls off with larger +graph sizes, our rounded quantum solutions maintain high approximation ratios, at least for the problem sizes probed +here. Notably, the conv SO(n)-based rounding on the quantum state is significantly more powerful and consistent +than the vertex-marginal rounding. +This feature is not unexpected since, as discussed in Section VII D, we are +maximizing an objective Hamiltonian with only two-body terms, whereas the single-vertex rounding uses strictly +one-body expectation values. Furthermore, as demonstrated in previous works [21, 25] the conv SO(n) constraints are +powerful in practice, and so we expect that the quantum rounding protocol which makes use of this structure enjoys +the same advantages. +Meanwhile, when varying the noise parameter σ, we observe that all methods are fairly consistent. In particular, the +conv SO(n)-based rounding only shows an appreciable decrease in approximation quality when the noise is considerable +(note that σ = 0.5 ≈ 10−0.3 is a relatively large amount of noise, since gugT +v is an orthogonal matrix and therefore +has matrix elements bounded in magnitude by 1). + +21 +0 +1 +2 +3 +4 +Total evolution time T +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Approximation ratio +Optimal objective value +Maximum quantum relaxed value +Quantum (CR) +Quantum (VR) +Classical SDP +Quantum relaxed value +FIG. 3. Demonstration of a typical instance of adiabatic state preparation for preparing relaxed quantum solutions. The initial +state is the product of Gaussian states corresponding to the rounded solution of the classical SDP. As the total evolution time T +increases, the evolution becomes more adiabatic, indicated by the convergence of the relaxed value to the maximum eigenvalue +(in units of the original problem’s optimal value). The rounded solutions of course can never exceed the original problem’s +optimal value. +B. +Quasi-adiabatic state preparation +Because it may be unrealistic to prepare the maximum eigenvector of �H, here we consider preparing states using +ideas from adiabatic quantum computation [63]. +Specifically, we wish to demonstrate that states whose relaxed +energy may be far from the maximum eigenvalue can still provide high-quality approximations after rounding. If +this is the case then we do not need to prepare very close approximations to the maximum eigenstate of �H, so the +rigorous conditions of adiabatic state preparation may not be required in this context. Hence we consider “quasi- +adiabatic” state preparation, wherein we explore how time-evolution speeds far from the adiabatic limit may still +return high-quality approximations. Our numerical experiments here provide a preliminary investigation into this +conjecture. +For simplicity of the demonstration, we consider a linear annealing schedule according to the time-dependent +Hamiltonian +H(t) = +Å +1 − t +T +ã +Hi + t +T Hf, +(106) +which prepares the state +|ψ(T)⟩ = T exp +Ç +−i +� T +0 +dt H(t) +å +|ψ(0)⟩ +(107) +for some T > 0, where T is the time-ordering operator. The final Hamiltonian Hf is the desired objective LNCG +Hamiltonian, +Hf = �H. +(108) +The initial Hamiltonian Hi is the parent Hamiltonian of the initial state, which we choose to be the approximation +obtained from the classical SDP, as it can be obtained classically in polynomial time. Let R1, . . . , Rm ∈ SO(n) be the +SDP solution. Our initial state is then the product of Gaussian states +|ψ(0)⟩ = +� +v∈[m] +|φ(Rv)⟩, +(109) +where each |φ(Rv)⟩ is the maximum eigenvector of the free-fermion Hamiltonian +F(Rv) = i +� +i,j∈[n] +[Rv]ij�γiγj. +(110) + +22 +4 +6 +8 +10 +Number of vertices m +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Approximation ratio +Optimal objective value +Maximum quantum relaxed value +Quantum (CR) +Quantum (VR) +Classical SDP +Quantum relaxed value +10 +3.3 10 +2.8 10 +2.3 10 +1.8 10 +1.3 10 +0.8 10 +0.3 +Noise strength +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +Approximation ratio +Maximum quantum relaxed value +Quantum (CR) +Quantum (VR) +Classical SDP +Quantum relaxed value +FIG. 4. Approximation ratios for solutions obtained from rounding the “adiabatically” evolved state |ψ(T)⟩ with fixed T = 1 +for all m, σ. Problem instances and visualization is the same as in Figure 2. We also include the maximum eigenvalue of +the relaxed Hamiltonian and the energy of the prepared unrounded state, to demonstrate how far |ψ(T)⟩ is from the exact +maximum eigenvector. (Left) Varying the number of vertices m in the graph. Note that the number of qubits required here is +2m. (Right) Varying the noise strength parameter σ. +Therefore the initial Hamiltonian Hi is a sum of such free-fermion Hamiltonians (here we include the even-subspace +projection since we are working with SO(n)): +Hi = +� +v∈[m] +Π0F(Rv)ΠT +0 . +(111) +As a Gaussian state, |φ(Rv)⟩ can be prepared exactly from a quantum circuit of O(n2) gates [64]. Note that since we +are working directly in the even subspace of n − 1 qubits here, this n-qubit circuit must be projected appropriately +using Π0. We discuss how to perform this circuit recompilation in Appendix C. We comment that this choice of +initial state is that of a mean-field state for non-number-preserving fermionic systems, for instance as obtained from +Hartree–Fock–Bogoliubov theory. Suitably, the final Hamiltonian we evolve into is non-number-preserving two-body +fermionic Hamiltonian. +In adiabatic state preparation, the total evolution time T controls how close the final state |ψ(T)⟩ is to the max- +imum eigenstate10 of the final Hamiltonian Hf. One metric of closeness is how the energy of the prepared state, +⟨ψ(T)|Hf|ψ(T)⟩, compares to the maximum eigenvalue of Hf. On the other hand, as a relaxation, this maximum +energy is already larger than the optimal objective value of the original problem. We showcase this in Figure 3, using +one random problem instance as a demonstrative (typical) example on a graph of m = 6 vertices (12 qubits). For +each total evolution time point T, we computed |ψ(T)⟩ by numerically integrating the time-dependent Schr¨odinger +equation, and we plot its relaxed energy as well as its rounded objective values. For large T we approach the maximum +eigenstate of �H as expected (thereby also demonstrating that the initial “mean-field” state |ψ(0)⟩ has appreciable +overlap). Particularly interesting is the behavior for relatively small total evolution times T, wherein the energy of +|ψ(T)⟩ is far from the maximum eigenenergy. Despite this, the approximation quality after rounding the state using +M is nearly exact around T ≈ 1. On the other hand, the approximation quality of vertex-marginal rounding is highly +inconsistent, which again we attribute to the fact that the single-vertex information is not directly seen by the final +Hamiltonian Hf. +Then in Figure 4 we plot the same 50 problem instances (per graph size/noise level) as in Figure 2, but using the +quasi-adibatically prepared state |ψ(T)⟩ where we have fixed T = 1 for all graph sizes. The classical SDP results are +the same as in Figure 4, and for reference we include the energy of the unrounded quantum state and the maximum +eigenvalue of the relaxed Hamiltonian (normalized with respect to the optimal objective value). Qualitatively, we +observe features similar to those seen in Figure 3. Namely, although the annealing schedule is too fast to prepare a close +10 We remind the reader that we are starting in the maximum eigenstate of the initial Hamiltonian, whereas in the physics literature, +adiabatic theorems are typically stated in terms of ground states. Of course, the two perspectives are equivalent by simply an overall +sign change (note that all Hamiltonians here are traceless). + +23 +approximation to the maximum eigenstate, the rounded solutions (using the conv SO(n)-based protocol) consistently +have high approximation ratios. Meanwhile, the vertex-rounded solutions are highly inconsistent, which reflects the +highly fluctuating behavior seen in Figure 3. +IX. +DISCUSSION AND FUTURE WORK +In this paper we have developed a quantum relaxation for a quadratic program over orthogonal and rotation ma- +trices, known as an instance of the little noncommutative Grothendieck problem. The embedding of the classical +objective is achieved by recognizing an intimate connection between the geometric-algebra construction of the orthog- +onal group and the structure of quantum mechanics, in particular the formalism of fermions in second quantization. +From this perspective, the determinant condition of SO(n) is succinctly captured by a simple linear property of the +state—its parity—and the convex bodies conv O(n) and conv SO(n) (relevant to convex relaxations of optimization +over orthogonal matrices) are completely characterized by density operators on n and n − 1 qubits, respectively. Rec- +ognizing that the reduced state on each vertex therefore corresponds to an element of this convex hull, we proposed +vertex-marginal rounding which classically rounds the measured one-body reduced density matrix of each vertex. +We additionally showed that these convex hulls are characterized by density operators on 2n and 2(n − 1) qubits as +well, where the linear functionals defining this PSD lift are the Hamiltonian terms appearing in our quantum relaxation. +This insight enables our second proposed rounding scheme, conv G-based edge rounding, which is inspired by the fact +that the a quantum Gram matrix M can be constructed from the expectation values of the quantum state which +obeys the same properties as the classical SDP of Saunderson et al. [21]. Numerically we observe that this approach +to quantum rounding is significantly more accurate and consistent than vertex rounding, and it consistently achieves +larger approximation ratios than the basic SDP relaxation. However, we are severely limited by the exponential +scaling of classically simulating quantum states; further investigations would be valuable to ascertain the empirical +performance of these ideas at larger scales. +The primary goal of this work was to formulate the problem of orthogonal-matrix optimization into a familiar quan- +tum Hamiltonian problem, and to establish the notion of a quantum relaxation for such optimization problems over +continuous-valued decision variables. A clear next step is to prove nontrivial approximation ratios from our quantum +relaxation. If such approximation ratios exceed known guarantees by classical algorithms, for example on certain +types of graphs, then this would potentially provide a quantum advantage for a class of applications not previously +considered in the quantum literature. We have proposed one standard, realistically preparable class of states—quasi- +adiabatic time evolution—but a variety of energy-optimizing ansatze exist in the literature, especially considering +that the constructed Hamiltonian is an interacting-fermion model. From this perspective, it would also be interesting +to see if a classical many-body method can produce states which round down to high-quality approximations, even +heuristically. Such an approach would constitute a potential example of a quantum-inspired classical algorithm. +From a broader perspective, the quantum formalism described here may also provide new insights into the compu- +tational hardness of the classical problem. First, the NP-hard thresholds for Problem (2) are not currently known. +However, by establishing the classical problem as an instance of Gaussian product state optimization on the many- +body Hamiltonian, it may be possible to import tools from quantum computational complexity to study the classical +problem. This idea also applies to the more general instances of noncommutative Grothendieck problems, +max +U,V ∈O(N) +� +i,j,k,l∈[N] +TijklUijVkl, +(112) +where the N × N × N × N tensor T specifies the problem input. It is straightforward apply our quantum relaxation +construction to this problem, yielding a 2N-qubit Hamiltonian whose terms are of the form Pij ⊗ Pkl. While Bri¨et et +al. [17] showed that the NP-hardness threshold of approximating this problem is 1/2, it remains an open problem to +construct an algorithm which is guaranteed to achieve this approximation ratio. +Although we have provided new approximation ratios for the instance of Problem (2) over SO(n), it is unclear +precisely how much harder the SO(n) problem is compared to the O(n) problem. The work by Saunderson et al. [24] +establishes a clear distinction between the representation sizes required for conv O(n) and conv SO(n), and this paper +has connected this structure to properties of quantum states on n qubits. However this does not yet establish a +difference of hardness for the corresponding quadratic programs. Again it would be interesting to see if the tools of +quantum information theory can be used to further understand this classical problem. For example, one might study +the NP-hardness threshold of Problem (112) where instead U, V ∈ SO(N) and leverage the quantum (or equivalently, +Clifford-algebraic) representation of SO(N). In such a setting, the size of the problem is given by a single parameter +N and so the exponentially large parametrization of conv SO(N) appears to signify a central difficulty of this problem. +We note that it is straightforward to extend our quantum relaxation to the unitary groups U(n) and SU(n), +essentially by doubling the number of qubits per vertex via the inclusions U(n) ⊂ O(2n) and SU(n) ⊂ SO(2n). + +24 +However this is likely an inefficient embedding, since the n-qubit Majorana operators already form a representation +of Cl(2n). It may therefore be possible to encode complex-valued matrices via a complexification of Q, using the +same amount of quantum space. It is interesting to note that Bri¨et et al. 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[24, 65]. +The Clifford algebra Cl(n) of Rn is a 2n-dimensional real vector space, equipped with an inner product ⟨·, ·⟩ : +Cl(n) × Cl(n) → R and a multiplication operation satisfying the anticommutation relation +eiej + ejei = −2δij11, +(A1) +where {e1, . . . , en} is an orthonormal basis of Rn and 11 is the multiplicative identity of the algebra. The basis elements +ei are called the generators of the Clifford algebra, in the sense that they generate all other basis vectors of Cl(n) as +eI := ei1 · · · eik, +I = {i1, . . . , ik} ⊆ [n]. +(A2) +By convention we order the indices i1 < · · · < ik, and the empty set corresponds to the identity, e∅ = 11. Taking +all subsets I ⊆ [n] and extending the inner product definition from Rn to Cl(n), it follows that {eI | I ⊆ [n]} is an +orthonormal basis with 2n elements. Specifically, we can write any element x ∈ Cl(n) as +x = +� +I⊆[n] +xIeI +(A3) +with each xI ∈ R, and the inner product on Cl(n) is11 +⟨x, y⟩ = +� +I⊆[n] +xIyI, +(A4) +11 Equipping an inner product to the vector representation of Cl(n) elements is achieved using the fact that algebra elements square to a +multiple of the identity. + +27 +where y = � +I⊆[n] yIeI. Hence Cl(n) is isomorphic as a Hilbert space to R2n. +Now we show how to realize the orthogonal group O(n) from this algebra. First observe the inclusion Rn = span{ei | +i ∈ [n]} ⊂ Cl(n). We shall identify the sphere Sn−1 ⊂ Rn as all u ∈ Rn satisfying ⟨u, u⟩ = 1. We then define the Pin +group as all possible products of Sn−1 elements: +Pin(n) := {u1 · · · uk | u1, . . . , uk ∈ Sn−1, 0 ≤ k ≤ n}. +(A5) +It is straightforward to check that this is indeed a group. Each x ∈ Pin(n) is also normalized, ⟨x, x⟩ = 1. In fact, an +equivalent definition of this group is all elements x ∈ Cl(n) satisfying xx = 11, where conjugation x is defined from +the linear extension of +eI := (−1)|I|eik · · · ei1. +(A6) +The Pin group is a double cover of O(n), which can be seen from defining a quadratic map Q : Cl(n) → Rn×n. This +map arises from the so-called twisted adjoint action, introduced by Atiyah et al. [65]:12 +v �→ α(x)vx, +x, v ∈ Cl(n), +(A7) +where the linear map α : Cl(n) → Cl(n) is the parity automorphism, defined by linearly extending +α(eI) := (−1)|I|eI. +(A8) +Then for any x ∈ Cl(n), the linear map Q(x) : Rn → Rn is defined as +Q(x)(v) := πRn(α(x)vx) +∀v ∈ Rn, +(A9) +where πRn is the projection from Cl(n) onto Rn. To show that Q(Pin(n)) = O(n), it suffices to recognize that, for +any u ∈ Sn−1, α(u)vu ∈ Rn is the reflection of the vector v ∈ Rn across the hyperplane normal to u. To see this, first +observe that uv + vu = −2⟨u, v⟩11, which follows from Eq. (A1) by linearity. Then +α(u)vu = uvu += (−vu − 2⟨u, v⟩11)u += v − 2⟨u, v⟩u, +(A10) +which is precisely the elementary reflection as claimed. By the Cartan–Dieudonn´e theorem, one can implement any +orthogonal transformation on Rn by composing k ≤ n such reflections about arbitrary hyperplanes u1, . . . , uk [66]. This +characterization coincides precisely with the definition of the Pin group provided in Eq. (A5), through the composition +of the linear maps Q(u1), . . . , Q(uk) on Rn. Hence for all x ∈ Pin(n), Q(x) is an orthogonal transformation on Rn. +The double cover property follows from the fact that Q is quadratic in x, so Q(x) = Q(−x). +The special orthogonal group arises from the subgroup Spin(n) ⊂ Pin(n) containing only even-parity Clifford +elements. First observe that Cl(n) is a Z2-graded algebra: +Cl(n) = Cl0(n) ⊕ Cl1(n), +(A11) +where +Cl0(n) := span{eI | |I| even}, +(A12) +Cl1(n) := span{eI | |I| odd}. +(A13) +By a Z2 grading we mean that for each x ∈ Cla(n) and y ∈ Clb(n), their product xy lies in Cla+b mod 2(n). We say +that elements in Cl0(n) (resp., Cl1(n)) have even (resp., odd) parity. In particular, this grading implies that Cl0(n) +is a subalgebra, hence its intersection with the Pin group is also a group, which defines +Spin(n) := Pin(n) ∩ Cl0(n) = {u1 · · · u2k | u1, . . . , u2k ∈ Sn−1, 0 ≤ k ≤ ⌊n/2⌋}. +(A14) +Just as the Pin group double covers O(n), so does the Spin group double cover SO(n). This is again a consequence +of the Cartan–Dieudonn´e theorem, wherein all rotations on Rn can be decomposed into an even number of (at most +n) arbitrary reflections. +12 Saunderson et al. [24] consider the standard adjoint action, which is sufficient for describing rotations. However, the “twist” due to α +is necessary to construct arbitrary orthogonal transformations. + +28 +Appendix B: Convex hull of orthogonal matrices and quantum states +Recall the following characterizations of the convex hulls: +conv O(n) = �X ∈ Rn×n | σ1(X) ≤ 1� +(B1) +conv SO(n) = +� +� +�X ∈ Rn×n +����� +� +i∈[n]\I +�σi(X) − +� +i∈I +�σi(X) ≤ n − 2 +∀I ⊆ [n], |I| odd +� +� +�, +(B2) +where {σi(X)}i∈[n] and {�σi(X}i∈[n] are the singular values and special singular values of X in descending order, +respectively. Note that σi(X) = �σi(X) for all i ≤ n − 1 and σn(X) = sign(det(X))σn(X). +1. +PSD lift of conv O(n) and conv SO(n) +In this section we show that Q(D(Cl(n))) = conv O(n) and Q(D(Cl0(n))) = conv SO(n), using the quantum +formalism described in Section V. +First we show that for all X ∈ conv O(n), there exists some ρ ∈ D(H2n) which generates X, essentially by the +convex extension of Q. Every X ∈ conv O(n) can be expressed as a convex combination (� +µ pµ = 1, pµ ≥ 0) of +orthogonal matrices Rµ ∈ O(n): +X = +� +µ +pµRµ. +(B3) +For each Rµ there exists some xµ ∈ Pin(n) such that [Rµ]ij = ⟨xµ|Pij|xµ⟩. Therefore the matrix elements of X can +be expressed as +Xij = tr +� +Pij +� +µ +pµ|xµ⟩⟨xµ| +� += tr(Pijρ), +(B4) +where ρ := � +µ pµ|xµ⟩⟨xµ| ∈ D(H2n). +Next we show the reverse direction, that for all ρ ∈ D(H2n), the matrix X := [tr(Pijρ)]i,j∈[n] is an element of +conv O(n). Recall that X ∈ conv O(n) if and only if σ1(X) ≤ 1. Therefore we take the singular value decomposition +of X = UΣV T and, using Pij = i�γiγj, each singular value is equal to +σk(X) = [U TXV ]kk += +� +i,j∈[n] +Uik tr(i�γiγjρ)Vjk += tr(iU† +(U,In)�γkU(U,In)U† +(In,V )γkU(In,V )ρ) += tr(i�γkγkρ′), +(B5) +where ρ′ := U(U,V )ρU† +(U,V ). Because i�γkγk has eigenvalues ±1, we see that σk(X) ≤ 1 for all k ∈ [n]. +For the restriction to conv SO(n), the first argument is essentially the same. One merely replaces Pij with �Pij, hence +� +µ pµ|xµ⟩⟨xµ| ∈ D(H2n−1). For the reverse direction, we instead employ the special singular value decomposition +which yields +�σk(X) = tr(i�γkγkρ′), +(B6) +where now ρ′ := U(U,�V )ρU† +(U,�V ) ∈ D(H2n−1). Note that we have not projected to the even subspace this time, as it is +more convenient to work in the full n-qubit space when handling the Gaussian unitaries. Instead, we will impose the +constraint that ρ only has support on the even-parity subspace, so tr(Z⊗nρ) = 1. Furthermore, because the special +singular value decomposition guarantees that det(U) det(�V ) = 1, U(U,�V ) is parity preserving so that tr(Z⊗nρ′) = 1 as +well. Now recall that X ∈ conv SO(n) if and only if +� +k∈[n]\I +�σk(X) − +� +k∈I +�σk(X) ≤ n − 2 +(B7) + +29 +for all subsets I ⊆ [n] of odd size. By linearity, +� +k∈[n]\I +�σk(X) − +� +k∈I +�σk(X) = tr +Ñ +ρ′ � +k∈[n] +(−1)zkZk +é +, +(B8) +where z = z1 · · · zn ∈ {0, 1}n is defined as zk = 1 if k ∈ I and zk = 0 otherwise, and we have used the fact that +i�γkγk = Zk. It therefore suffices to examine the spectrum of Az := � +k∈[n](−1)zkZk: +Az|b⟩ = +Ñ +� +k∈[n] +(−1)[z⊕b]k +é +|b⟩, +b ∈ {0, 1}n, +(B9) +where ⊕ denotes addition modulo 2. As we are only interested in the subspace spanned by even-parity states, we +restrict attention to the eigenvalues for which |b| mod 2 = 0. Because |I| is odd, so too is |z|, hence |z ⊕ b| mod 2 = 1. +This implies that there must be at least one term in the sum of Eq. (B9) which is negative, so it can only take integer +values at most n − 2. This establishes Eq. (B7), hence X ∈ conv SO(n). +2. +Relation to conv SO(n)-based semidefinite relaxation +Here we provide details for our claim that the relaxed quantum solution obeys the same constraints as the classical +SDP which uses the exponentially large representation of conv SO(n). Recall that this relaxation can be formulated +as +max +M∈Rmn×mn +� +(u,v)∈E +⟨Cuv, Muv⟩ +subject to +� +� +� +� +� +M ⪰ 0, +Mvv = In +∀v ∈ [m], +Muv ∈ conv SO(n) +∀u, v ∈ [m]. +(B10) +We will show that the Gram matrix M constructed from the measurements of a quantum state ρ, defined in Sec- +tion VII B as +[Muv]ij = +� +� +� +� +� +δij +u = v, +1 +n tr(Γ(u,v) +ij +ρ) +u < v, +[Mvu]ji +u > v, +(B11) +obeys the constraints of Eq. (B10). Specifically, when the marginals of ρ on each vertex are even-parity states (recall +this is equivalent to replacing Γij with �Γij), we obtain the conv SO(n) condition, whereas when the parity of ρ is not +fixed then Muv ∈ conv O(n). +First, we show that M is positive semidefinite for all quantum states. +Lemma B.1. Let M ∈ Rmn×mn be defined as in Eq. (B11). For all ρ ∈ D(H⊗m +2n ), M ⪰ 0. +Proof. We prove the statement by a sum-of-squares argument. To see where the fact of 1/n appears in the quantum +definition of M above, we first construct a matrix M′ ⪰ 0 which turns out to simply be M′ = nM. +For each k ∈ [n] define the Hermitian operator +Ak = +� +v∈V +� +i∈[n] +c(v) +i +P (v) +ik , +(B12) +where c(v) +i +∈ R are arbitrary coefficients. Consider its square, +A2 +k = +Ñ +� +v∈V +� +i∈[n] +c(v) +i +P (v) +ik +é2 += +� +v∈V +� +i,j∈[n] +c(v) +i +c(v) +j P (v) +ik P (v) +jk + +� +u,v∈V +u̸=v +� +i,j∈[n] +c(u) +i +c(v) +j P (u) +ik +⊗ P (v) +jk . +(B13) + +30 +Note that the terms with u ̸= v feature the two-vertex operators as desired, while the diagonal terms of the sum +contain products of the Pauli operators acting on the same vertex. Because Pik = i�γiγk, the diagonal terms reduce +to (suppressing superscripts here) +� +i,j∈[n] +cicjPikPjk = − +� +i,j∈[n] +cicj�γiγk�γjγk += +� +i,j∈[n] +cicj�γi�γj += +� +i∈[n] +c2 +i I2n + +� +1≤i 0 for all finite n, so the rounding algorithm guarantees a strictly smaller +approximation ratio for the problem over SO(n) than over O(n). +The proof of Theorem E.1 requires two lemmas regarding the expected value of random Gaussian matrices under +the rounding operator P. Analogously, our proof of Theorem E.2 requires a modification of those lemmas when P is +replaced by �P. +Lemma E.3 (Adapted from [16, Lemma 5]). Let M, N ∈ Rn×mn obey MM T = NN T = In. For Z ∈ Rmn×n with +i.i.d. entries drawn from N(0, n−1), we have +E +î �P(MZ)(NZ)Tó += E +î +(MZ) �P(NZ)Tó += αSO(n)MN T. +(E9) +This lemma is proved with the help of the following lemma. +Lemma E.4 (Adapted from [16, Lemma 6]). Let Z1 ∈ Rn×n with i.i.d. entries drawn from N(0, 1/n). Then +E +î �P(Z1)ZT +1 +ó += E +î +Z1 �P(Z1)Tó += αSO(n)In. +(E10) + +37 +Before we prove these two lemmas, we will use them to prove Theorem E.2. The proof idea here is entirely analogous +to the original argument of Theorem E.1 from Ref. [16], but with the appropriate replacements of P by �P. Nonetheless +we sketch the proof below for completeness. +Proof (of Theorem E.2). We wish to lower bound the average rounded value +E[f(Q1, . . . , Qm)] = E +� +� � +(u,v)∈E +⟨Cuv, �P(XuZ) �P(XvZ)T⟩ +� +� +(E11) +in terms of the relaxed value � +(u,v)∈E⟨Cuv, XuXT +v ⟩. Assuming we have such a lower bound with ratio 0 < α2 ≤ 1, +this leads to a chain of inequalities establishing the desired approximation ratio to the original problem: +E +� +� � +(u,v)∈E +⟨Cuv, �P(XuZ) �P(XvZ)T⟩ +� +� ≥ α2 +� +(u,v)∈E +⟨Cuv, XuXT +v ⟩ +≥ α2 +max +R1,...,Rm∈O(n) +� +(u,v)∈E +⟨Cuv, RuRT +v ⟩ +≥ α2 +max +R1,...,Rm∈SO(n) +� +(u,v)∈E +⟨Cuv, RuRT +v ⟩, +(E12) +where the second inequality follows from the fact that the relaxation provides an upper bound to the original problem, +and the third inequality is a consequence of SO(n) ⊂ O(n). The task is then to determine such an α which satisfies +the first inequality of Eq. (E12). The core argument is a generalization of the Rietz method [49], which proceeds by +constructing a positive semidefinite matrix S ∈ Rmn×mn whose (u, v)th block is defined as +Suv := +Ä +XuZ − α−1 �P(XuZ) +äÄ +XvZ − α−1 �P(XvZ) +ä +T. +(E13) +The expected value of this matrix is +E[Suv] = E +î +XuZ(XvZ)T − α−1 �P(XuZ)(XvZ)T − α−1(XuZ) �P(XvZ)T + α−2 �P(XuZ) �P(XvZ)Tó += Xu E�ZZT�Xv − α−1 E +î �P(XuZ)(XvZ)Tó +− α−1 E +î +(XuZ) �P(XvZ)Tó ++ α−2 E +î �P(XuZ) �P(XvZ)Tó +. +(E14) +Because ZZT is a Wishart matrix with covariance matrix In/n, we have E[ZZT] = In. Meanwhile, E +î �P(XuZ) �P(XvZ)Tó +is the quantity we wish to bound. To compute the expected values of the two cross terms, we invoke Lemma E.3 +which holds because XuXT +u = XvXT +v = In: +E +î �P(XuZ)(XvZ)Tó += E +î +(XuZ) �P(XvZ)Tó += αSO(n)XuXT +v . +(E15) +Thus, setting α = αSO(n), we obtain +E[Suv] = XuXT +v − XuXT +v − XuXT +v + α−2 +SO(n) E +î �P(XuZ) �P(XvZ)Tó += −XuXT +v + α−2 +SO(n) E +î �P(XuZ) �P(XvZ)Tó +. +(E16) +Finally, using the fact that C, S ⪰ 0, we have that ⟨C, S⟩ ≥ 0 and so E⟨C, S⟩ ≥ 0, which implies that +E +� +� � +(u,v)∈E +⟨Cuv, �P(XuZ) �P(XvZ)T⟩ +� +� ≥ α2 +SO(n) +� +(u,v)∈E +⟨Cuv, XuXT +v ⟩. +(E17) +Then by Eq. (E12) the claim follows. +We now establish the value of +αSO(n) = E +� +� 1 +n +� +i∈[n−1] +σi(Z1) +� +� +(E18) + +38 +from Lemmas E.3 and E.4. Because Lemma E.3 is somewhat technical and the argument is virtually unchanged by +replacing P with �P, we refer the reader to Ref. [16] for proof details. Instead, we simply note that the only part of +the proof for Lemma E.3 which does depend on the change to �P is the final result, wherein it is established that +E +î +(MZ) �P(NZ)Tó += E +î +Z1 �P(Z1)Tó +MN T, +(E19) +where Z1 ∈ Rn×n has entries i.i.d. from N(0, 1/n). Thus proving Lemma E.4 is the key component in establishing +the value of the approximation ratio α2 +SO(n). +Proof (of Lemma E.4). Consider the singular value decomposition of Z1 = UΣV T ∈ Rn×n. Its special singular value +decomposition can be written as Z1 = U(ΣJUV T)(V JUV T)T, where JUV T is the n × n diagonal matrix +JUV T := +ïIn−1 +0 +0 +det UV T +ò +. +(E20) +Note that JUV T = JUJV . Using the fact that the (special) rounding operator returns +�P(Z1) = UJUV TV T, +(E21) +we have +�P(Z1)ZT +1 = UJUJV ΣU T. +(E22) +Because Z1 is a random Gaussian matrix with i.i.d. entries, its singular values and left- and right-singular vectors are +distributed independently [67]. In particular, both U and V are distributed according to the Haar measure on O(n). +The expected value of Eq. (E22) can therefore be split into three independent averages: +E +Z1∼N (0,In/n) +î �P(Z1)ZT +1 +ó += +E +Σ∼D +E +U∼O(n) +ï +UJU +E +V ∼O(n)[JV ]ΣU T +ò +. +(E23) +(We shall comment on the distribution D of singular values later.) Because O(n) is evenly divided into its unconnected +(+1)- and (−1)-determinant components, the average determinant vanishes: EV ∼O(n)[det V ] = 0. This leaves us with +E +î �P(Z1)ZT +1 +ó += E�UΣU T�, +(E24) +where +Σ = +� +���� +σ1(Z1) +... +σn−1(Z1) +0 +� +���� . +(E25) +The Haar average over U ∼ O(n) in Eq. (E24) is well-known [68] to be proportional to the identity, E[UΣU T] = λIn, +and the constant of proportionality can be determined by considering its trace: +nλ = tr(λIn) = tr�E[UΣU T]� += E[tr Σ] += E +� +� � +i∈[n−1] +σi(Z1) +� +�. +(E26) +Hence λ = αSO(n) and so +E +î �P(Z1)ZT +1 +ó += αSO(n)In. +(E27) +The corresponding statement for E +î +Z1 �P(Z1)Tó +follows completely analogously, essentially by interchanging the roles +of U and V . The entire argument is equivalent because U and V are i.i.d. + +39 +To numerically evaluate αSO(n) we can use the linearity of expectation, +αSO(n) = αO(n) − 1 +n E[σn(Z1)]. +(E28) +The distribution of singular values of random Gaussian matrices can be analyzed from the theory of Wishart matrices. +In particular, Z1ZT +1 = UΣ2U T is a Wishart matrix with covariance matrix In/n, so the distribution of singular values Σ +is the square root of the Wishart distribution of eigenvalues. The quantity αO(n) in terms of the marginal distribution +p(avg) +n +(x) of Wishart eigenvalues x ∈ (0, ∞) was studied in Ref. [16], yielding the expression +αO(n) = +1 +√n +� ∞ +0 +p(avg) +n +(x)√x dx. +(E29) +Note the factor of n−1/2, which is introduced because the distribution p(avg) +n +(x) is normalized to have unit variance. +An explicit expression of p(avg) +n +(x) can be found in Refs. [16, Lemma 21] and [69, Eq. (16)]. +For our newly derived approximation ratio αSO(n), we need to additionally evaluate the expected smallest singu- +lar value of this Wishart distribution. This minimum-eigenvalue distribution was studied in Ref. [70], wherein an +analytical expression was derived (again assuming unit variance): +p(min) +n +(x) = +n +2n−1/2 +Γ(n) +Γ(n/2) +e−xn/2 +√x +U +Ån − 1 +2 +, −1 +2, x +2 +ã +. +(E30) +Here, U(a, b, z) with a > 0 and b < 1 is the Tricomi confluent hypergeometric function, the unique solution to the +differential equation +z d2U +dz2 + (b − z)dU +dz − aU = 0 +(E31) +with boundary conditions U(a, b, 0) = Γ(1−b)/Γ(1+a−b) and limz→∞ U(a, b, z) = 0. The expression for the average +smallest singular value is therefore +E[σn(Z1)] = +1 +√n +� ∞ +0 +p(min) +n +(x)√x dx. +(E32) +Altogether, we arrive at the integral expression for +αSO(n) = +1 +√n +� ∞ +0 +ï +p(avg) +n +(x) − 1 +np(min) +n +(x) +ò√x dx. +(E33) +Appendix F: LNCG Hamiltonian symmetries +Here we demonstrate the local O(n) symmetry discussed in Section VII D. Consider an edge term +Huv = +� +i,j∈[n] +[Cuv]ij +� +k∈[n] +P (u) +ik +⊗ P (v) +jk . +(F1) +Because Pik = i�γiγk and the sum over k is independent of Cuv, we can factor out each γ(u) +k +⊗ γ(v) +k +and rewrite the +Hamiltonian term as as +Huv = − +� +i,j∈[n] +[Cuv]ij +Ä +�γ(u) +i +⊗ �γ(v) +j +ä +Ñ +� +k∈[n] +γ(u) +k +⊗ γ(v) +k +é +. +(F2) + +40 +The operator � +k∈[n] γ(u) +k +⊗ γ(v) +k +is invariant to any orthogonal transformation V ∈ O(n) which acts identically on +both vertices: +U⊗2 +(In,V ) +Ñ +� +k∈[n] +γ(u) +k +⊗ γ(v) +k +é +(U⊗2 +(In,V ))† = +� +k∈[n] +Ñ +� +ℓ∈[n] +Vkℓγ(u) +ℓ +é +⊗ +Ñ +� +ℓ′∈[n] +Vkℓ′γ(v) +ℓ′ +é += +� +ℓ,ℓ′∈[n] +Ñ +� +k∈[n] +[V T]ℓkVkℓ′ +é +γ(u) +ℓ +⊗ γ(v) +ℓ′ += +� +ℓ,ℓ′∈[n] +δℓℓ′γ(u) +ℓ +⊗ γ(v) +ℓ′ += +� +ℓ∈[n] +γ(u) +ℓ +⊗ γ(v) +ℓ +. +(F3) +Because U(In,V ) acts trivially on all �γi, it follows that +U⊗2 +(In,V )Huv(U⊗2 +(In,V ))† = Huv +(F4) +for each (u, v). Finally, this symmetry can be straightforwardly extended to all m vertices: +U⊗m +(In,V )H(U⊗m +(In,V ))† = H. +(F5) +Now we investigate some consequences of this continuous symmetry. The following lemma is particularly important, +as it necessitates the use of the one-body perturbation ζH1 to break this symmetry when preparing of eigenstates of +H. +Lemma F.1. Let |ψ⟩ be a nondegenerate eigenstate of H. Then for each single-vertex marginal σv := tr¬v |ψ⟩⟨ψ|, +v ∈ [m], we have +Q(σv) = 0. +(F6) +Proof. Consider the expansion of its density matrix |ψ⟩⟨ψ| in the Majorana operator basis, up to the relevant one-body +expectation values: +|ψ⟩⟨ψ| = 1 +dm +Ñ +11⊗m + +� +v∈[m] +� +i,j∈[n] +[Q(σv)]iji�γ(v) +i +γ(v) +j ++ · · · +é +, +(F7) +where we recall that [Q(σv)]ij = ⟨ψ|i�γ(v) +i +γ(v) +j +|ψ⟩. Due to the symmetry [Eq. (F5)], for every V ∈ O(n) the state +|ψ(V )⟩ = U⊗m +(In,V )|ψ⟩ is also an eigenvector of H with the same eigenvalue. The one-body expectation values of |ψ⟩ are +therefore transformed as +U⊗m +(In,V ) +Ñ +� +v∈[m] +� +i,j∈[n] +[Q(σv)]iji�γ(v) +i +γ(v) +j +é +(U⊗m +(In,V ))† = +� +v∈[m] +� +i,j∈[n] +[Q(σv)]iji +� +i′∈[n] +Vii′�γ(v) +i′ γ(v) +j += +� +v∈[m] +� +i′,j∈[n] +[V TQ(σv)]i′ji�γ(v) +i′ γ(v) +j +. +(F8) +Now suppose that |ψ⟩ is nondegenerate. Then we have that |ψ⟩⟨ψ| = |ψ(V )⟩⟨ψ(V )| for all V ∈ O(n), and in particular +we can take the Haar integral over O(n) of this identity: +� +O(n) +dµ(V )|ψ(V )⟩⟨ψ(V )| = +� +O(n) +dµ(V )|ψ⟩⟨ψ| = |ψ⟩⟨ψ|, +(F9) +where µ is the normalized Haar measure satisfying µ(O(n)) = 1. Because the Haar integral over linear functions +vanishes, i.e., +� +O(n) dµ(V ) Vij = 0 [68], it follows that +� +O(n) +dµ(V ) V TQ(σv) = 0. +(F10) + +41 +Furthermore, because i�γ(v) +i +γ(v) +j +are linearly independent (as elements of an operator basis), the equality |ψ⟩⟨ψ| = +� +O(n) dµ(V )|ψ(V )⟩⟨ψ(V )| implies that +Q(σv) = +� +O(n) +dµ(V ) V TQ(σv) = 0 +(F11) +for all v ∈ [m]. +Appendix G: The Pin group from quantum circuits +In the main text we showed that each x ∈ Pin(n) corresponds to the eigenstates of a family of free-fermion Hamilto- +nians, which are (pure) fermionic Gaussian states. Here we provide an alternative perspective of this correspondence, +using quantum circuits which prepare such states. +Recall that every x ∈ Pin(n) can be written as +x = u1 · · · uk +(G1) +for some k ≤ n, where we may expand each uj ∈ Sn−1 in the standard basis as +uj = +� +i∈[n] +v(j) +i ei +(G2) +for some unit vector v(j) ∈ Rn. The product of these unit vectors can be expressed using the right-multiplication +operator ρuj acting on the identity element, +x = e∅x += e∅u1 · · · uk += (ρuk · · · ρu1)(e∅) +(G3) +On the other hand, consider the so-called Clifford loader [71], a circuit primitive defined (in our notation) as +Γ(v) = +� +i∈[n] +viγi +(G4) +for any unit vector v ∈ Rn. It is straightforward to check that this operator is Hermitian and unitary, and Ref. [71] +provides an explicit circuit constructions based on two-qubit Givens rotation primitives.14 Using the relation γi = ρiα +and acting this circuit on the vacuum state |0n⟩ ≡ |e∅⟩, we see that the state +|x⟩ = Γ(v(k)) · · · Γ(v(1))|0n⟩ += (ρukα · · · ρu1α)|e∅⟩ += (−1)( +k +2)(ρuk · · · ρu1)|e∅⟩ +(G5) +indeed is equivalent to x ∈ Pin(n), up to a global sign (recall that α2 = 11 and α|e∅⟩ = |e∅⟩). In other words, the +Clifford loader is precisely the quantum-circuit representation of generators of the Pin group. +It is worth noting that in Ref. [71] they construct “subspace states” from this composition of Clifford loaders. In +the language of fermions, subspace states are Slater determinants: free-fermion states with fixed particle number. +Preparing Slater determinants in this fashion requires that the unit vectors v(1), . . . , v(k) be linearly independent (and +thus, without loss of generality, they can be made orthonormal while preserving the subspace that they span, hence +the alternative name). However, the definition of the Pin group demands all possible unit vectors in such products, +not just those which are linearly independent. Indeed, one can see that if the state |x⟩ is a Slater determinant, then +its trace is an integer, as +tr[Q(x)] = ⟨x| +� +i∈[n] +i�γiγi|x⟩ = ⟨x|(nI2n − 2N)|x⟩ = n − 2k ∈ {−n, . . . , n}, +(G6) +where N = � +i∈[n] a† +iai is the total number operator. Clearly not all orthogonal matrices have integer trace, so Slater +determinants are insufficient to cover all of Pin(n). To reach the remaining elements, we note that if the unit vectors +are linearly dependent, then one can show that the |x⟩ = Γ(v(k)) · · · Γ(v(1))|0n⟩ does not have fixed particle number, +so ⟨x|N|x⟩ is not necessarily an integer. +14 Givens rotations themselves are representations of fermionic Gaussian transformations acting on two modes at a time. + +42 +Appendix H: Measurement schemes +In this section we comment on the efficient schemes available for measuring the relevant expectation values. This +is important even in the context of a phase-estimation approach, as one needs to obtain the values of the decision +variables to perform the rounding procedure. +1. +Tomography of edge marginals +To measure the energy (for variational approaches) or to perform edge rounding, we require the expectation values +of the two-body observables +Γ(u,v) +ij += +� +k∈[n] +P (u) +ik +⊗ P (v) +jk , +G = O(n), +(H1) +�Γ(u,v) +ij += +� +k∈[n] +�P (u) +ik +⊗ �P (v) +jk , +G = SO(n), +(H2) +for each (u, v) ∈ E and i, j ∈ [n]. When considering G = O(n), because each P (u) +ik +⊗ P (v) +jk +is a fermionic two-body +operator, we can straightforwardly apply the partial tomography schemes developed for local fermionic systems, such +as Majorana swap networks [60] or classical shadows [61]. +In either case, the measurement circuits required are +fermionic Gaussian unitaries and the sample complexity is O(N 2/ϵ2), where N = n|V | is the total number of qubits +and ϵ > 0 is the desired estimation precision of each expectation value. +2. +Tomography of vertex marginals +The vertex-rounding procedure requires the expectation values of only single-qudit observables P (v) +ij +or �P (v) +ij +on each +vertex v ∈ V , i, j ∈ [n]. In this case the observables being measured commute across vertices, so it suffices to talk +about the tomography of a single vertex, as the same process can be executed in parallel across all vertices. Again, +because these operators are fermionic one-body observables, the same fermionic partial tomography technology [60, 61] +can be applied here, incurring a sampling cost of O(n/ϵ2). In fact, further constant-factor savings can be achieved in +the one-body setting by using the measurement scheme introduced in Ref. [72]. This scheme requires only particle- +conserving fermionic Gaussian unitaries, which can be compiled with only half the depth of the more general Gaussian +unitaries required of the previous two methods. Note that each operator Pij is of the form of either XX or Y Y when +|i − j| = 1, and so they correspond precisely to the observables measured to reconstruct the real part of the fermionic +one-body reduced density matrix [72]. +3. +Estimating observables via gradient method +Ref. [62] introduces a quantum algorithm for estimating a large collection of (generically noncommuting) M ob- +servables {Oj | j ∈ [M]} to precision ϵ by encoding their expectation values into the gradient of a function. This +function is implemented as a quantum circuit which prepares the state of interest and applies � +O( +√ +M/ϵ) gates of +the form c-e−iθOj, controlled on O(M log(1/ϵ)) ancilla qubits. Finally, using the algorithm of Ref. [73] for gradient +estimation, one calls this circuit � +O( +√ +M/ϵ) times to estimate the encoded expectation values (the notation � +O(·) sup- +presses polylogarithmic factors). Although this approach demands additional qubits and more complicated circuitry, +it has the striking advantage of a quadratically improved scaling in the number of state preparations with respect to +estimation error ϵ, compared to the refinement of sampling error in tomographic approaches. In our context, we have +either M = n3|E| or M = n2|V | observables of interest (satisfying a technical requirement of having their spectral +norms bounded by 1), corresponding to the measurement of edge or vertex terms respectively. The gates required are +then simply controlled Pauli rotations. + diff --git a/QtAzT4oBgHgl3EQfz_6d/content/tmp_files/load_file.txt b/QtAzT4oBgHgl3EQfz_6d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6219c367b4032b074980f33dab8ed746a2ce8e7a --- /dev/null +++ b/QtAzT4oBgHgl3EQfz_6d/content/tmp_files/load_file.txt @@ -0,0 +1,1623 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf,len=1622 +page_content='Quantum relaxation for quadratic programs over orthogonal matrices Andrew Zhao1, 2, ∗ and Nicholas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Rubin1, † 1Google Quantum AI, San Francisco, CA 94105, USA 2Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87106, USA (Dated: January 6, 2023) Quadratic programming over the (special) orthogonal group encompasses a broad class of opti- mization problems such as group synchronization, point-set registration, and simultaneous localiza- tion and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a natural generalization of quadratic combinatorial optimization where, instead of binary decision variables, one optimizes over orthogonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this work, we establish an embed- ding of this class of LNCG problems over the orthogonal group onto a quantum Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This embedding is accomplished by identifying orthogonal matrices with their double cover (Pin and Spin group) elements, which we represent as quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We connect this construction to the theory of free fermions, which provides a physical interpretation of the derived LNCG Hamiltonian as a two-body interacting-fermion model due to the quadratic nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Determin- ing extremal states of this Hamiltonian provides an outer approximation to the original problem, analogous to classical relaxations of the problem via semidefinite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When optimizing over the special orthogonal group, our quantum relaxation naturally obeys additional, powerful constraints based on the convex hull of rotation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The classical size of this convex-hull rep- resentation is exponential in matrix dimension, whereas the quantum representation requires only a linear number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, to project the relaxed solution into the feasible space, we em- ploy rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Through numerical experiments we provide evidence that this quantum relaxation can produce high-quality approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Problem statement 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Applications 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Group synchronization 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Generalized orthogonal Procrustes problem 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Summary 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum Hamiltonian relaxation 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum rounding 8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum formalism for optimization over orthogonal matrices 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Qubit representation of the Clifford algebra 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The quadratic mapping as quantum expectation values 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Fermionic representation of the construction 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notation 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Linear optimization as free-fermion models 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Mixed states and the convex hull 14 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum relaxation for the quadratic problem 14 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Rounding algorithms 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Approximation ratio for rounding the classical SDP 17 ∗ azhao@unm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='edu † nickrubin@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='01778v1 [quant-ph] 4 Jan 2023 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum Gram matrix 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' conv SO(n)-based rounding 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Vertex-marginal rounding 19 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Numerical experiments 19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Exact eigenvectors 20 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quasi-adiabatic state preparation 21 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Discussion and future work 23 Acknowledgments 24 References 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Clifford algebras and the orthogonal group 26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Convex hull of orthogonal matrices and quantum states 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' PSD lift of conv O(n) and conv SO(n) 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Relation to conv SO(n)-based semidefinite relaxation 29 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Details for working in the even-parity subspace 33 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Small n examples 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The Ising model from the O(1) setting 35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The projected operators of the SO(3) setting 35 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Classical approximation ratio for SO(n) 35 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' LNCG Hamiltonian symmetries 39 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The Pin group from quantum circuits 41 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Measurement schemes 42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Tomography of edge marginals 42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Tomography of vertex marginals 42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Estimating observables via gradient method 42 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' INTRODUCTION Finding computational tasks where a quantum computer could have a large speedup is a primary driver for the field of quantum algorithm development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While some examples of quantum advantage are known, such as quantum simulation [1, 2], prime number factoring [3], and unstructured search [4], generally speaking computational advantages for industrially relevant calculations are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Specifically in the field of optimization, which has attracted a large amount of attention from quantum algorithms researchers due to the ubiquity and relevance of the computational problems, substantial quantum speedups, even on model problems, are difficult to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This difficulty is in part because it is not obvious a priori how the unique features of quantum mechanics—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', entanglement, unitarity, and interference—can be leveraged towards a computational advantage [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this work we take steps toward understanding how to apply quantum computers to optimization problems by demonstrating that the class of optimization problems involving rotation matrices as decision variables has a natural quantum formulation and efficient embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Examples of such problems include the joint alignment of points in Euclidean space by isometries, which has applications within the contexts of structural biology via cryogenic electron microscopy (cryo-EM) [7, 8] and NMR spectroscopy [9], computer vision [10, 11], robotics [12, 13], and sensor network localization [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The central difficulty in solving these problems is twofold: first, the set of orthogonal transformations O(n) is nonconvex, making the optimization landscape challenging to navigate in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Second, the objectives of these problems are quadratic in the decision variables, making them examples of quadratic programming under orthogonality constraints [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this paper we specifically focus on the problem considered by Bandeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [16], which is a special case of the real little noncommutative Grothendieck (LNCG) problem [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While significant progress has been made in classical algorithms development for finding approximate solutions, for example by semidefinite relaxations [16, 18–21], guaranteeing high-quality solutions remains difficult in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This paper 3 therefore provides a quantum formulation of the optimization problem, as a first step in exploring the potential use of a quantum computer to obtain more accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The difficulty of the LNCG problem becomes even more pronounced when restricting the decision variables to the group of rotation matrices SO(n) [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One promising approach to resolving this issue is through the convex relaxation of the problem, studied by Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' They identified that the convex hull of rotation matrices, conv SO(n), is precisely the feasible region of a semidefinite program (SDP) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore, standard semidefinite relaxations of the quadratic optimization problem can be straightforwardly augmented with this convex- hull description as an additional constraint [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' They prove that when the problem is defined over particular types of graphs, this enhanced SDP is exact, and for more general instances of the problem they numerically demonstrate that it yields significantly higher-quality approximations than the basic SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The use of this convex hull has since been explored in related optimization contexts [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notably however, the semidefinite description of conv SO(n) is exponentially large in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Roughly speaking, this reflects the complexity of linearizing a nonlinear determinant constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One such representation is the so-called positive-semidefinite (PSD) lift of conv SO(n), which is defined through linear functionals on the trace-1, PSD matrices of size 2n−1 × 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One may immediately recognize this description as the set of density operators on n − 1 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this paper we investigate this statement in detail and make a number of connections between the optimization of orthogonal/rotation matrices and the optimization of quantum states, namely fermionic states in second quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The upshot is that these connections provide us with a relaxation of the quadratic program into a quantum Hamiltonian problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Although this relaxation admits solutions (quantum states) which lie outside the feasible space of the original problem, we show that it retains much of the important orthogonal-group structure due to this natural embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The notion of quantum relaxations have been previously considered in the context of combinatorial optimization (such as the Max- Cut problem), wherein quantum rounding protocols were proposed to return binary decision variables from the relaxed quantum state [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In a similar spirit, in this paper we consider rounding protocols which return orthogonal/rotation matrices from our quantum relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Within the broader context of quantum information theory, our work here also provides an alternative perspective to relaxations of quantum Hamiltonian problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' There is a growing interest in classical methods for approximating quantum many-body problems based on SDP relaxations [29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In that context, rounding procedures are more difficult to formulate because the space of quantum states is exponentially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For instance, the algorithm may only round to a subset of quantum states with efficient classical descriptions such as product states [29–31, 33, 36] or low-entanglement states [32, 34], effectively restricting the approximation from representing the true ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Nonetheless, these algorithms can still obtain meaningful approximation ratios of the optimal energy, indicating that such states can at least capture some qualitative properties of the generically entangled ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Our quantum relaxation can be viewed as working in the opposite direction: we construct a many-body Hamiltonian where the optimal solution to the underlying classical quadratic program is essentially a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore, we propose preparing an approximation to the ground state of the Hamiltonian,1 which is then rounded to the nearest product state corresponding to the original classical solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is not unlike quantum approaches to binary optimization such as quantum annealing or the quantum approximate optimization algorithm [6, 39–42], which explore a state space outside the classical feasible region before projectively measuring, or rounding, the quantum state to binary decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We furthermore provide numerical evidence that the physical qualitative similarity between optimal product and entangled states may translate into quantitative accuracy for the classical optimization problem, in a context beyond discrete combinatorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, we remark that Grothendieck-type problems and inequalities have a considerable historical connection to quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Tsirelson [43] employed Clifford algebras to reformulate the commutative Grothendieck inequality into a statement about classical XOR games with entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Regev and Vidick [44] later introduced the notion of quantum XOR games, which they studied through the generalization of such ideas to noncommutative Grothendieck inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The mathematical work of Haagerup and Itoh [45] studied Grothendieck-type inequalities as the norms of operators on C∗-algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' their analysis makes prominent use of canonical anticommutation relation algebras over fermionic Fock spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quadratic programming with orthogonality constraints has also been applied for classical approximation algorithms for quantum many-body problems, for instance by Bravyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recasting noncom- mutative Grothendieck problems into a quantum Hamiltonian problem may therefore provide new insights into these connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The rest of this paper is organized as follows: Section II provides a formal description of the optimization problem that we study in this paper and reviews known complexity results of related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Section III we describe two well-known applications of the problem: the group synchronization problem and the generalized orthogonal Procrustes problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Section IV provides a summary of our quantum relaxation which embeds the optimization problem into a 1 While the physical problem typically considers the ground-state problem, this paper takes the convention of maximizing objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 4 Hamiltonian, and two accompanying rounding protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Section V we derive an embedding of orthogonal matrices into quantum states via the Pin and Spin groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We elaborate on the connection to fermionic theories and provide a quantum perspective on the convex hull of the orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' From this embedding, Section VI then establishes the quantum Hamiltonian relaxation of the quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Section VII describes both classical and quantum rounding protocols for relaxations of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notably, for the classical SDP we derive an approximation ratio for SO(n) rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, in Section VIII we demonstrate numerical experiments on random instances of the group synchronization problem for SO(3) on three-regular graphs and report the performance of various classical and quantum rounding protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For our simulations of the quantum relaxation, we consider two classes of quantum states: maximal eigenstates of the Hamiltonian and quasi-adiabatically evolved states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We close in Section IX with a discussion on future lines of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' PROBLEM STATEMENT In this paper we consider the class of little noncommutative Grothendieck (LNCG) problems over the orthogonal group, as studied previously by Bandeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='2 Let (V, E) be an undirected graph with m = |V | vertices and edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For integer n ≥ 1, let C ∈ Rmn×mn which for notation we partition into n × n blocks as C = � �� C11 · · C1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Cm1 · · · Cmm � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (1) The quadratic program we wish to solve is of the form max R1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=',Rm∈G � (u,v)∈E ⟨Cuv, RuRT v ⟩, (2) where G is either the orthogonal group O(n) := {R ∈ Rn×n | RTR = In}, (3) or the special orthogonal group SO(n) := {R ∈ Rn×n | RTR = In, det R = 1} (4) on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Here, ⟨A, B⟩ = tr(ATB) denotes the Frobenius inner product on the space of real matrices and In is the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that when G = O(1) = {±1}, Problem (2) reduces to combinatorial optimization of the form max x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=',xm∈{±1} � (u,v)∈E Cuvxuxv, (5) where now C ∈ Rm×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is sometimes referred to as the commutative instance of the little Grothendieck problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Problem (2) can therefore be viewed as a natural generalization of quadratic binary optimization to the noncommu- tative matrix setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We now comment on the known hardness results of these optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The commutative problem (5) is already NP-hard in general, as can be seen by the fact that the Max-Cut problem can be expressed in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In particular, Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [46] proved that, assuming the Unique Games conjecture, it is NP-hard to approximate the optimal Max-Cut solution to better than a fraction of (2/π) minθ∈[0,π] θ 1−cos θ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This value coincides with the approximation ratio achieved by the celebrated Goemans–Williamson (GW) algorithm for rounding the semidefinite relaxation of the problem [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' More generally, consider the fully connected graph and let C ⪰ 0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Nesterov [48] showed that GW rounding guarantees an approximation ratio of 2/π ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='636 in this setting, which Alon and Naor [49] showed matches the integrality gap of the semidefinite program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Khot and Naor [50] later demonstrated that this approximation ratio is also Unique-Games-hard to exceed, and finally Bri¨et et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [17] strengthened this result to be unconditionally NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For the noncommutative problem (2) that we are interested in, less is known about its hardness of approximability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, it is a subclass of more general optimization problems for which some results are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The most general 2 The authors also consider the complex-valued problem over the unitary group, which is outside the scope of this present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 5 instance is the “big” noncommutative Grothendieck problem, for which Naor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [20] provided a rounding procedure of its semidefinite relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Their algorithm achieves an approximation ratio of at least 1/2 √ 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='353 in the real- valued setting, and 1/2 in the complex-valued setting (wherein optimization is over the unitary group instead of the orthogonal group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This 1/2 result was later shown to be tight by Bri¨et et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [17] for both the real- and complex- valued settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' in fact, they show that this is the NP-hardness threshold of a special case of the problem, called the little noncommutative Grothendieck problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 However, the threshold for Problem (2), which is an special case of LNCG, is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Algorithmically, Bandeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [16] demonstrated constant approximation ratios for Problem (2) when C ⪰ 0 and G = O(n) or U(n) via an (n×n)-dimensional generalization of GW rounding, along with matching integrality gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' These approximation ratios exceed 1/2, indicating that this subclass is quantitatively less difficult than the general instance of the LNCG problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Although the optimization of rotation matrices is of central importance to many applications, we are unaware of any general approximation ratio guarantees for the G = SO(n) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' APPLICATIONS Before describing our quantum relaxation, here we motivate the practical interest in Problem (2) by briefly discussing some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Throughout, let G = O(n) or SO(n) and (V, E) be a graph as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Group synchronization The group synchronization problem over orthogonal transformations has applications in a variety of disciplines, including structural biology, robotics, and wireless networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For example, in structural biology the problem appears as part of the cryogenic electron microscopy (cryo-EM) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' There, one uses electron microscopy on cryogenically frozen samples of a molecular structure to obtain a collection of noisy images of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The images are noisy due to an inherently low signal-to-noise ratio, and furthermore they feature the structure in different, unknown orientations (represented by rotation matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One approach to solving the group synchronization problem yields best-fit estimates for these orientations via least-squares minimization [51], from which one can produce a model of the desired 3D structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='4 See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [8] for a further overview, and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [11] for a survey of other applications of group synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The formal problem description is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To each vertex v ∈ [m] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , m} we assign an unknown but fixed element gv ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' An interaction between each pair of vertices connected by an edge (u, v) ∈ E is modeled as guv = gugT v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, measurements of the interactions are typically corrupted by some form of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For instance, one may consider an additive noise model of the form Cuv = guv + σWuv, where σ ≥ 0 characterizes the strength of the noise and each Wuv ∈ Rn×n has independently, normally distributed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We would like to recover each gv given only access to the matrices Cuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore, as a proxy to the recovery problem one may cast the solution as the least-squares minimizer min R∈Gm � (u,v)∈E ∥Cuv − RuRT v ∥2 F , (6) where ∥A∥F = � ⟨A, A⟩ is the Frobenius norm and we employ the notation R ≡ (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It is straightforward to see that the minimzer of this problem is equivalent to the maximizer of max R∈Gm � (u,v)∈E ⟨Cuv, RuRT v ⟩, (7) which is precisely in the form of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Generalized orthogonal Procrustes problem Procrustes analysis has applications in fields such as shape and image recognition, as well as sensory analysis and market research on n-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this problem, one has a collection of point clouds, each representing for 3 See Section 6 of Bri¨et et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [17] for the precise relation between the big and little NCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 4 Note that other loss functions are also considered in the literature, which may not necessarily have a reformulation as Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 6 instance the important features of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One wishes to determine how similar these images are to each other collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is achieved by simultaneously fitting each pair of point clouds to each other, allowing for arbitrary orthogonal transformations on each cloud to best align the individual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [52] for a comprehensive review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Consider m sets of K points in Rn, Sv = {xv,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , xv,K} ⊂ Rn for each v ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We wish to find an orthogonal transformation Rv ∈ G for each Sv that best aligns all sets of points simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' That is, for each k ∈ [K] and u, v ∈ [m] we wish to minimize the Euclidean distance ∥RT uxu,k − RT v xv,k∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Taking least-squares minimization as our objective, we seek to solve min R∈Gm � u,v∈[m] � k∈[K] ∥RT uxu,k − RT v xv,k∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (8) From the relation between the vector 2-norm and matrix Frobenius norm, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (8) can be formulated as max R∈Gm � u,v∈[m] ⟨Cuv, RuRT v ⟩, (9) where each Cuv ∈ Rn×n is defined as Cuv = � k∈[K] xu,kxT v,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (10) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' SUMMARY We now provide a high-level overview of the main contributions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We provide summary cartoon in Figure 1, depicting the quantum embedding of the problem and the quantum rounding protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Let (V, E) be a graph where we label the vertices by V = [m], and denote the objective function of Problem (2) by f(R) := � (u,v)∈E ⟨Cuv, RuRT v ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (11) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum Hamiltonian relaxation First, consider the setting in which R = (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm) ∈ O(n)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We embed this problem into a Hamiltonian by placing n qubits on each vertex v ∈ [m], resulting in a total Hilbert space H⊗m 2n of mn qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Define the n-qubit Pauli operators Pij := � � � � � −XiZi+1 · · · Zj−1Xj i < j, Zi i = j, −YjZj+1 · · · Zi−1Yi i > j, (12) where Zi := I⊗(i−1) 2 ⊗ Z ⊗ I⊗(n−i) 2 (similarly for Xi, Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The Hamiltonian H := � (u,v)∈E � i,j∈[n] [Cuv]ij � k∈[n] P (u) ik ⊗ P (v) jk (13) defines our quantum relaxation of the objective f over O(n)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The notation A(v) denotes the operator A acting only on the Hilbert space of vertex v, and we overload this notation to indicate either the n-qubit operator or mn-qubit operator acting trivially on the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When the context is clear we typically omit writing the trivial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For optimization over SO(n)m, we consider instead the (n − 1)-qubit Pauli operators �Pij := Π0PijΠT 0 , (14) Π0 = 1 √ 2 Ä ⟨+| ⊗ I⊗(n−1) 2 + ⟨−| ⊗ Z⊗(n−1)ä , (15) 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A cartoon description of the quantum and classical encodings of an LNCG problem, followed by classical and quantum rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Left) The description of the problem that we consider, which is described by a graph ([m], E) and n × n matrices Cuv for each edge (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We wish to assign elements of O(n) or SO(n) to each vertex such that the quadratic form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (11) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Center top) The description of the standard classical relaxation of the LNCG problem as an mn × mn PSD matrix M ⪰ 0, which is optimized using a semidefinite program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Right top) The classical rounding procedure, which returns a collection of orthogonal matrices from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Center bottom) A description of our quantum formulation of the LNCG problem as a two-body interacting Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' On each vertex we place a d-dimensional Hilbert space, and the Hamiltonian corresponds to interaction terms Huv on the edges (L(H) is the set of linear operators on a Hilbert space H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The classical solution of the LNCG problem lies in a subset of the full Hilbert space containing separable Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Right bottom) Our proposed quantum rounding protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One protocol requires knowledge of the two-body reduced density matrices across edges, while the other uses the one-body reduced density matrices on each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' where Π0 : H2n → H2n−1 represents the projection onto the even-parity subspace of H2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The construction of the relaxed Hamiltonian for SO(n) is then analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (13): �H := � u,v∈E � i,j∈[n] [Cuv]ij � k∈[n] �P (u) ik ⊗ �P (v) jk , (16) where now the relaxed quantum problem is defined over m(n − 1) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' These Hamiltonians serve as relaxations to Problem (2) in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First, we show that for every R ∈ O(n), there is an n-qubit state |φ(R)⟩ which is the maximum eigenstate of F(R) = � i,j∈[n] RijPij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (17) In particular, F(R) is a free-fermion Hamiltonian, so |φ(R)⟩ is a fermionic Gaussian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If R ∈ SO(n), then furthermore |φ(R)⟩ is an even-parity state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', ⟨φ(R)|Z⊗n|φ(R)⟩ = 1, so it is only supported on a subspace of dimension 2n−1 (the image of Π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This correspondence establishes a reformulation of the classical optimization problem as a constrained Hamiltonian problem: max R∈Gm f(R) = max |ψ⟩=� v∈[m] |φ(Rv)⟩ Rv∈G ⟨ψ|H|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (18) Dropping these constraints on |ψ⟩ implies the inequalities max R∈O(n)m f(R) ≤ max ρ∈D(H⊗m 2n ) tr(Hρ), (19) max R∈SO(n)m f(R) ≤ max ρ∈D(H⊗m 2n−1) tr( �Hρ), (20) where D(H) denotes the set of density operators on a Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This establishes the quantum Hamiltonian relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 8 Algorithm 1: conv G-based rounding of edge marginals Data: Quantum state ρ ∈ D(H⊗m d ) over a graph of m vertices, each with local Hilbert space of dimension d = 2n if G = O(n), or d = 2n−1 if G = SO(n) Result: Orthogonal matrices on each vertex, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm ∈ G M ← Imn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' for u ̸= v ∈ [m] do for (i, j) ∈ [n]2 do if G = O(n) then [Muv]ij ← 1 n tr(Γ(u,v) ij ρ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' else if G = SO(n) then [Muv]ij ← 1 n tr(�Γ(u,v) ij ρ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' end end end for v ∈ [m] do Rv ← arg minY ∈G ∥Y − M1v∥F ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' end Algorithm 2: Rounding vertex marginals Data: Quantum state ρ ∈ D(H⊗m d ) over a graph of m vertices, each with local Hilbert space of dimension d = 2n if G = O(n), or d = 2n−1 if G = SO(n) Result: Orthogonal matrices on each vertex, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm ∈ G for v ∈ [m] do Qv ← 0 ∈ Rn×n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' for (i, j) ∈ [n]2 do if G = O(n) then [Qv]ij ← tr(P (v) ij ρ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' else if G = SO(n) then [Qv]ij ← tr( �P (v) ij ρ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' end end end for v ∈ [m] do Rv ← arg minY ∈G ∥Y − Qv∥F ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' end B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum rounding In order to recover orthogonal matrices from a relaxed quantum solution ρ, we propose two rounding procedures, summarized in Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' These rounding procedures operate on local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', single- or two-vertex observables) expectation values of ρ stored in classical memory, which can be efficiently estimated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', by partial state tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Algorithm 1 is inspired by constructing a quantum analogue of the PSD variable appearing in semidefinite relax- ations to Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Consider the mn × mn matrix of expectation values M := � ���� In T12 · · T1m T21 In · · T2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Tm1 Tm2 · · · In � ���� , (21) 9 where the off-diagonal blocks are defined as Tuv := 1 n � �� tr(Γ(u,v) 11 ρ) · · · tr(Γ(u,v) 1n ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' tr(Γ(u,v) n1 ρ) · · · tr(Γ(u,v) nn ρ) � �� = T T vu, (22) Γ(u,v) ij := � k∈[n] P (u) ik ⊗ P (v) jk (23) when G = O(n), and we replace the operators Pij with �Pij when G = SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We show that M satisfies the following properties for all states ρ: M ⪰ 0, (24) Muv ∈ conv G ∀u, v ∈ [m], (25) where conv G is the convex hull of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Thus when G = SO(n), M obeys the same constraints as the conv SO(n)- based semidefinite relaxation proposed by Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, whereas the classical representation of the conv SO(n) constraints requires at least matrices of size 2n−1 × 2n−1 for each edge, our quantum state automatically satisfies these constraints (using only n − 1 qubits per vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Algorithm 2 uses the single-vertex information tr(P (v) ij ρ) of ρ, as opposed to the two-vertex information tr(Γ(u,v) ij ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We consider this rounding procedure due to the fact that, if ρ is a pure Gaussian state satisfying the constraint of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (18), then the matrix of expectation values Qv := � �� tr(P (v) 11 ρ) · · · tr(P (v) 1n ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' tr(P (v) n1 ρ) · · · tr(P (v) nn ρ) � �� (26) lies in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' On the other hand, for arbitrary density matrices we have the relaxation Qv ∈ conv O(n), and again when we replace P (v) ij with �P (v) ij then Qv ∈ conv SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Both rounding procedures use the standard projection of the matrices X ∈ conv G (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', the matrices Tuv or Qv measured from the quantum state) to some R ∈ G by finding the nearest (special) orthogonal matrix according to Frobenius-norm distance: R = arg min Y ∈G ∥X − Y ∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (27) This can be solved efficiently as a classical postprocessing step, essentially by computing the singular value decom- position of X = UΣV T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When G = O(n), the solution is R = UV T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When G = SO(n), we instead use the so-called special singular value decomposition of X = U �Σ�V T, where �Σ = ΣJ and �V = V J, with J being the diagonal matrix J = ïIn−1 0 0 det(UV T) ò , (28) assuming that the singular values σi(X) are in descending order, σ1(X) ≥ · · · ≥ σn(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (27) is R = U �V T ∈ SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' QUANTUM FORMALISM FOR OPTIMIZATION OVER ORTHOGONAL MATRICES Our key insight into encoding orthogonal matrices into quantum states comes from the construction of the orthogonal group from a Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We review this mathematical construction in Appendix A and only discuss the main aspects here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The Clifford algebra Cl(n) is a 2n-dimensional real vector space equipped with an inner product and multiplication operation satisfying the anticommutation relation eiej + ejei = −2δij11, (29) where e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , en is an orthonormal basis for Rn and 11 is the multiplicative identity of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The orthogonal group is then realized through a quadratic map Q : Cl(n) → Rn×n and the identification of a subgroup Pin(n) ⊂ Cl(n) 10 such that Q(Pin(n)) = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notably, the elements of Pin(n) have unit norm (with respect to the inner product on Cl(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The special orthogonal group, meanwhile, is constructed by considering only the even-parity elements of Cl(n), denoted by Cl0(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The group Spin(n) = Pin(n) ∩ Cl0(n) then yields Q(Spin(n)) = SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because the Clifford algebra Cl(n) is a 2n-dimensional vector space, we observe that it can be identified with a Hilbert space of n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='5 In this section we explore this connection in detail, showing how to represent orthogonal matrices as quantum states and how the mapping Q acts as a linear functional on those states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Qubit representation of the Clifford algebra First we describe the canonical isomorphism between Cl(n) and H2n := (R2)⊗n as Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We denote the standard basis of Cl(n) by {eI := ei1 · · · eik | I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , ik} ⊆ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' By convention we assume that the elements of I are ordered as i1 < · · · < ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Each basis element eI maps onto to a computational basis state |b⟩, where b = b1 · · · bn ∈ {0, 1}n, via the correspondence eI ≡ � i∈[n] |bi⟩, where bi = ® 1 if i ∈ I, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (30) The inner products on both spaces coincide since this associates one orthonormal basis to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This correspondence also naturally equates the grade |I| of the Clifford algebra with the Hamming weight |b| of the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The notion of parity, |I|mod2 = |b|mod2, is therefore preserved, so Cl0(n) corresponds to the subspace of H2n with even Hamming weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To represent the multiplication of algebra elements in this Hilbert space, we use the fact that left- and right- multiplication are linear automorphisms on Cl(n), which are denoted by λx(y) = xy, ρx(y) = yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (31) The action of the algebra can therefore be represented on H2n as linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We shall use the matrix represen- tation provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24], as it precisely coincides with the n-qubit computational basis described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because of linearity, it suffices to specify left- and right-multiplication by the generators ei, which are the operators λi ≡ Z⊗(i−1) ⊗ (−iY ) ⊗ I⊗(n−i) 2 , (32) ρi ≡ I⊗(i−1) 2 ⊗ (−iY ) ⊗ Z⊗(n−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (33) It will also be useful to write down the parity automorphism α(eI) = (−1)|I|eI under this matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' As the notion of parity is equivalent between Cl(n) and H2n, α is simply the n-qubit parity operator, α ≡ Z⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (34) It will also be useful to represent the subspace Cl0(n) explicitly as an (n − 1)-qubit Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is achieved by the projection from Cl(n) to Cl0(n), expressed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24] as the 2n−1 × 2n matrix Π0 := 1 √ 2 Ä ⟨+| ⊗ I⊗(n−1) 2 + ⟨−| ⊗ Z⊗(n−1)ä .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (35) It is straightforward to check that Π0|b⟩ = 0 if |b| mod 2 = 1, and that its image is a 2n−1-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The quadratic mapping as quantum expectation values The quadratic map Q : Cl(n) → Rn×n is defined as Q(x)(v) := πRn(α(x)vx) ∀x ∈ Cl(n), v ∈ Rn, (36) where πRn is the projector from Cl(n) to Rn and the conjugation operation is eI = (−1)|I|eik · · · ei1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This map associates Clifford algebra elements with orthogonal matrices via the relations Q(Pin(n)) = O(n) and Q(Spin(n)) = 5 In fact, n rebits suffice since Cl(n) is a real vector space, but to keep the presentation straightforward we will not make such a distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 11 SO(n) (see Appendix A for a review of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In the standard basis of Rn, the linear map Q(x) : Rn → Rn has the matrix elements [Q(x)]ij = ⟨ei, Q(x)(ej)⟩ = ⟨ei, α(x)ejx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (37) Using the linear maps λi, ρj of left- and right-multiplication by ei, ej, as well as the conjugation identity ⟨x, yz⟩ = ⟨xz, y⟩ in the Clifford algebra, these matrix elements of Q(x) can be rearranged as [Q(x)]ij = ⟨ei, α(x)ejx⟩ = ⟨eix, α(x)ej⟩ = ⟨λi(x), ρj(α(x))⟩ = ⟨x, λ† i(ρj(α(x)))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (38) We now transfer this expression to the quantum representation developed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First, define the following n-qubit Pauli operators as the composition of the linear maps appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (38): Pij := λ† iρjα = � � � � � −I⊗(i−1) 2 ⊗ X ⊗ Z⊗(j−i−1) ⊗ X ⊗ I⊗(n−j) 2 i < j, I⊗(i−1) 2 ⊗ Z ⊗ I⊗(n−i) 2 i = j, −I⊗(j−1) 2 ⊗ Y ⊗ Z⊗(i−j−1) ⊗ Y ⊗ I⊗(n−i) 2 i > j, (39) where the expressions in terms of Pauli matrices follow from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (32) to (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then we may rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (38) as [Q(x)]ij = ⟨x|Pij|x⟩, (40) where |x⟩ ∈ H2n is the quantum state identified with x ∈ Cl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence, the matrix elements of Q(x) ∈ Rn×n possess the interpretation as expectation values of a collection of n2 Pauli observables {Pij}i,j∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Furthermore, recall that Q(x) ∈ O(n) if and only if x ∈ Pin(n), and Q(x) ∈ SO(n) if and only if x ∈ Spin(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because Spin(n) = Pin(n)∩Cl0(n), one can work in the even-parity sector directly by projecting the operators as �Pij := Π0PijΠT 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (41) These are (n − 1)-qubit Pauli operators, and we provide explicit expressions in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When necessary, we may specify another map �Q : Cl0(n) → Rn×n, [ �Q(x)]ij := ⟨x| �Pij|x⟩, (42) for which �Q(Spin(n)) = SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In general, these double covers are only a subset of the unit sphere in Hd (d = 2n or 2n−1), so not all quantum states mapped by Q yield orthogonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Section V C we characterize the elements of Pin(n) and Spin(n) as a class of well-studied quantum states, namely, pure fermionic Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Fermionic representation of the construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notation First we establish some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A system of n fermionic modes, described by the creation operators a† 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , a† n, can be equivalently represented by the 2n Majorana operators γi = ai + a† i, (43) �γi = −i(ai − a† i), (44) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' These operators form a representation for the Clifford algebra Cl(2n), as they satisfy6 γiγj + γjγi = �γi�γj + �γj�γi = 2δij11, (45) γi�γj + �γjγi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (46) 6 Note that we adopt the physicist’s convention here, which takes the generators to be Hermitian, as opposed to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A1) wherein they square to −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 12 The Jordan–Wigner mapping allows us to identify this fermionic system with an n-qubit system via the relations γi = Z⊗(i−1) ⊗ X ⊗ I⊗(n−i) 2 , (47) �γi = Z⊗(i−1) ⊗ Y ⊗ I⊗(n−i) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (48) We will work with the two representations interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A central tool for describing noninteracting fermions is the Bogoliubov transformation γ �→ Oγ, where O ∈ O(2n) and γ := ��γ1 · · · �γn γ1 · · · γn �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (49) This transformation is achieved by fermionic Gaussian unitaries, which are equivalent to matchgate circuits on qubits under the Jordan–Wigner mapping [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In particular, we will make use of a subgroup of such unitaries corre- sponding to O(n) × O(n) ⊂ O(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For any U, V ∈ O(n), let U(U,V ) be the fermionic Gaussian unitary with the adjoint action U(U,V )�γiU† (U,V ) = � j∈[n] Uij�γj, (50) U(U,V )γiU† (U,V ) = � j∈[n] Vijγj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (51) In contrast to arbitrary O(2n) transformations, these unitaries do not mix between the γ- and �γ-type Majorana operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Linear optimization as free-fermion models Applying the representation of Majorana operators under the Jordan–Wigner transformation, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (47) and (48), to the Clifford algebra automorphisms, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (32) to (34), we see that λ† i = i�γi and ρjα = γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore the Pauli operators Pij defining the quadratic map Q are equivalent to fermionic one-body operators, Pij = i�γiγj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (52) Consider now a linear objective function ℓ(X) := ⟨C, X⟩ for some fixed C ∈ Rn×n, which we wish to optimize over O(n): max X∈O(n) ℓ(X) = max X∈O(n)⟨C, X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (53) Because we require X ∈ O(n), it is equivalent to search over all x ∈ Pin(n) through Q: max X∈O(n)⟨C, X⟩ = max x∈Pin(n)⟨C, Q(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (54) Writing out the matrix elements explicitly, we see that the objective takes the form ℓ(X) = � i,j∈[n] Cij[Q(x)]ij = � i,j∈[n] Cij⟨x|Pij|x⟩ = ⟨x|F(C)|x⟩, (55) where we have defined the noninteracting fermionic Hamiltonian F(C) := � i,j∈[n] CijPij = i � i,j∈[n] Cij�γiγj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (56) The linear optimization problem is therefore equivalent to solving a free-fermion model, max X∈O(n)⟨C, X⟩ = max x∈Pin(n)⟨x|F(C)|x⟩, (57) 13 the eigenvectors of which are fermionic Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' As such, this problem can be solved efficiently by a classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In fact, the known classical algorithm for solving the optimization problem is exactly the same as that used for diagonalizing F(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We now review the standard method to diagonalize F(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Consider the singular value decomposition of C = UΣV T, which is computable in time O(n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This decomposition immediately reveals the diagonal form of the Hamiltonian: F(C) = i � i,j∈[n] [UΣV T]ij�γiγj = i � k∈[n] σk(C) Ñ � i∈[n] [U T]ki�γi éÑ � j∈[n] [V T]kjγj é = U† (U,V ) Ñ � k∈[n] σk(C)i�γkγk é U(U,V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (58) Because i�γkγk = Zk, it follows that the eigenvectors of F(C) are the fermionic Gaussian states |φb⟩ = U† (U,V )|b⟩, b ∈ {0, 1}n, (59) with eigenvalues Eb = � k∈[n] (−1)bkσk(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (60) The maximum energy is E0n = tr Σ since all singular values are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The corresponding eigenstate |φ0n⟩ is the maximizer of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (57), so it corresponds to an element φ0n ∈ Pin(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It is straightforward to see this by recognizing that [Q(φ0n)]ij = ⟨φ0n|i�γiγj|φ0n⟩ = [UV T]ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The fact that Q(φ0n) ∈ O(n) if and only if φ0n ∈ Pin(n) concludes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Indeed, the standard classical algorithm [56] for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (53) uses precisely the same decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' From the cyclic property of the trace and the fact that O(n) is a group, we have max X∈O(n)⟨UΣV T, X⟩ = max X′∈O(n)⟨Σ, X′⟩, (61) where we have employed the change of variables X′ := U TXV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Again, because Σ has only nonnegative entries, ⟨Σ, X′⟩ achieves its maximum, tr Σ, when X′ = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This implies that the optimal solution is X = UV T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that this problem is equivalent to minimizing the Frobenius-norm distance, since arg min X∈O(n) ∥C − X∥2 F = arg min X∈O(n) �∥C∥2 F + ∥X∥2 F − 2⟨C, X⟩� = arg max X∈O(n) ⟨C, X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (62) Now suppose we wish to optimize ℓ over SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this setting, one instead computes X = U �V T from the special singular value decomposition of C = U �Σ�V T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This ensures that det(X) = 1 while maximizing ℓ(X), as only the smallest singular value σn(C) has its sign potentially flipped to guarantee the positive determinant constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This sign flip also has a direct analogue within the free-fermion perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recall that the determinant of Q(x) ∈ O(n) is given by the parity of x ∈ Pin(n), or equivalently the parity of the state |x⟩ in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note also that all fermionic states are eigenstates of the parity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To optimize over SO(n), we therefore seek the maximal eigenstate |φb⟩ of F(C) which has even parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If ⟨φ0n|Z⊗n|φ0n⟩ = 1 then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' On the other hand, if ⟨φ0n|Z⊗n|φ0n⟩ = −1 then we need to flip only a single bit in 0n to reach an even-parity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The smallest change in energy by such a flip is achieved from changing the occupation of the mode corresponding to the smallest singular value of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The resulting eigenstate |φ0n−11⟩ is then the even-parity state with the largest energy, E0n−11 = tr Σ − 2σn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, we point out that all elements of Pin(n) are free-fermion states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To see this, observe that C is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We can therefore construct the family of Hamiltonians {F(C) | C ∈ O(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Clearly, the maximum ⟨C, X⟩ = n within this family is achieved when X = C, each of which corresponds to a fermionic Gaussian state |φ⟩ satisfying F(C)|φ⟩ = n|φ⟩ and Q(φ) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We note that this argument generalizes the mathematical one presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24], which only considered the eigenvectors lying in Spin(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Mixed states and the convex hull First we review descriptions of the convex hull of orthogonal and rotation matrices, the latter of which was charac- terized by Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The convex hull of O(n) is the set of all matrices with operator norm bounded by 1, conv O(n) = �X ∈ Rn×n | σ1(X) ≤ 1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (63) On the other hand, the convex hull of SO(n) has a more complicated description in terms of special singular values: conv SO(n) = � � �X ∈ Rn×n ����� � i∈[n]\\I �σi(X) − � i∈I �σi(X) ≤ n − 2 ∀I ⊆ [n], |I| odd � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (64) Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24] establish that this convex body is a spectrahedron, the feasible region of a semidefinite program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The representation that we will be interested in is called a PSD lift: conv SO(n) = � � � � � � �� ⟨ �P11, ρ⟩ · · · ⟨ �P1n, ρ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' ⟨ �Pn1, ρ⟩ · · · ⟨ �Pnn, ρ⟩ � �� ����� ρ ⪰ 0, tr ρ = 1 � � � � � , (65) where the 2n−1 × 2n−1 matrices �Pij are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='7 Recall that the density operators on a Hilbert space H form the convex hull of its pure states: D(H) := conv{|ψ⟩⟨ψ| | |ψ⟩ ∈ H, ⟨ψ|ψ⟩ = 1} = {ρ ∈ L(H) | ρ ⪰ 0, tr ρ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (66) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (65) one immediately recognizes that the PSD lift of conv SO(n) corresponds to D(H2n−1), where we recognize that H2n−1 ∼= Cl0(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Furthermore, the projection of the lift is achieved through the convexification of the map Q : Cl(n) → Rn×n, where the fact that Q is quadratic in Cl(n) translates to being linear in D(Cl(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Specifically, by a slight abuse of notation we shall extend the definition of Q to act on density operators as Q(ρ) = � µ pµQ(xµ), where ρ = � µ pµ|xµ⟩⟨xµ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (67) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (65) is the statement that Q(D(Cl0(n))) = conv SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Appendix B 1 we show that this statement straightforwardly generalizes for Q(D(Cl(n))) = conv O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We prove this using the fermionic representation developed in Section V C 2, and furthermore use these techniques to provide an alternative derivation for the PSD lift of conv SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The core of our argument is showing that the singular-value conditions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (63) and (64) translate into bounds on the largest eigenvalue of corresponding n-qubit observables: σi(X) = tr(i�γiγiρ) ≤ 1, (68) � i∈[n]\\I �σi(X) − � i∈I �σi(X) = tr � �ρ0 Ñ � i∈[n]\\I i�γiγi − � i∈I i�γiγi é� � ≤ n − 2, (69) where ρ ∈ D(Cl(n)) and ρ0 ∈ D(Cl0(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The physical interpretation here is that not all pure quantum states map onto to orthogonal or rotation matrices (which is clear from the fact that fermionic Gaussian states are only a subset of quantum states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, all density operators do map onto to their convex hulls, and the distinction between conv O(n) and conv SO(n) can be automatically specified by restricting the support of ρ to the even-parity subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' QUANTUM RELAXATION FOR THE QUADRATIC PROBLEM We now arrive at the primary problem of interest in this work, the little noncommutative Grothendieck problem over the (special) orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While the linear problem of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (53) can be solved classically in polynomial 7 Technically, Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24] use the definition �Pij = −Π0λiρjΠT 0 because they employ the standard adjoint representation, which differs from our use of the twisted adjoint representation which includes the parity automorphism α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However since α(x) = x for all x ∈ Cl0(n), both definitions of �Pij coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 15 time, quadratic programs are considerably more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Here, we use the quantum formalism of the Pin and Spin groups developed above to construct a quantum relaxation of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then in Section VII we describe rounding procedures to recover a collection of orthogonal matrices from the quantum solution to this relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recall the description of the input to Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Let (V, E) be a graph, and associate to each edge (u, v) ∈ E a matrix Cuv ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We label the vertices as V = [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We wish to maximize the objective f(R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm) = � (u,v)∈E ⟨Cuv, RuRT v ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (70) over (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm) ∈ O(n)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First, expand this expression in terms of matrix elements: � (u,v)∈E ⟨Cuv, RuRT v ⟩ = � (u,v)∈E � i,j∈[n] [Cuv]ij � k∈[n] [Ru]ik[RT v ]kj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (71) From the quadratic mapping Q : Cl(n) → Rn×n, we know that for each R ∈ G there exists some φ ∈ Pin(n) such that Rij = ⟨φ|Pij|φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence we can express the matrix product as [Ru]ik[RT v ]kj = ⟨φu|Pik|φu⟩⟨φv|Pjk|φv⟩ = ⟨φu ⊗ φv|Pik ⊗ Pjk|φu ⊗ φv⟩, (72) which is now the expectation value of a 2n-qubit Pauli operator with respect to a product state of two Gaussian states |φu⟩, |φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To extend this over the entire graph, we define a Hilbert space of m registers of n qubits each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For each edge (u, v) ∈ E we introduce the Hamiltonian terms Huv := � i,j∈[n] [Cuv]ijΓ(u,v) ij , (73) where Γ(u,v) ij := Ñ � k∈[n] P (u) ik ⊗ P (v) jk é � w∈V \\{u,v} I(w) 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (74) To simplify notation, we shall omit the trivial support � w∈V \\{u,v} I(w) 2n when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The problem is now reformulated as optimizing the mn-qubit Hamiltonian H := � (u,v)∈E Huv = � (u,v)∈E � i,j∈[n] [Cuv]ij � k∈[n] P (u) ik ⊗ P (v) jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (75) The exact LNCG problem over O(n) then corresponds to max R∈Gm f(R) = max |ψ⟩∈H⊗m 2n ⟨ψ|H|ψ⟩ subject to � � � � � ⟨ψ|ψ⟩ = 1, |ψ⟩ = � v∈[m] |φv⟩, |φv⟩ = U(Rv,In)|0n⟩, Rv ∈ O(n) ∀v ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (76) The hardness of this problem is therefore related to finding the optimal separable state for local Hamiltonians, which is NP-hard in general [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Dropping these constraints on the state provides a relaxation of the problem, since max |ψ⟩∈H⊗m 2n , ⟨ψ|ψ⟩=1 ⟨ψ|H|ψ⟩ ≥ max R∈Gm f(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (77) We point out here that the Hamiltonian terms Huv can be interpreted as two-body fermionic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that there is an important distinction between two-body fermionic operators (Clifford-algebra products of four Majorana operators) and two-body qudit operators (tensor products of two qudit Pauli operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recall that Pij = i�γiγj is one-body in the fermionic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While the operators P (u) ik ⊗ P (v) jk appear to mix both notions, here they in fact coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To see this, we consider a global algebra of Majorana operators {γi+(v−1)n, �γi+(v−1)n | i ∈ [n], v ∈ [m]} 16 acting on a Hilbert space of mn fermionic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While it is not true that the local single-mode Majorana operators map onto the global single-mode operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', γ(v) i � w∈V \\{v} I(w) 2n ̸= γi+(v−1)n, (78) the local two-mode Majorana operators in fact do correspond to global two-mode operators: �γ(v) i γ(v) j � w∈V \\{v} I(w) 2n = �γi+(v−1)nγj+(v−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (79) Thus, taking the tensor product of two local two-mode Majorana operators on different vertices is equivalent to taking the product of two global two-mode Majorana operators: �γ(u) i γ(u) j ⊗ �γ(v) k γ(v) l � w∈V \\{u,v} I(w) 2n = �γi+(u−1)nγj+(u−1)n�γk+(v−1)nγl+(v−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (80) Therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (75) can be equivalently expressed as a Hamiltonian with two-body fermionic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, when we wish to optimize over (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm) ∈ SO(n)m, it is straightforward to see that we can simply replace the terms Pij with �Pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Defining �Γ(u,v) ij := Ñ � k∈[n] �P (u) ik ⊗ �P (v) jk é � w∈V \\{u,v} I(w) 2n−1, (81) �Huv := � i,j∈[n] [Cuv]ij�Γ(u,v) ij , (82) the quantum relaxation for the SO(n) problem is given by the m(n − 1)-qubit Hamiltonian �H := � (u,v)∈E �Huv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (83) VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' ROUNDING ALGORITHMS Optimizing the energy of a local Hamiltonian is a well-studied problem, both from the perspective of quantum and classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this section we will assume that such an algorithm has been used to produce the state ρ ∈ D(H⊗m d ) which (approximately) maximizes the energy tr(Hρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We wish to round this state into the feasible space, namely the set of product states of Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We do so by rounding the expectation values of ρ appropriately, such that we return some valid approximation R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this section we propose two approaches to perform this quantum rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The first uses insight from the fact that our quantum relaxation is equivalent to a classical semidefinite relaxation with additional constraints based on the convex hull of the orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is approach is particularly advantageous when optimizing over SO(n), as conv SO(n) has a matrix representation exponential in n (its PSD lift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To build the semidefinite variable from the quantum state, we require measurements of the expectation values of the two-vertex operators Γ(u,v) ij = � k∈[n] P (u) ik ⊗ P (v) jk for each pair of vertices (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We refer this procedure as conv SO(n)-based rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 Our second rounding protocol uses the expectation values of P (v) ij of each vertex v directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this case, rather than expectation values of two-vertex operators as before, we only require the information of single-vertex marginals ρv := tr¬v(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore we call this approach vertex-marginal rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If ρ is produced by a deterministic classical algorithm, then the relevant expectation values can be exactly computed (to machine precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However if the state is produced by a randomized algorithm, or is otherwise prepared by a quantum computer, then we can only estimate the expectation values to within statistical error by some form of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In the quantum setting, this can be achieved either by partial state tomography [60, 61] or a more sophisticated measurement protocol [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='9 See Appendix H for further comments on this quantum measurement aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The rounding algorithms then operate entirely as classical postprocessing after estimating the necessary expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 8 This rounding can also be applied to the optimization problem over O(n) as well, but we are particularly interested in the conv SO(n) constraints due to their exponentially large classical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 9 For the present discussion we do not consider the effects of finite sampling, although we expect that rounding is fairly robust to such errors since it will always return a solution in the feasible space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Approximation ratio for rounding the classical SDP Before describing our quantum rounding protocols, we first review classical relaxations and rounding procedures for Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The standard semidefinite relaxation can be expressed as the SDP max M∈Rmn×mn⟨C, M⟩ subject to ® M ⪰ 0, Mvv = In ∀v ∈ [m], (84) where C ∈ Rmn×mn is the matrix with n × n blocks Cuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If an additional nonconvex constraint rank(M) = n is imposed, then the solution would be exact: M = RRT = � ���� In R1RT 2 · · R1RT m R2RT 1 In · · R2RT m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' RmRT 1 RmRT 2 · · · In � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (85) Problem (84) is there a relaxation of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, the solution M ∈ Rmn×mn is still PSD, so it can be decomposed as M = XXT, where X = � �� X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Xm � �� , Xv ∈ Rn×mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (86) The rounding algorithm of Bandeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [16] then computes, for each v ∈ [m], Ov = P(XvZ) := arg min Y ∈O(n) ∥Y − XvZ∥F , (87) where Z is an mn × n Gaussian random matrix whose entries are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' from N(0, 1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When optimizing over G = O(n), this rounded solution guarantees (in expectation) an approximation ratio of α2 O(n) = E � � 1 n � i∈[n] σi(Z1) � � 2 , (88) where Z1 is a random n × n matrix with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' entries from N(0, 1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Appendix E we extend the argument used to obtain this result for the optimization problem over G = SO(n), and we show a corresponding approximation ratio of α2 SO(n) = E � � 1 n � i∈[n−1] σi(Z1) � � 2 (89) where the only change to the rounding algorithm is that we project to the nearest SO(n) element via �P, which is defined as �P(X) := arg min Y ∈SO(n) ∥Y − X∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (90) Note that singular values are nonnegative, and in particular we show that E[σn(Z1)] > 0 for all finite n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence it follows that α2 SO(n) < α2 O(n), which provides evidence for the claim that solving for rotations is generally a more difficult problem (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [23, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3] for a brief discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For small values of n, the numerical values of these approximation ratios are (computed using Mathematica): α2 O(2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6564, α2 SO(2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3927, (91) α2 O(3) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6704, α2 SO(3) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='5476, (92) α2 O(4) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6795, α2 SO(4) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (93) 18 In Appendix E we provide an integral expression for αSO(n) which can be evaluated for arbitrary n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For the problem over SO(n), Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [21] propose augmenting this SDP by adding the constraints that each block of M lies in conv SO(n): max M∈Rmn×mn⟨C, M⟩ subject to � � � � � M ⪰ 0, Mvv = In ∀v ∈ [m], Muv ∈ conv SO(n) ∀u, v ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (94) Although they do not prove approximation guarantees for this enhanced SDP, they first show that, if one reintroduces the rank constraint on M, then the convex constraint Muv ∈ conv SO(n) in fact suffices to guarantee the much stronger condition Muv ∈ SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then, when dropping the rank constraint (but leaving the conv SO(n) constraint) they show that the relaxed problem is still exact over certain types of graphs, such as tree graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Finally, they provide numerical evidence that even when the relaxation is not exact, it returns substantially more accurate approximations than the standard SDP (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quantum Gram matrix Analogous to the classical SDP solution M, we can form a matrix M ∈ Rmn×mn from the expectation values of ρ as M := � ���� In T12 · · T1m T21 In · · T2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Tm1 Tm2 · · · In � ���� , (95) where Tuv := 1 n � �� tr(Γ(u,v) 11 ρ) · · · tr(Γ(u,v) 1n ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' tr(Γ(u,v) n1 ρ) · · · tr(Γ(u,v) nn ρ) � �� (96) and Tvu = T T uv for u < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Just as ⟨C, M⟩ gives the relaxed objective value (up to rescaling and constant shifts), here we have that ⟨C, M⟩ = 2 n tr(Hρ) + tr(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Appendix B we show that for any quantum state, M satisfies the following properties: � � � � � M ⪰ 0, Mvv = In ∀v ∈ [m], Muv ∈ conv O(n) ∀u, v ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (97) Furthermore, we show that when ρ is supported only on the even subspace of each single-vertex Hilbert space (or equivalently, if we replace Γ(u,v) ij with �Γ(u,v) ij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (96)), then 1 n � �� tr(�Γ(u,v) 11 ρ) · · · tr(�Γ(u,v) 1n ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' tr(�Γ(u,v) n1 ρ) · · · tr(�Γ(u,v) nn ρ) � �� ∈ conv SO(n) ∀ρ ∈ D(H⊗m 2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (98) Therefore when optimizing the relaxed Hamiltonian �H for the SO(n) setting, we are guaranteed to automatically satisfy the conv SO(n) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' conv SO(n)-based rounding Given the construction of the M from quantum expectation values, we proceed to round the Gram matrix as in the classical SDP with conv SO(n) constraints [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This consists of computing the matrices Rv = �P(M1v), (99) 19 where the projection to SO(n) can be efficiently computed from the special singular value decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', �P(X) = U �V T (100) (recall Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (28)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Our choice of rounding using the first n × mn “row” of M amounts to fixing R1 = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We note that the same rounding procedure can naturally be applied to the O(n) setting as well, replacing �P with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Vertex-marginal rounding The single-vertex marginals are obtained by tracing out the qudits associated to all but one vertex v ∈ [m], ρv = tr¬v(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (101) As ρv ∈ D(H⊗m d ), from Section V C 3 we have that Q(ρv) ∈ conv G, where we linearly extend the definition of Q to Q(ρv) := � �� tr(P (v) 11 ρ) · · · tr(P (v) 1n ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' tr(P (v) n1 ρ) · · · tr(P (v) nn ρ) � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (102) The rounding scheme we propose here then projects Q(ρv) to G using either P or �P: Rv = arg min Y ∈G ∥Y − Q(ρv)∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (103) We point out that the relaxed Hamiltonian only has two-vertex terms which we seek to maximize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Appendix F we show that H commutes with U⊗m (In,V ) for all V ∈ O(n), which we further show implies that H may possess eigenstates whose single-vertex marginals obey Q(σv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This indicates that there may exist eigenstates of H whose single-vertex marginals yield no information, despite the fact that their two-vertex marginals are nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In our numerical studies, we observe that breaking this symmetry resolves this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We accomplish this by including small perturbative one-body terms which correspond to the trace of Q(σv): H1 = � v∈[m] � i∈[n] P (v) ii , (104) Note that this trace quantity is importantly invariant with respect to the choice of basis for Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We then augment the objective Hamiltonian with H1, defining H′(ζ) := H + ζH1 (105) where ζ > 0 is a small regularizing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While this one-body perturbation does not correspond to any terms in the original quadratic objective function, any arbitrarily small ζ > 0 suffices to break the O(n) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Furthermore, the rounding procedure always guarantees that the solution is projected back into the feasible space Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When G = SO(n) we define �H′(ζ) analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' NUMERICAL EXPERIMENTS To explore the potential of our quantum relaxation and rounding procedures, we performed numerical experiments on randomly generated instances of the group synchronization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because the Hilbert-space dimension grows exponentially in both m and n, our classical simulations here are limited to small problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, optimizing over rotations in R3 (requiring only two qubits per vertex) is highly relevant to many practical applications, so here we focus on the problem of SO(3) group synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For example, this problem appears in the context of cryo-EM as described in Section III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To model the problem, we generated random instances by selecting random three-regular graphs ([m], E), uniformly randomly sampling m rotations g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , gm ∈ SO(n), and then constructing Cuv = gugT v + σWuv for each (u, v) ∈ E, where the Gaussian noise matrix Wuv ∈ Rn×n has i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' elements drawn from N(0, 1) and σ ≥ 0 represents the strength (standard deviation) of this noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While the classical conv SO(n)-based SDP is not guaranteed to find the optimal solution, the problems studied here were selected for such that this enhanced SDP in fact does solve the exact problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We verify this property 20 4 6 8 10 Number of vertices m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='0 Approximation ratio Quantum (CR) Quantum (VR) Classical SDP 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 Noise strength 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='0 Approximation ratio Quantum (CR) Quantum (VR) Classical SDP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Approximation ratios for solutions obtained from rounding the maximum eigenvector of the relaxed Hamiltonian � H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Violin plots show the distribution of approximation ratios over 50 randomly generated instance, and with the median being indicated by the center marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' CR refers to rounding according to the conv SO(n)-based scheme (Section VII C), while VR denotes the vertex-marginal rounding scheme (Section VII D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The classical SDP solution was rounded by the standard randomized algorithm [16], and we report the best solution over 1000 rounding trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Left) Varying the number of vertices m in the graph (random 3-regular graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that the number of qubits required here is 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Right) Varying the noise strength parameter σ which defines the problem via Cuv = gugT v + σWuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' by confirming that rank(M) = n before rounding on each problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In this way we are able to calculate an approximation ratio for the other methods (as it is not clear how to solve for the globally optimal solution in general, even with an exponential-time classical algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The methods compared here include our quantum relaxation with conv SO(n)-based rounding (denoted CR), vertex-marginal rounding (VR), and the classical SDP (without conv SO(n) constraints but using the �P projection to guarantee that the rounded solutions are elements of SO(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' When using the vertex-rounding method, we employ �H′(ζ) as the objective Hamiltonian with ζ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Exact eigenvectors First, we consider the solution obtained by rounding the maximum eigenvector of �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Although the hardness of preparing such a state is equivalent that of the ground-state problem, this nonetheless provides us with a benchmark for the ultimate approximation quality of our quantum relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In Figure 2 we plot the approximation ratio of the rounded quantum states and compare to that of the classical SDP on the same problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Each violin plot was constructed from the results of 50 random instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The results here demonstrate that, while the approximation quality of the classical SDP quickly falls off with larger graph sizes, our rounded quantum solutions maintain high approximation ratios, at least for the problem sizes probed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Notably, the conv SO(n)-based rounding on the quantum state is significantly more powerful and consistent than the vertex-marginal rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This feature is not unexpected since, as discussed in Section VII D, we are maximizing an objective Hamiltonian with only two-body terms, whereas the single-vertex rounding uses strictly one-body expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Furthermore, as demonstrated in previous works [21, 25] the conv SO(n) constraints are powerful in practice, and so we expect that the quantum rounding protocol which makes use of this structure enjoys the same advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Meanwhile, when varying the noise parameter σ, we observe that all methods are fairly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In particular, the conv SO(n)-based rounding only shows an appreciable decrease in approximation quality when the noise is considerable (note that σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='5 ≈ 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 is a relatively large amount of noise, since gugT v is an orthogonal matrix and therefore has matrix elements bounded in magnitude by 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 21 0 1 2 3 4 Total evolution time T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='6 Approximation ratio Optimal objective value Maximum quantum relaxed value Quantum (CR) Quantum (VR) Classical SDP Quantum relaxed value FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Demonstration of a typical instance of adiabatic state preparation for preparing relaxed quantum solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The initial state is the product of Gaussian states corresponding to the rounded solution of the classical SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' As the total evolution time T increases, the evolution becomes more adiabatic, indicated by the convergence of the relaxed value to the maximum eigenvalue (in units of the original problem’s optimal value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The rounded solutions of course can never exceed the original problem’s optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Quasi-adiabatic state preparation Because it may be unrealistic to prepare the maximum eigenvector of �H, here we consider preparing states using ideas from adiabatic quantum computation [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Specifically, we wish to demonstrate that states whose relaxed energy may be far from the maximum eigenvalue can still provide high-quality approximations after rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If this is the case then we do not need to prepare very close approximations to the maximum eigenstate of �H, so the rigorous conditions of adiabatic state preparation may not be required in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence we consider “quasi- adiabatic” state preparation, wherein we explore how time-evolution speeds far from the adiabatic limit may still return high-quality approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Our numerical experiments here provide a preliminary investigation into this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For simplicity of the demonstration, we consider a linear annealing schedule according to the time-dependent Hamiltonian H(t) = Å 1 − t T ã Hi + t T Hf, (106) which prepares the state |ψ(T)⟩ = T exp Ç −i � T 0 dt H(t) å |ψ(0)⟩ (107) for some T > 0, where T is the time-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The final Hamiltonian Hf is the desired objective LNCG Hamiltonian, Hf = �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (108) The initial Hamiltonian Hi is the parent Hamiltonian of the initial state, which we choose to be the approximation obtained from the classical SDP, as it can be obtained classically in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Let R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Rm ∈ SO(n) be the SDP solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Our initial state is then the product of Gaussian states |ψ(0)⟩ = � v∈[m] |φ(Rv)⟩, (109) where each |φ(Rv)⟩ is the maximum eigenvector of the free-fermion Hamiltonian F(Rv) = i � i,j∈[n] [Rv]ij�γiγj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (110) 22 4 6 8 10 Number of vertices m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='75 Approximation ratio Optimal objective value Maximum quantum relaxed value Quantum (CR) Quantum (VR) Classical SDP Quantum relaxed value 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='3 Noise strength 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='75 Approximation ratio Maximum quantum relaxed value Quantum (CR) Quantum (VR) Classical SDP Quantum relaxed value FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Approximation ratios for solutions obtained from rounding the “adiabatically” evolved state |ψ(T)⟩ with fixed T = 1 for all m, σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Problem instances and visualization is the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We also include the maximum eigenvalue of the relaxed Hamiltonian and the energy of the prepared unrounded state, to demonstrate how far |ψ(T)⟩ is from the exact maximum eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Left) Varying the number of vertices m in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that the number of qubits required here is 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (Right) Varying the noise strength parameter σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore the initial Hamiltonian Hi is a sum of such free-fermion Hamiltonians (here we include the even-subspace projection since we are working with SO(n)): Hi = � v∈[m] Π0F(Rv)ΠT 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (111) As a Gaussian state, |φ(Rv)⟩ can be prepared exactly from a quantum circuit of O(n2) gates [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that since we are working directly in the even subspace of n − 1 qubits here, this n-qubit circuit must be projected appropriately using Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We discuss how to perform this circuit recompilation in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We comment that this choice of initial state is that of a mean-field state for non-number-preserving fermionic systems, for instance as obtained from Hartree–Fock–Bogoliubov theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Suitably, the final Hamiltonian we evolve into is non-number-preserving two-body fermionic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In adiabatic state preparation, the total evolution time T controls how close the final state |ψ(T)⟩ is to the max- imum eigenstate10 of the final Hamiltonian Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One metric of closeness is how the energy of the prepared state, ⟨ψ(T)|Hf|ψ(T)⟩, compares to the maximum eigenvalue of Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' On the other hand, as a relaxation, this maximum energy is already larger than the optimal objective value of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We showcase this in Figure 3, using one random problem instance as a demonstrative (typical) example on a graph of m = 6 vertices (12 qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For each total evolution time point T, we computed |ψ(T)⟩ by numerically integrating the time-dependent Schr¨odinger equation, and we plot its relaxed energy as well as its rounded objective values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For large T we approach the maximum eigenstate of �H as expected (thereby also demonstrating that the initial “mean-field” state |ψ(0)⟩ has appreciable overlap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Particularly interesting is the behavior for relatively small total evolution times T, wherein the energy of |ψ(T)⟩ is far from the maximum eigenenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Despite this, the approximation quality after rounding the state using M is nearly exact around T ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' On the other hand, the approximation quality of vertex-marginal rounding is highly inconsistent, which again we attribute to the fact that the single-vertex information is not directly seen by the final Hamiltonian Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then in Figure 4 we plot the same 50 problem instances (per graph size/noise level) as in Figure 2, but using the quasi-adibatically prepared state |ψ(T)⟩ where we have fixed T = 1 for all graph sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The classical SDP results are the same as in Figure 4, and for reference we include the energy of the unrounded quantum state and the maximum eigenvalue of the relaxed Hamiltonian (normalized with respect to the optimal objective value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Qualitatively, we observe features similar to those seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Namely, although the annealing schedule is too fast to prepare a close 10 We remind the reader that we are starting in the maximum eigenstate of the initial Hamiltonian, whereas in the physics literature, adiabatic theorems are typically stated in terms of ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Of course, the two perspectives are equivalent by simply an overall sign change (note that all Hamiltonians here are traceless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 23 approximation to the maximum eigenstate, the rounded solutions (using the conv SO(n)-based protocol) consistently have high approximation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Meanwhile, the vertex-rounded solutions are highly inconsistent, which reflects the highly fluctuating behavior seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK In this paper we have developed a quantum relaxation for a quadratic program over orthogonal and rotation ma- trices, known as an instance of the little noncommutative Grothendieck problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The embedding of the classical objective is achieved by recognizing an intimate connection between the geometric-algebra construction of the orthog- onal group and the structure of quantum mechanics, in particular the formalism of fermions in second quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' From this perspective, the determinant condition of SO(n) is succinctly captured by a simple linear property of the state—its parity—and the convex bodies conv O(n) and conv SO(n) (relevant to convex relaxations of optimization over orthogonal matrices) are completely characterized by density operators on n and n − 1 qubits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Rec- ognizing that the reduced state on each vertex therefore corresponds to an element of this convex hull, we proposed vertex-marginal rounding which classically rounds the measured one-body reduced density matrix of each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We additionally showed that these convex hulls are characterized by density operators on 2n and 2(n − 1) qubits as well, where the linear functionals defining this PSD lift are the Hamiltonian terms appearing in our quantum relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This insight enables our second proposed rounding scheme, conv G-based edge rounding, which is inspired by the fact that the a quantum Gram matrix M can be constructed from the expectation values of the quantum state which obeys the same properties as the classical SDP of Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Numerically we observe that this approach to quantum rounding is significantly more accurate and consistent than vertex rounding, and it consistently achieves larger approximation ratios than the basic SDP relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, we are severely limited by the exponential scaling of classically simulating quantum states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' further investigations would be valuable to ascertain the empirical performance of these ideas at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The primary goal of this work was to formulate the problem of orthogonal-matrix optimization into a familiar quan- tum Hamiltonian problem, and to establish the notion of a quantum relaxation for such optimization problems over continuous-valued decision variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' A clear next step is to prove nontrivial approximation ratios from our quantum relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' If such approximation ratios exceed known guarantees by classical algorithms, for example on certain types of graphs, then this would potentially provide a quantum advantage for a class of applications not previously considered in the quantum literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We have proposed one standard, realistically preparable class of states—quasi- adiabatic time evolution—but a variety of energy-optimizing ansatze exist in the literature, especially considering that the constructed Hamiltonian is an interacting-fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' From this perspective, it would also be interesting to see if a classical many-body method can produce states which round down to high-quality approximations, even heuristically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Such an approach would constitute a potential example of a quantum-inspired classical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' From a broader perspective, the quantum formalism described here may also provide new insights into the compu- tational hardness of the classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First, the NP-hard thresholds for Problem (2) are not currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, by establishing the classical problem as an instance of Gaussian product state optimization on the many- body Hamiltonian, it may be possible to import tools from quantum computational complexity to study the classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This idea also applies to the more general instances of noncommutative Grothendieck problems, max U,V ∈O(N) � i,j,k,l∈[N] TijklUijVkl, (112) where the N × N × N × N tensor T specifies the problem input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It is straightforward apply our quantum relaxation construction to this problem, yielding a 2N-qubit Hamiltonian whose terms are of the form Pij ⊗ Pkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' While Bri¨et et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [17] showed that the NP-hardness threshold of approximating this problem is 1/2, it remains an open problem to construct an algorithm which is guaranteed to achieve this approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Although we have provided new approximation ratios for the instance of Problem (2) over SO(n), it is unclear precisely how much harder the SO(n) problem is compared to the O(n) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The work by Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24] establishes a clear distinction between the representation sizes required for conv O(n) and conv SO(n), and this paper has connected this structure to properties of quantum states on n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However this does not yet establish a difference of hardness for the corresponding quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Again it would be interesting to see if the tools of quantum information theory can be used to further understand this classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For example, one might study the NP-hardness threshold of Problem (112) where instead U, V ∈ SO(N) and leverage the quantum (or equivalently, Clifford-algebraic) representation of SO(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In such a setting, the size of the problem is given by a single parameter N and so the exponentially large parametrization of conv SO(N) appears to signify a central difficulty of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We note that it is straightforward to extend our quantum relaxation to the unitary groups U(n) and SU(n), essentially by doubling the number of qubits per vertex via the inclusions U(n) ⊂ O(2n) and SU(n) ⊂ SO(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 24 However this is likely an inefficient embedding, since the n-qubit Majorana operators already form a representation of Cl(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It may therefore be possible to encode complex-valued matrices via a complexification of Q, using the same amount of quantum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It is interesting to note that Bri¨et et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [17] in fact utilize a “complex extension” of Clifford algebras when considering Problem (112) over the unitary group, although the usage is different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Ryan Babbush, Bill Huggins, Robin Kothari, Jarrod McClean, Chaithanya Rayudu, and Jun Takahashi for helpful discussions and feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' AZ thanks Akimasa Miyake for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Kerenidis and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Prakash, Quantum machine learning with subspace states, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='00054 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [72] Google AI Quantum and Collaborators, Hartree-Fock on a superconducting qubit quantum computer, Science 369, 1084 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [73] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Gily´en, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Arunachalam, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Wiebe, Optimizing quantum optimization algorithms via faster quantum gradient computation, in Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, 2019) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 1425–1444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Appendix A: Clifford algebras and the orthogonal group In this appendix we review the key components for constructing the orthogonal and special orthogonal groups from a Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Our presentation of this material broadly follows Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The Clifford algebra Cl(n) of Rn is a 2n-dimensional real vector space, equipped with an inner product ⟨·, ·⟩ : Cl(n) × Cl(n) → R and a multiplication operation satisfying the anticommutation relation eiej + ejei = −2δij11, (A1) where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , en} is an orthonormal basis of Rn and 11 is the multiplicative identity of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The basis elements ei are called the generators of the Clifford algebra, in the sense that they generate all other basis vectors of Cl(n) as eI := ei1 · · · eik, I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , ik} ⊆ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A2) By convention we order the indices i1 < · · · < ik, and the empty set corresponds to the identity, e∅ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Taking all subsets I ⊆ [n] and extending the inner product definition from Rn to Cl(n), it follows that {eI | I ⊆ [n]} is an orthonormal basis with 2n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Specifically, we can write any element x ∈ Cl(n) as x = � I⊆[n] xIeI (A3) with each xI ∈ R, and the inner product on Cl(n) is11 ⟨x, y⟩ = � I⊆[n] xIyI, (A4) 11 Equipping an inner product to the vector representation of Cl(n) elements is achieved using the fact that algebra elements square to a multiple of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 27 where y = � I⊆[n] yIeI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence Cl(n) is isomorphic as a Hilbert space to R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Now we show how to realize the orthogonal group O(n) from this algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First observe the inclusion Rn = span{ei | i ∈ [n]} ⊂ Cl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We shall identify the sphere Sn−1 ⊂ Rn as all u ∈ Rn satisfying ⟨u, u⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We then define the Pin group as all possible products of Sn−1 elements: Pin(n) := {u1 · · · uk | u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , uk ∈ Sn−1, 0 ≤ k ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A5) It is straightforward to check that this is indeed a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Each x ∈ Pin(n) is also normalized, ⟨x, x⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In fact, an equivalent definition of this group is all elements x ∈ Cl(n) satisfying xx = 11, where conjugation x is defined from the linear extension of eI := (−1)|I|eik · · · ei1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A6) The Pin group is a double cover of O(n), which can be seen from defining a quadratic map Q : Cl(n) → Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This map arises from the so-called twisted adjoint action, introduced by Atiyah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [65]:12 v �→ α(x)vx, x, v ∈ Cl(n), (A7) where the linear map α : Cl(n) → Cl(n) is the parity automorphism, defined by linearly extending α(eI) := (−1)|I|eI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A8) Then for any x ∈ Cl(n), the linear map Q(x) : Rn → Rn is defined as Q(x)(v) := πRn(α(x)vx) ∀v ∈ Rn, (A9) where πRn is the projection from Cl(n) onto Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To show that Q(Pin(n)) = O(n), it suffices to recognize that, for any u ∈ Sn−1, α(u)vu ∈ Rn is the reflection of the vector v ∈ Rn across the hyperplane normal to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To see this, first observe that uv + vu = −2⟨u, v⟩11, which follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A1) by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Then α(u)vu = uvu = (−vu − 2⟨u, v⟩11)u = v − 2⟨u, v⟩u, (A10) which is precisely the elementary reflection as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' By the Cartan–Dieudonn´e theorem, one can implement any orthogonal transformation on Rn by composing k ≤ n such reflections about arbitrary hyperplanes u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , uk [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This characterization coincides precisely with the definition of the Pin group provided in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A5), through the composition of the linear maps Q(u1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , Q(uk) on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Hence for all x ∈ Pin(n), Q(x) is an orthogonal transformation on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The double cover property follows from the fact that Q is quadratic in x, so Q(x) = Q(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' The special orthogonal group arises from the subgroup Spin(n) ⊂ Pin(n) containing only even-parity Clifford elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First observe that Cl(n) is a Z2-graded algebra: Cl(n) = Cl0(n) ⊕ Cl1(n), (A11) where Cl0(n) := span{eI | |I| even}, (A12) Cl1(n) := span{eI | |I| odd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A13) By a Z2 grading we mean that for each x ∈ Cla(n) and y ∈ Clb(n), their product xy lies in Cla+b mod 2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We say that elements in Cl0(n) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', Cl1(n)) have even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=', odd) parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' In particular, this grading implies that Cl0(n) is a subalgebra, hence its intersection with the Pin group is also a group, which defines Spin(n) := Pin(n) ∩ Cl0(n) = {u1 · · · u2k | u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' , u2k ∈ Sn−1, 0 ≤ k ≤ ⌊n/2⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (A14) Just as the Pin group double covers O(n), so does the Spin group double cover SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This is again a consequence of the Cartan–Dieudonn´e theorem, wherein all rotations on Rn can be decomposed into an even number of (at most n) arbitrary reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 12 Saunderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' [24] consider the standard adjoint action, which is sufficient for describing rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' However, the “twist” due to α is necessary to construct arbitrary orthogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 28 Appendix B: Convex hull of orthogonal matrices and quantum states Recall the following characterizations of the convex hulls: conv O(n) = �X ∈ Rn×n | σ1(X) ≤ 1� (B1) conv SO(n) = � � �X ∈ Rn×n ����� � i∈[n]\\I �σi(X) − � i∈I �σi(X) ≤ n − 2 ∀I ⊆ [n], |I| odd � � �, (B2) where {σi(X)}i∈[n] and {�σi(X}i∈[n] are the singular values and special singular values of X in descending order, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that σi(X) = �σi(X) for all i ≤ n − 1 and σn(X) = sign(det(X))σn(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' PSD lift of conv O(n) and conv SO(n) In this section we show that Q(D(Cl(n))) = conv O(n) and Q(D(Cl0(n))) = conv SO(n), using the quantum formalism described in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First we show that for all X ∈ conv O(n), there exists some ρ ∈ D(H2n) which generates X, essentially by the convex extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Every X ∈ conv O(n) can be expressed as a convex combination (� µ pµ = 1, pµ ≥ 0) of orthogonal matrices Rµ ∈ O(n): X = � µ pµRµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B3) For each Rµ there exists some xµ ∈ Pin(n) such that [Rµ]ij = ⟨xµ|Pij|xµ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore the matrix elements of X can be expressed as Xij = tr � Pij � µ pµ|xµ⟩⟨xµ| � = tr(Pijρ), (B4) where ρ := � µ pµ|xµ⟩⟨xµ| ∈ D(H2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Next we show the reverse direction, that for all ρ ∈ D(H2n), the matrix X := [tr(Pijρ)]i,j∈[n] is an element of conv O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recall that X ∈ conv O(n) if and only if σ1(X) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Therefore we take the singular value decomposition of X = UΣV T and, using Pij = i�γiγj, each singular value is equal to σk(X) = [U TXV ]kk = � i,j∈[n] Uik tr(i�γiγjρ)Vjk = tr(iU† (U,In)�γkU(U,In)U† (In,V )γkU(In,V )ρ) = tr(i�γkγkρ′), (B5) where ρ′ := U(U,V )ρU† (U,V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because i�γkγk has eigenvalues ±1, we see that σk(X) ≤ 1 for all k ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For the restriction to conv SO(n), the first argument is essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' One merely replaces Pij with �Pij, hence � µ pµ|xµ⟩⟨xµ| ∈ D(H2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For the reverse direction, we instead employ the special singular value decomposition which yields �σk(X) = tr(i�γkγkρ′), (B6) where now ρ′ := U(U,�V )ρU† (U,�V ) ∈ D(H2n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Note that we have not projected to the even subspace this time, as it is more convenient to work in the full n-qubit space when handling the Gaussian unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Instead, we will impose the constraint that ρ only has support on the even-parity subspace, so tr(Z⊗nρ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Furthermore, because the special singular value decomposition guarantees that det(U) det(�V ) = 1, U(U,�V ) is parity preserving so that tr(Z⊗nρ′) = 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Now recall that X ∈ conv SO(n) if and only if � k∈[n]\\I �σk(X) − � k∈I �σk(X) ≤ n − 2 (B7) 29 for all subsets I ⊆ [n] of odd size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' By linearity, � k∈[n]\\I �σk(X) − � k∈I �σk(X) = tr Ñ ρ′ � k∈[n] (−1)zkZk é , (B8) where z = z1 · · · zn ∈ {0, 1}n is defined as zk = 1 if k ∈ I and zk = 0 otherwise, and we have used the fact that i�γkγk = Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' It therefore suffices to examine the spectrum of Az := � k∈[n](−1)zkZk: Az|b⟩ = Ñ � k∈[n] (−1)[z⊕b]k é |b⟩, b ∈ {0, 1}n, (B9) where ⊕ denotes addition modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' As we are only interested in the subspace spanned by even-parity states, we restrict attention to the eigenvalues for which |b| mod 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because |I| is odd, so too is |z|, hence |z ⊕ b| mod 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This implies that there must be at least one term in the sum of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B9) which is negative, so it can only take integer values at most n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' This establishes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B7), hence X ∈ conv SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Relation to conv SO(n)-based semidefinite relaxation Here we provide details for our claim that the relaxed quantum solution obeys the same constraints as the classical SDP which uses the exponentially large representation of conv SO(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Recall that this relaxation can be formulated as max M∈Rmn×mn � (u,v)∈E ⟨Cuv, Muv⟩ subject to � � � � � M ⪰ 0, Mvv = In ∀v ∈ [m], Muv ∈ conv SO(n) ∀u, v ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B10) We will show that the Gram matrix M constructed from the measurements of a quantum state ρ, defined in Sec- tion VII B as [Muv]ij = � � � � � δij u = v, 1 n tr(Γ(u,v) ij ρ) u < v, [Mvu]ji u > v, (B11) obeys the constraints of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Specifically, when the marginals of ρ on each vertex are even-parity states (recall this is equivalent to replacing Γij with �Γij), we obtain the conv SO(n) condition, whereas when the parity of ρ is not fixed then Muv ∈ conv O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' First, we show that M is positive semidefinite for all quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Let M ∈ Rmn×mn be defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For all ρ ∈ D(H⊗m 2n ), M ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' We prove the statement by a sum-of-squares argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' To see where the fact of 1/n appears in the quantum definition of M above, we first construct a matrix M′ ⪰ 0 which turns out to simply be M′ = nM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' For each k ∈ [n] define the Hermitian operator Ak = � v∈V � i∈[n] c(v) i P (v) ik , (B12) where c(v) i ∈ R are arbitrary coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Consider its square, A2 k = Ñ � v∈V � i∈[n] c(v) i P (v) ik é2 = � v∈V � i,j∈[n] c(v) i c(v) j P (v) ik P (v) jk + � u,v∈V u̸=v � i,j∈[n] c(u) i c(v) j P (u) ik ⊗ P (v) jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' (B13) 30 Note that the terms with u ̸= v feature the two-vertex operators as desired, while the diagonal terms of the sum contain products of the Pauli operators acting on the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtAzT4oBgHgl3EQfz_6d/content/2301.01778v1.pdf'} +page_content=' Because Pik = i�γiγk, the diagonal terms reduce to (suppressing superscripts here) � i,j∈[n] cicjPikPjk = − � i,j∈[n] cicj�γiγk�γjγk = � i,j∈[n] cicj�γi�γj = � i∈[n] c2 i I2n + � 1≤i 0 do +10 +if ri.dist = 0 then +11 +ri.dist = ri.dist + 1 +12 +if ri.dist = 1 and ri.state = explore then +13 +wait for 2φ rounds +14 +if any settled robot visits with rj.special = 0 and rj.id is smaller than the +settled robot already present with the group then +15 +ri.state = backtrack +16 +update φ = max{φ, δ(u)} +17 +move through ri.portentered +18 +decrement ri.dist value to 0 +19 +else +20 +ri.state = explore +21 +update φ = max{φ, δ(u)} +22 +ri.portentered = (ri.portentered + 1)modδ +23 +if ri.portentered = rj.virtualparent then +24 +ri.state = backtrack +25 +move through ri.portentered +26 +else +27 +move through ri.portentered +28 +increment ri.dist to 2 +29 +if ri.dist = 2 and ri.state = explore then +30 +wait for 2φ rounds +31 +update φ = max{φ, δ(u)} +32 +if no settled robot visit the group of unsettled robots then +33 +the robot ri with lowest id settles on that node and sets +ri.parent = ri.portentered and ri.special = 1 +34 +each unsettled robot makes ri.portentered = (ri.portentered + 1)modδ +and ri.dist = 0 +35 +if ri.portentered = parent pointer of the settled robot at the current node +then +36 +ri.state = backtrack +37 +move through ri.portentered +38 +else +39 +ri.state = backtrack +40 +move through ri.portentered +41 +if ri.state = backtrack then +42 +wait for 2φ rounds +43 +decrement ri.dist by 1 +44 +ri.portentered = (ri.portentered + 1)modδ +45 +if the settled robot rj has rj.parent = −1 then +46 +if ri.portentered = 0 then +47 +the unsettled robots settle at the root +48 +else +49 +ri.state = explore +50 +move through ri.portentered +51 +else +52 +if portentered = parent or virtualparent then +53 +ri.state = backtrack +54 +move through portentered +55 +else +56 +ri.state = explore +57 +move through portentered +58 +increment ri.dist by 1 +15 + +of rj.count. Else if this meeting is done at a neighboring node of the node where +rj settled in phase 1, it changes rj.stage = 2, goes back to its original position and +waits for rL with rL.terminate = 1 to arrive there. +The above description shows that, any settled robot r in phase 2 increments +their variable r.count′ only after becoming act settled, i.e., only after the corre- +sponding round of stage 1 when r became settled and started incrementing its +r.count. Since rL works as the unsettled robot by following the algorithm of unset- +tled robots in stage 1 and its id is the largest, rL will settle in phase 2 again at the +end and will terminate. By that time, all settled robots count value match with +their count′ value, and all terminate. With this, below we provide the algorithm of +stage 2 in a formal way. +When the robot rL with rL.terminate = 1 and rL.stage = 2 reaches the root +u, it initializes the value of φ = δ(u), and begins the traversal in order to terminate +the settled robots. Similar to the algorithm described above for the unsettled group +of robots in stage 1, every time rL reaches a node, say, v, it waits for 2φ rounds +at that node. Also, it updates the value of φ with max{φ, δ(v)}. After reaching a +new node, decisions are made based on the following cases: +• When rL.dist = 1 and rL.state = explore +– If any settled robot visits with rj.act settled = 1, rj.special = 0 and +rj.id is smaller than the id of act settled robot already present with rL +then rL backtracks via rL.portentered +– if any settled robot visits with rj.act settled = 1 and rj.special = 0 +but rj.id > the id of act settled robot already present with rL or no +act settled = 1 robot visits then rL is set to explore and it moves +through the incremented value of portentered +• When rL.dist = 2 and rL.state = explore +– If no settled robot with rj.act settled = 1 visits then this indicates that +it is the original position where rL is settled and hence set rL.act settled = +1. Terminate rL. +– if any settled robot visits with rj.act settled = 1 then rL backtracks +through rL.portentered +– if there is a robot rj present at the node with rj.stage = 2 but rj.act settled = +0 then rj.parent is set to rL.portentered and rj.special = 1. +• When rL.state = backtrack +– If the reached node is the root node and there are some ports to be +explored i.e. (rL.portentered + 1)modδ ̸= 0 then rL.state is updated +to explore, else rL and the remaining settled robots are terminated at +this node. +– If the reached node is other than the root node then rL.portentered +is compared with the value of rj.parent or rj.virtualparent. If all the +16 + +ports are explored then the state is changed to backtrack otherwise rL +explores through the incremented value of rL.portentered. +The pseudo-code of the algorithm for the last settled robot rL is given as Algo- +rithm 4. The Algorithm 4 uses the Algorithm 3 as a subroutine, where Algorithm +3 is the pseudo-code of what the last settled robot rL does after reaching the root. +The robots with rj.settled = 1 and rj.act settled = 0/1 proceeds based on the +following cases. +• If rj is a settled robot with rj.settled = 1 but rj.act settled = 0, when it +meets any robot ri with ri.stage = 2 then it returns to its original position +and waits there. It now becomes aware that the termination stage has started. +Thus, it does not increment the value of rj.count. +• If rj.act settled = 1 and rj.special = 0 then it continues the back and forth +movement through all its neighbors and increments the value of rj.count′ by +1 as and when it meets rL. When rj.count becomes equal to rj.count′, it +terminates at its original position where it was settled. +• If rj.act settled = 1 and rj.special = 1 then it moves with the robot rL to +the neighboring node through the updated value of rL.portentered and waits +with rL. It increments the value of rj.count′ by 1. Now two cases arise: +– If no act settled robot visits with id lower than rj.id then rj sets +rj.virtualparent = rL.portentered +– Else move to the original settled position and set rj.special = 0 +The pseudo-code of the algorithm for each settled robot rj with rj.act settled = 1 +is given in the Algorithm 5. +The pseudo-code of the Distance-2-Dispersion with Termination is given in the +Algorithm 6. Figure 2 shows the run of stage 1 of our algorithm. +3.2 +Analysis of the Algorithm +Definition 3.1. Tree Edge: An edge (u, v) is said to be a tree edge if the group of +unsettled robots in stage 1 reaches v through (u, v) such that either the settled robot +at node u (if exists) stores the parent pointer of the node v or the minimum id robot +among the group of unsettled robots settles at v. +Remark 3.1. A settled robot ri in stage 1 stores the parent pointer for its adjacent +node u at some round t only if ri.special = 1, rj.dist = 1 for any unsettled robot +rj at u and no settled robot with id lower than ri.id visits rj during the 2φ rounds +i.e., during the waiting period. +Theorem 3.1. By the end of the Algorithm 6, there are no two robots that are +settled at adjacent nodes. +17 + +Algorithm 3: Help Termination(): Algorithm for the robot rL +1 set φ = δu, rL.state = explore and rL.portentered = 0 +2 move through rL.portentered +3 rL.dist = rL.dist + 1 +4 for phase > 0 do +5 +if rL.dist = 1 and rL.state = explore then +6 +wait for 2φ rounds +7 +if any settled robot visits with rj.act settled = 1, rj.special = 0, and rj.id < +id of act settled robot already present with rL then +8 +rL.state = backtrack +9 +update φ= max{φ, δ(u)} +10 +move through rL.portentered +11 +decrement rL.dist value to 0 +12 +if any settled robot visits with rj.act settled = 1, rj.special = 0, but rj.id > id +of act settled robot already present with rL or no rj.act settled = 1 robot +visits then +13 +rL.state = explore +14 +update φ = max{φ, δ(u)} +15 +rL.portentered = (rL.portentered + 1)modδ +16 +if rL.portentered = rj.virtualparent then +17 +rL.state = backtrack +18 +move through rL.portentered +19 +else +20 +move through rL.portentered +21 +increment rL.dist to 2 +22 +if rL.dist = 2 and rL.state = explore then +23 +wait for 2φ rounds +24 +update φ= max{φ, δ(u)} +25 +if no settled robot with rj.act settled = 1 visits then +26 +set rL.act settled = 1 +27 +terminate rL +28 +if any settled robot visits with rj.act settled = 1 then +29 +set rL.state = backtrack +30 +move through rL.portentered +31 +decrement the value of rL.dist +32 +if there is a robot rj present at the node with rj.stage = 2 then +33 +set rL.dist = 0 +34 +set rL.portentered = (rL.portentered + 1)modδ +35 +move through rL.portentered +36 +if rL.state = backtrack then +37 +wait for 2φ rounds +38 +set rL.portentered = (rL.portentered + 1)modδ +39 +if the act settled robot rj has rj.parent = −1 then +40 +if rL.portentered = 0 then +41 +set rL.act settled = 1 +42 +terminate rL and all the remaining settled robots at this node +43 +else +44 +rL.state = explore +45 +move through rL.portentered +46 +else +47 +if rL.portentered = rj.parent or rj.virtualparent then +48 +set rL.state = backtrack +49 +move through rL.portentered +50 +else +51 +rL.state = explore +52 +move through rL.portentered +53 +increment rL.dist by 1 +18 + +(a) +(b) +(c) +(d) +Figure 2: (a) The initial configuration with four robots at v1. (b) One robot, +say r, settles at v1 and updates r.special = 1. The remaining robots leave +through port 0 and the settled robot moves along with the group to the +neighboring node v2. The robot r updates the value r.count = 1. They wait +for 2φ rounds and since no settled robot visit the group, they move further +to node v3 and update ri.dist = 2 while r moves back to v1 after storing 0 as +the parent pointer of v2 in r.virtualparent and updates r.special = 0. The +group of unsettled robots wait for 2φ rounds at v3 and due to the visit by the +settled robot r they backtrack to the previous node v2 while r updates the +value of r.count = 2 From v2, it explores v5 and since no settled robot visit +v5 during the wait of 2φ rounds, one robot, say r′ settles there. The group +backtracks to v2. The settled robot r′ updates r′.count = 1. The settled +robot r increments the value of r.count as well. +The group of unsettled +robots further backtracks to v1. (c) Now the group of unsettled robots as +well as r, move through port 1 and explore v3. From v3 it explores v2 while r +goes back to v1 after storing 2 as the parent pointer of v3 in r.virtualparent. +As and when the settled robots r and r′ meets the group of unsettled robots +at any node, they increment their value of count. During the wait of 2φ +rounds, r visits v2 and thus, the group backtracks to the node v3. Then the +group explores v4 and due similar reasons, the group backtracks to to v3. +After waiting 2φ rounds, it learns the parent pointer from r and backtracks +to v1. (d) Now the group moves through port 2 to explore v4 and it further +moves to explore v3. +Due to the visit of settled robot at v1, the group +backtracks to v4 and then further backtracks to v1. As v1 is the root and +tall the ports of v1 are explored, the unsettled robots settle here at v1. At +the end of this stage 1, the value of r.count = 15 while r′.count = 2. +19 + +V5 +C +V1 +0 +00.2 +1 +1 +2 +0 +2 +0 +V4 +V2 +0 +1 +V3V5 +V1 +0 +1 +1 +0 +2 +0 +04 +V2 +0 +1 +0305 +V1 +0 +2 +1 +14 +1 +2 +2 +4 +V2 +0 +1 +V305 +V1 +0 +2 +1 +1 +2 +2 +04 +V2 +0 +0 +1 +V3Algorithm 4: Algorithm for the robot rL with rL.terminate = 1 +to reach the root node +1 if rL.terminate = 1 and settled at a node other than the root node then +2 +set rL.stage = 2, rL.state = backtrack +3 +move through rL.parent +4 +while reached node is not the root node do +5 +wait for 2φ rounds to get the parent port information +6 +move through the parent pointer of that node +7 +call Help Termination() +8 else if rL.terminate = 1 and rL is settled at the root node then +9 +call Help Termination() +Algorithm 5: Algorithm for each robot rj with rj.settled = 1 and +rj.act settled = 1 during the termination stage +1 if rj.act settled = 1 and rj.special = 0 then +2 +do the back and forth movement through all its ports +3 +if rj meets rL with rL.terminate = 1 then +4 +wait with rL till it leaves that node +5 +set rj.count′ = rj.count′ + 1 +6 +move to its original position and then resume the back and forth movement +7 +if rj.count == rj.count′ then +8 +terminate rj +9 else if rj.act settled = 1 and rj.special = 1 then +10 +move along with rL through the updated value of rL.portentered and wait till rL +leaves +11 +increment the value of rj.count′ +12 +if no act settled robot visits with id lower than rj.id then +13 +Set rj.virtualparent = rL.portentered +14 +move to the original position via rj.portentered +15 +else +16 +move to the original position via rj.portentered +17 +update rj.special = 0 +Proof. The Algorithm 1 for the settled robots in stage 1 guarantees that the settled +robots show their presence by back-and-forth movement to their one-hop neighbors. +Thus, when the group of unsettled robots visits a node and wait for 2φ rounds, then +they encounter the settled robot, if present, in its one-hop neighbor. And according +to the Algorithm 2 for unsettled robots in stage 1, no unsettled robot settles if some +settled robot meets the unsettled robot in some node. Also according to Algorithm +5, the settled robots in stage 2 settle at nodes where they get settled in stage 1. +This guarantees that no two adjacent nodes are occupied by the robots. +Lemma 3.1. Multiple robots can settle only at the root. +Proof. When the robots complete the traversal of the graph and do not find any +node to settle satisfying the conditions of the D-2-D problem, they finally reach +the root of the graph to continue Algorithm 2 and traverse through the root of +the graph. The robots can easily recognize the root as the parent pointer of the +20 + +Algorithm 6: D-2-D with Termination +1 if ri.settle = 0 then +2 +call Algorithm 2 +3 else if ri.settle = 1 and ri.act settled = 0 then +4 +call Algorithm 1 +5 else if ri.settle = 1 and ri.terminate = 1 then +6 +call Algorithm 4 +7 else if ri.settle = 1 and ri.act settled = 1 then +8 +call Algorithm 5 +root is −1. After following the algorithm from the root and subsequently exploring +through all outgoing edges, robots backtrack to root only if they don’t find nodes +to settle. In this case, they settle at the root. Thus, algorithm 2 leads multiple +robots to settle only at the root. +Lemma 3.2. If multiple robots settle at the root in stage 1, then it is guaranteed +that each node is visited by a group of unsettled robots at least once. +Proof. Let us suppose there is a node u which is not visited at all but at least one +of its one-hop neighbors, say v, is visited. This implies, that every time the group +reached v, either it backtracked from v or it explored all the ports except the port +joining v with u. The latter case is not possible as the Algorithm 2 increments the +value of portentered unless its value is equal to the value of the parent pointer. +Now without loss of generality let us consider the case when v is visited by the +group from the node that contains the lowest id settled robot among the ids of the +settled robots at one-hop neighbors of v. The Algorithm 2 ensures that the group +of unsettled robots backtracks from v only when all the ports of v are explored. +And hence u must be explored and this is a contradiction to the existence of such +a node u. +Theorem 3.2. By the end of the Algorithm 6, multiple robots settled at the root +implies no vacant node left such that none of its neighbors contains a settled robot. +Proof. Let us suppose there is a vacant node u in the graph such that no settled +robot is present in any of its one-hop neighbors in the stage 1. Lemma 1 proves +that node u is visited at least once. According to our algorithm for unsettled robots +in stage 1, i.e. Algorithm 2, when the group visited u, each of the robots rj in the +group must set rj.dist = 2. During the waiting period, there were no settled robots +in the neighbors of u to visit u. Hence the minimum id robot must have settled +there. This contradicts the presence of such a node in the graph. +Observation 3.1. If multiple robots settle in the root, it follows from Theorem 3.1 +and Theorem 3.2 that the nodes with settled robots form a maximal independent set. +Theorem 3.3. D-2-D with termination can be run by the robots with O(log ∆) +additional memory. +21 + +Proof. The variables ri.state, ri.stage, ri.settled, ri.act settled and ri.special re- +quires 1 bit of memory while ri.dist requires 2 bits of memory. +The variables +ri.parent, ri.portentered and ri.virtualparent requires O(log ∆) bits of memory. +The settled robot at a node v with δ(v) ≤ ∆ can meet the group of unsettled robots +at at most (∆ + 1) nodes including node v and there can be at most O(∆2) asso- +ciated edges with these nodes. Since the group of unsettled robots visits any edge +at most 4 times, the variable ri.count can take maximum value that is in O(∆2). +Similarly, in stage 2 the act settled robot at a node v with δ(v) ≤ ∆ can meet rL +at (∆ + 1) nodes and thus, ri.count′ can take maximum value that is in O(∆2). +Therefore, O(log∆) is the amount of memory needed by the robots to store the +information relating to these variables. As a result, each robot only needs O(log ∆) +bits of additional memory to run the algorithm. +Lemma 3.3. When the group of unsettled robots in stage 1 are in explore state +and ri.dist = 1 then there is exactly one settled robot present along with the group +which has ri.special = 1. +Proof. It is easy to observe this from the description of ri.special variable of a +settled robot as mentioned in table 1. As no node except the root contains multiple +settled robots. Also the root contains multiple robots only when no robots are +left to settle, i.e. no robot is in the explore state in stage 1 anymore. Hence, the +statement follows. +Lemma 3.4. Every tree edge in stage 1 is traversed exactly twice by the group of +unsettled robots. +Proof. Without loss of generality, according to Definition 3.1, let u has a settled +robot. the tree edge (u, v) has either a settled robot at v, or a settled robot at +u that stores the parent pointer for node v during the exploration of edge (u, v). +This ensures that v is visited for the first time as we have its parent pointer stored. +Thus, the edge (u, v) is traversed twice once in the explore state and the next in +the backtrack state. As mentioned in Algorithm 1, the parent pointer of node v is +saved by robot ru settled at node u only when no robot visits v with id < ru.id. +Hence the robots do not backtrack from v with the objective of exploring node v +from another node with lower id robot settled on it. This proves the edge (u, v) is +traversed exactly two times. +Lemma 3.5. Every non-tree edge is traversed at most four times by the group of +unsettled robots. +Proof. Let (u, v) be a non-tree edge. According to Definition 3.1, the robots back- +track from node v and the parent pointer for v is not yet stored. Till this round, +the edge (u, v) has been traversed twice. The robots reach v from the smallest id +settled robot in its neighborhood to explore v later. At that time edge (v, u) is +traversed again. Hence, every non-tree edge is traversed at most four times. +Lemma 3.6. The graph induced by the tree edges is connected and cycle free. +22 + +Proof. Consider a rooted configuration on a graph G with root u such that degree +of u is at least 2 and also k ≥ 1. First we show that the tree edges form a connected +component. It is easy to see that first two tree edges form a connected component. +Let e1, e2, ..., eh be the first h tree edges and they form a connected component. Let +there be still a group of unsettled robots that is doing the traversal. Let eh = uw for +some nodes u, w such that the tree edge was formed when the group of unsettled +robot visited w from u. If ww′ becomes a tree edge for some neighbor w′ of w +then we are done, i.e., the (h + 1)th tree edge also remains in the same connected +component. Else, if no more associated edge of w becomes a tree edge, the group +backtracks from w via a tree edge and reaches a new node v, say. Again, either +an adjacent edge of v becomes a new tree edge (in which case we are done), or it +backtracks through another tree edge. And in this way the group continues to stay +on a path consisting of tree edges until if finds a new tree edge, or it completes the +exploration and all the robots of the group settles at the root. Whatever be the +case, the tree edges form a connected component. +Now we prove that the induced graph is cycle free. Let us assume on contrary +that there is a cycle consisting of the tree edges. Let u1, u2, ..., up, u1 be the cycle +consisting of the tree edges. W.l.o.g., assume that uiui+1 be the last tree edge due +to which cycle is formed and group of unsettled robots moved from ui to ui+1. +Now we have two cases: either there is a settled robot at ui+1 or there is no settled +robot at ui+1. In case there is a settled robot at ui+1, then the group of unsettled +robots should have done a backtrack from ui+1 to ui and hence uiui+1 can not be a +tree edge. This is a contradiction to our assumption. So, let us assume there is no +settled robot at ui+1. Definition 3.1 implies there will be settled robots both at ui +and ui+2. Now, ui+1 is at one hop distance from these two settled robots and the +exploration is being done from ui to ui+1. Either of the two settled robots at ui and +ui+2 have smaller id. If the settled robot at ui+2 has smaller id then the robots will +backtrack from ui+1 to ui and thus uiui+1 will not be a tree edge. However, if the +settled robot at ui has smaller id then while exploring the node ui+2 and traversing +from ui+2 to ui+1, the group of unsettled robots must have backtracked due to +presence of a smaller id settled robot at ui thus forming ui+2ui+1 as the non tree +edge. Thus, we see that uiui+1 and ui+2ui+1 cannot be tree edges simultaneously. +Hence, our assumption of the presence of a cycle consisting of all the tree edges is +wrong and the graph induced by the tree edges is connected and cycle free. +Lemma 3.7. By the time stage 2 finishes, each robot terminates. +Proof. Since the robot rL with rL.terminate = 1 replicates the group of unsettled +robots in stage 1 and all the robots with rL.act settled = 1 replicates the settled +robots in stage 1, so, the number of times each settled robot meets with the group +of unsettled robots in stage 1 is same as the number of times each act settled robot +meets with rL. As mentioned in section 3.1, the stage 2 is replay of stage 1. So the +correctness of stage 1 implies the correctness of stage 2. And hence for each settled +robot ri except rL, ri.count = ri.count, and terminates. Finally, rL settles at the +node where it settled at the end of stage 1 and terminates. +23 + +Theorem 3.4. The Algorithm 6 achieves D-2-D with termination in 2∆(8m−3n+ +3) rounds on arbitrary rooted graphs. +Proof. It is clear from Lemma 3.4 and Lemma 3.5 that every edge is traversed +at most 4 times except the tree edges. +Also from Lemma 3.6, there can be at +most (n − 1) tree edges. So the total number of edge traversal is no more than +4(m − (n − 1)) + 2(n − 1) = 4m − 2n + 2. After each edge traversal, the robots wait +for 2φ rounds and φ ≤ ∆. So at most 2∆(4m − 2n + 2) rounds are required for +all the robots to settle. Thus Stage 1 is completed within 2∆(4m − 2n + 2) many +rounds. After the last robot settles, it may take at most 2∆(n − 1) rounds to reach +the root node in the worst-case. Now, the remaining part of stage 2 is replica of +the stage 1 of our algorithm. Thus, it takes 2∆(8m − 3n + 3) many rounds in order +to achieve D-2-D with termination +4 +Lower Bound +In this section we discuss the lower bound on number of rounds of D-2-D problem +considering robots do not have more than O(log ∆) additional memory. We start +by defining view of a node to a robot. +Definition 4.1. View: View of a node v to a robot is the information of whether +there is a settled robot at any of its one hop neighbor or not, including v. +Next we prove the theorem by constructing a class of graphs. The idea is that, +each graph in the class is a regular graph of degree n−1 and has 2n nodes. We start +with two robots, one of which settles first and the other looks for a node to settle. +The graphs are such that, unless the unsettled robot reaches two particular nodes, +it will not be able to differentiate the graph with a clique. So, before reaching one of +those nodes, if it decides to settle, that will lead to a wrong solution. We show that, +with limited memory, finding one of those nodes requires at least Ω(m∆) rounds. +Theorem 4.1. The lower bound on number of rounds of D-2-D problem on arbi- +trary graphs is Ω(m∆) considering robots have no more than O(log ∆) additional +memory. +Proof. We will prove this using a class of graphs where we show that there will be at +least one graph for which the robots require at least ∆m +12 many rounds to complete +D-2-D. Let us consider two cliques of n vertices but with one edge missing from +each of them. Let v1, v2, ..., vn be the vertices of the first clique Q1 and u1, u2, +..., un be the vertices of the second clique Q2. Let v1v2 be the missing edge from +the first clique and u1u2 be missing from the second clique. We join v1 with u1 +and v2 with u2. Now, the graph G has 2n nodes with ∆ = n − 1. Considering all +possible different port-numbering of this graph gives us a graph class G which has +cardinality equal to [(n − 1)!]2n. Let two robots r1 and r2 are initially present at +vj where j ̸= 1, 2. Let us assume that there exists an algorithm A which solves +D-2-D in time less than m∆ +12 . Let r1 settles first and at node w. We can claim that +24 + +there will be at least |G | +2 +graphs where, w /∈ {v1, v2, u1, u2}. W.l.o.g. let w = vi be +some vertex of Q1. Let us denote |G | +2 +by N. +As the robots have O(log ∆) memory, they can remember only a constant many +port numbers at a time. We provide r2 more power by letting it know that there +is a node to settle within two hop distance of vi. The robot r2 aims to explore all +the ∆(∆ − 1) many two hop neighbors. There are enough graphs(in particular, N +4 ) +wherein the robot r2 needs to explore at least ∆(∆−1) +2 +many vertices before exploring +u1 or u2. Unless it reaches u1 or u2 and has the view, r2 can not distinguish any +graph of our graph class from a clique of n nodes. +Let the sequence in which the nodes are explored is as follows {vi1, vi2,..., +vi ∆(∆−1) +2 +}. When r2 reaches vi1, it needs to know the view of the graph. If vi1 is +reached from vi directly, then getting the view takes only one round as r2 under- +stands it is one hop away from vi. Else, if vi1 is not reached directly from vi, then +it is easy to see that, in at least half of the graphs, r2 needs at least ∆ +2 rounds to +get the view. So, there exists enough instances(in particular at least +N +4.2) where r2 +requires ∆ +2 rounds to find the view. Similarly, after reaching vi2 there exists at least +N +4.22 many graphs where ∆ +2 many rounds will be required to find the view of that +node. In similar fashion, at vi ∆(∆−1) +2 +there exists at least +N +4.2 +∆(∆−1) +2 +many graphs. +Now +N +4.2 +∆(∆−1) +2 +is a function of n and the value becomes more than 1 for all n ≥ 3. +Hence, there is at least one graph where robot r2 needs to spend at least +∆(∆−1) +2 +. ∆ +2 rounds to settle. For n ≥ 3, ∆ ≥ M +3 where M = 2n. Thus, ∆(∆−1) +2 +. ∆ +2 ≥ +M +3 . (∆−1) +2 +. ∆ +2 = m. ∆−1 +6 +≥ m∆ +12 . This proves there is at least one such instance in +the class G where the robot r2 requires m∆ +12 many rounds to complete D-2-D, else +both r1 and r2 settles either on Q1 or on Q2 and this leads to wrong D-2-D. This +completes the proof. +5 +Conclusion and Future Work +We propose a variant of the dispersion problem and provide an algorithm that +solves it for the rooted initial configuration with O(log ∆) additional memory per +robot and in 2∆(8m−3n+3) synchronous rounds. We also provide a Ω(m∆) lower +bound of the problem on number of rounds. In some cases, we guarantee forming +a maximal independent set by the robots which can be of independent interest. It +will be interesting to see how to solve the problem for arbitrary initial configuration +of the robots. +References +[1] Ankush Agarwalla, John Augustine, William K. Moses Jr., Sankar Madhav +K., and Arvind Krishna Sridhar. Deterministic dispersion of mobile robots in +dynamic rings. In ICDCN, pages 19:1–19:4. ACM, 2018. +25 + +[2] John Augustine and William K. Moses Jr. Dispersion of mobile robots: A +study of memory-time trade-offs. In ICDCN, pages 1:1–1:10, 2018. +[3] Lali Barri`ere, P Flocchini, E M Barrameda, and N Santoro. Uniform scattering +of autonomous mobile robots in a grid. In IPDPS, pages 1–8, 2009. +[4] Archak Das, Kaustav Bose, and Buddhadeb Sau. Memory optimal dispersion +by anonymous mobile robots. In CALDAM, pages 426–439, 2021. +[5] Shantanu Das. Graph explorations with mobile agents. 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Sci., 887:111–121, 2021. +[18] S. Vamshi Samala, S. Pramanick, D. Pattanayak, and P. S. Mandal. Filling +MIS vertices by myopic luminous robots. CoRR, abs/2107.04885, 2021. +26 + +[19] Masahiro Shibata, Toshiya Mega, Fukuhito Ooshita, Hirotsugu Kakugawa, and +Toshimitsu Masuzawa. Uniform deployment of mobile agents in asynchronous +rings. J. Parallel Distributed Comput., 119:92–106, 2018. +[20] Takahiro Shintaku, Yuichi Sudo, Hirotsugu Kakugawa, and Toshimitsu Ma- +suzawa. Efficient dispersion of mobile agents without global knowledge. In +SSS, volume 12514, pages 280–294, 2020. +27 + diff --git a/S9E4T4oBgHgl3EQfLgwd/content/tmp_files/load_file.txt b/S9E4T4oBgHgl3EQfLgwd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..21009bb8c09934ffe3e966b9f0ded641d3fec42b --- /dev/null +++ b/S9E4T4oBgHgl3EQfLgwd/content/tmp_files/load_file.txt @@ -0,0 +1,1087 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf,len=1086 +page_content='Distance-2-Dispersion: Dispersion with Further Constraints Tanvir Kaur ∗ Kaushik Mondal∗ Abstract The aim of the dispersion problem is to place a set of k(≤ n) mo- bile robots in the nodes of an unknown graph consisting of n nodes such that in the final configuration each node contains at most one robot, starting from any arbitrary initial configuration of the robots on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this work we propose a variant of the dispersion prob- lem where we start with any number of robots, and put an additional constraint that no two adjacent nodes contain robots in the final con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' That is, the distance between any two nodes with robots must be at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We name this problem as Distance-2-Dispersion, in short, D-2-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, even if the number of robots k is less than n, it might be the case that it is not possible for each robot to find a distinct node to reside, maintaining our added constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To be more specific, if a maximal independent set is already formed by the nodes which contain a robot each, then other robots, if any, who are searching for a node to seat, will not find one to seat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence we allow multiple robots to seat on some nodes only if there is no place to seat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If k ≥ n, it is guaranteed that the nodes with robots form a maximal independent set of the underlying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The graph G = (V, E) has n nodes and m edges, where nodes are anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It is a port labelled graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', each node u assigns a distinct port number to each of its incident edges from a range [0, δ−1] where δ is the degree of the node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots have unique ids in the range [1, L], where L ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Co-located robots can communicate among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We provide an algorithm that solves D-2-D starting from a rooted configuration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', initially all the robots are co-located) and terminate after 2∆(8m − 3n + 3) synchronous rounds using O(log∆) memory per robot without using any global knowledge of the graph parameters m, n and ∆, the maximum degree of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We also ∗Department of Mathematics, Indian Institute of Technology Ropar, India tanvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='20maz0001@iitrpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='in, kaushik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='mondal@iitropar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='04938v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='DC] 12 Jan 2023 provide Ω(m∆) lower bound on the number of rounds for the D-2-D problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Keywords— Mobile robots, Anonymous graphs, Dispersion, Deterministic al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 1 Introduction The aim of the dispersion problem is to place a set of k(≤ n) mobile robots in the nodes of an unknown graph consisting of n nodes such that in the final configu- ration each node contains at most one robot, starting from any arbitrary initial configuration of the robots on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This problem was introduced in the year 2018 by Augustine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Later, this problem is studied under various models and with different assumptions in the literature[4, 16, 14, 17, 20, 12, 8, 15, 10, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The main tool used for dispersion is Depth-First-Search traversal and since the robots are equipped with memory, they store the important information required to complete dispersion without getting stuck in a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' A natural question arises what will happen if there are some extra constraints imposed on the dispersion problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As an example, no robot can settle in the one-hop neighborhood of an already settled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This led to the generation of Distance-2-Dispersion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this problem, k robots arbitrarily placed on the graph need to attain a config- uration such that no two adjacent nodes are occupied by the settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also, an unsettled robot can settle at node that already contains a settled robot, only if for the unsettled robot there is no other node to settle maintaining the added con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' With this, there can be many nodes without a settled robot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', no robot to store any information at those nodes, and the graph is a zero storage one, thus the problem becomes interesting if one aims to solve with less memory requirement at each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1 Model and the problem Let G be an arbitrary connected undirected graph with n nodes, m edges and maximum degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The nodes are anonymous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', they have no id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It is a port labelled graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', each node u assigns a distinct port number to each of its edges from a range [0, δ(u) − 1] where δ(u) is degree of node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Port numbers that are assigned at the two ends of any edge are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Nodes do not have any memory and hence G is a zero storage graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' A total of k movable entities are present in the system, which are called robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Each robot has a unique id from the range [1, L] and each robot knows its id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In some round, if two or more robots are at a single node, we call them co-located and such robots can share information via message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ∗Any robot present on ∗This is known as the Face-to-Face communication model and has already been con- sidered while studying problems related to mobile robots including exploration [5, 6] and 2 some node knows the degree of that node as well as the port-numbers associated with each of the edges corresponding to that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, if some robot needs to leave its current node through any particular port number, it can do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Besides this, whenever any robot moves from a node u to another node v, it learns the port number through which it enters the node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Our algorithm proceeds in synchronous rounds where in each round robots per- form the following steps in order: (i) co-located robots may exchange messages (ii) robots may compute based on available information (iii) robots may move through an edge to some adjacent node from the current node based on its computation in step (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We further assume that all the robots start the algorithm at the same time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', from the same round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The time complexity of the algorithm is measured as the number of synchronous rounds required by the robots to complete the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We also study the amount of memory required per robot to run the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The problem: Distance-2-Dispersion(D-2-D): Given a set of k ≥ 1 robots placed arbitrarily in a port labelled graph G with n nodes and m edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' the robots need to achieve a configuration by the end of the algorithm where each robot needs to settle at some node satisfying the following two conditions: (i) no two adjacent nodes can be occupied by settled robots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' and (ii) a robot can settle in a node where there is already a settled robot only if no more unoccupied node is present for the robot to settle satisfying condition (i) The conditions ensure that the distance between any pair of settled robots is at least 2 unless both are settled at the same node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence, the nodes with settled robots form an independent set of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' And with enough robots, we get a maximal independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Our contribution: We solve the D-2-D problem for rooted† initial configuration on arbitrary graphs in 2∆(8m−3n+3) rounds using O(log∆) memory per robot in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' All the settled robots terminate even without any global knowledge re- garding any of the graph parameters m, n or ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In Section 4, we provide a Ω(m∆) lower bound of the D-2-D problem on the number of rounds considering robots do not have more than O(log∆) memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also, if k ≥ n, it is guaranteed that the nodes with settled robots form a maximal independent set, which can itself be an in- teresting topic to study in the domain of distributed computing with mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 Related work Dispersion is the most related problem to our problem as we consider similar model that is considered to solve the dispersion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Augustine et al introduced the dispersion problem in [2] for the rooted configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' They proved the memory requirement by the robots for any deterministic algorithm to achieve dispersion on a graph is Ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The lower bound for any algorithm to perform dispersion dispersion [2, 13] †The configuration where all the robots are initially placed on a single node of the graph 3 on any graph is Ω(D), where D is the diameter of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For rooted graphs, with the knowledge of m, n, they gave an algorithm that requires O(log n) memory by the robots to complete dispersion in O(m) rounds[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Kshemkalyani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [11] improved the lower bound of running time to Ω(k) where k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' They developed an algorithm for synchronous system which solves dispersion in O(m) rounds using O(k log ∆) bits at each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, for an asynchronous system they developed an algorithm which requires O(max(log k, log ∆)) bits of memory with the time complexity O((m − n)k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Later Kshemkalyani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [9] significantly improved the result and provided a deterministic algorithm for dispersion in arbitrary graphs in synchronous setting that runs in O(min(m, k∆) · log l) rounds, where l ≤ k 2, using O(log n) bits of memory at each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Their intuitive idea was to run DFS traversals in parallel to minimize time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots required the knowledge of m, n, k and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Shintaku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' then presented a dispersion algorithm that does not require such global knowledge [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Their algorithm solves the dispersion problem on arbitrary graphs in O(min(m, k∆) · log l) rounds using Θ(log(k + ∆)) bits of memory at each robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Recently, Kshemkalyani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [13] provided an algorithm that is optimal in both time and memory in arbitrary anonymous graphs of constant degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' They presented an algorithm which solves dispersion in O(min(m, k∆)) time with Θ(log(k + ∆)) bits at each robot improving the time bound of the best previously known algorithm by O(log l) where l ≤ k 2 and matching asymptotically the single-source DFS traversal bounds[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' They extend the idea of [9] by making the larger size DFS traversal to subsume the smaller size DFS thus avoiding the need of revisiting the nodes of subsumed traversal more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' D-2-D, in some sense, is also related to the problem of scattering or uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Scattering has been worked mainly for grids[3] and rings[7, 19] though with anonymous robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Finally, as in some cases, our algorithm forms a maximal independent set, we cite the following study on forming maximal independent set with movable entities, though it is done with stronger model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Vamshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' presented the problem of finding maximal independent set(MIS) using my- opic luminous robots[18] of an arbitrary connected graph where the robots have prior knowledge of ∆, O(log ∆) bits of persistent memory and at least 3 hops vis- ibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Authors also used colors to represent different states and worked under semi-synchronous as well as asynchronous schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='3 D-2-D vs dispersion: the challenges In the previous works on the dispersion problem, the algorithms use the depth- first search (DFS) traversal with limited memory of the robots [2, 4, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The key idea to achieve dispersion from any rooted configuration is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' At the starting node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', the root, the robot with the lowest id settles down and the remaining unsettled robots leave the root to visit one of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The minimum id robot from this group of unsettled robots settles here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In any round, whenever an unoccupied node is visited by the group of robots for the first time, the minimum id robot settles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' A settled robot on a node represents that the 4 node has already been visited by the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Using this key idea, DFS traversal for anonymous graphs is feasible by robots since settled nodes help to track cycles and already visited nodes thus ensuring the safe dispersion of the anonymous graph by the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For instance, if the group of robots visit a node v from a node u during exploration, and find out that a settled robot is already present at v then as per DFS the group must backtrack to u and explore other unexplored neighbors of u if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Whenever it finds an empty node, the smallest id robot of the group settles down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The remaining unsettled robots continue the DFS traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, to do the backtracking successfully, the robots need to remember the path they have already explored and this needs high memory requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To avoid this, the algorithm instructs each settled robot to store the information required to backtrack from the respective node where the robot is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Observe that, the group never backtracks from a node where no robot is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To be more specific, each settled robot remembers the parent pointer as the port number it used while entering into the node it is settled at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of backtracking robots can use this parent pointer and can successfully backtrack from this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus dispersion can be achieved with O(log ∆) memory per robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In D-2-D, the main motive is to achieve dispersion such that no two adjacent nodes have settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In other words, the distance between any two settled robots must be at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To do so, we may face the following challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Since there is no settled robot present in the one-hop neighborhood of any settled robot, the information regarding the parent pointer of those neighbor- ing nodes is difficult to be stored and subsequently used while backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This may lead to high memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots can settle at a node u if and only if there are no settled robots at any of the neighbors of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As the maximum degree ∆ can be large, this may lead to high time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 2 Warm-Up: D-2-D with O(∆ log ∆) Memory per Robot In this section, we provide an informal discussion on a straightforward solution of the rooted D-2-D without bothering about the memory requirement per robot or the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Our algorithm is based on the depth-first search traversal (albeit with some modification) that solves the dispersion problem as we discussed above in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Later in Section 3, we improve over this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' While solving the dispersion problem, encountering a settled robot while doing forward exploration, implies the presence of cycle to the moving group of unsettled robots and then backtracking is done with the help of stored parent pointers at the settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We do the following modification to solve our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As the group may need to backtrack from an unoccupied node in our D-2-D problem, it is required to store the parent pointers of the unoccupied nodes too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that all the neighboring nodes of any occupied node must be unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We instruct 5 each settled robot to remember the parent ports of all its neighbors including itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Basically each settled robots work as virtuallysettled robot at its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The notion of virtually settled means that although there is no settled robot present at that node, yet no robot from the visiting group can occupy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, to store as well as to provide the stored parent pointers, each settled robot must meet the moving group of robots whenever the group reaches one of its neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To achieve this, each settle robot does a back and forth movement from its position to its neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To be more specific, let a robot r settles in round T at a node u of degree δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It visits u(0) in round T + 1, comes back to u in round T + 2, visits u(1) in round T + 3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It visits u(0) again after u(δ − 1) is visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r stops only when it meets the group of unsettled robots at some node, say v = u(p), and at some round say T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It stays with the group at u(p) till the group leaves u(p), say in round T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Then r comes back to u in round T ′′ + 1 and again starts visiting its neighbors one by one as described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that only this does not solve the problem as the group of moving robots may reach a neighboring node of some occupied node but at that time the respective settled robot may visit its another neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To solve this issue, the algorithm instructs the moving group to wait for 2∆ rounds at each node v which ensures that the moving settled robot must meet the group within this time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For simplicity, let us assume that all robots know ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='‡ If v is occupied, then the settled robot must meet the group within 2 rounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' else if v is a neighbor of an occupied node, then the settled robot that is working as a virtually settled robot must meet the group within the 2∆ waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now we provide the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If the group is in forward exploration phase, the following are the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group meets at least one virtually settled robot and finds that none of the virtually settled robots who meet the group has the parent pointer for this node, the group understands that it is visiting this node for the first time and continues the DFS traversal in the forward exploration phase after providing the parent pointer to each of the virtually settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This is possible as all the settled robots that meet this group wait with this group till the group leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group meets at least one virtually settled robot and finds that at least one of the virtually settled robots comes with the parent pointer for this node, the group understands that this node is already explored earlier, and subsequently backtracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group finds that the node is occupied, it goes to backtrack phase, and backtracks with the help of the parent pointer stored at the settled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group sees no settled or virtually settled robots, then the minimum id robot from the group settles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ‡In the next section we remove this assumption in our main algorithm using the idea that whenever the group of unsettled robots visits a new node, each unsettled robot updates the value of maximum degree seen by itself till now, as the value of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 6 If the group is in backtracking phase, it must meet at least one virtually settled robot or a settled robot at the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group checks if all the ports associated to this node is already explored or not, by looking at the parent pointer and the port though which it just backtracks to this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The following are the decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If all ports are explored, it continues the backtracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If some port remains to be explored, it changes to forward exploration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Figure 1 represents the implementation of the algorithm on an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 1: (a) The initial configuration with four robots at v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (b) One robot settles at v1 while the remaining unsettled robots do the DFS traversal through v2, v3, v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' None of them settles at any of these nodes as the robot settled at v1 works as the virtually settled robot at these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot at v1 maintains the parent pointer for the group at all its neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' From v1 the robots backtrack due to the presence of a settled robot on that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (c) The unsettled robots continue the DFS traversal and when the group reaches v5, a robot settles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (d) The remaining robots continue the traversal, finish the traversal at v1, then do it again and finally achieve the desired configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' To store the parent pointer for each of the neighbors, each settled robot requires O(∆ log ∆) memory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', log ∆ memory per port number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This leads to high mem- ory requirements and waiting for 2∆ rounds for the moving group at each newly explored node leads to a high run time of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Both of these issues are mentioned when we discussed about the challenges in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The run time of this algorithm becomes 2∆(4m − 2n + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This is because we just run the DFS traversal for dispersion that takes 4m − 2n + 2 time as done in [2, 9] assuming all the unoccupied neighbors of each occupied node are virtually occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also, we keep all the necessary information with the settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After completing 7 V5 C V1 0 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 1 1 2 0 2 0 V4 V2 0 1 V305 V1 0 0 2 1 1 2 2 0 04 V2 0 1 V3V1 0 2 1 1 2 0 2 0 04 V2 0 1 V3V5 V1 0 00: 2 1 1 2 0 2 0 V4 V2 0 1 V3the DFS traversal once, if some robots are left unsettled then they settles at the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that as the settled robots keep moving, it is desirable that the robots can terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, here we do not discuss about the termination of the algorithm, but it can be achieved in a similar way we do in Section 3 in our main algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 3 D-2-D with Termination: O(log ∆) Memory per Robot In this section, we present an algorithm that improves the memory requirement of the algorithm discussed in section 2 as well as we include termination of the robots without any global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' First we provide a high level idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The idea is not to do the usual DFS traversal and in turn, no settled robot needs to maintain parent pointers for all its unoccupied neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Each settled robot stores the parent pointer corresponding to only one of its neighboring nodes along with the parent pointer corresponding to the node where it is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This bounds the path from the last settled robot to the node where the unsettled group is currently present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If the length of this path is high, then in the worst case, all the nodes in this path may remain unoccupied as all of them can be the neighbor of some particular occupied node v, and in this case the robot that is settled at v, needs to remember all the parent pointer of the unoccupied nodes present in that path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We restrict the depth of the traversal during forward exploration by 2 from the node where a robot settled last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now, the question is how does the information corresponding to only one neighbor suffice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Similar to the previous algorithm, each settled robot visits each of its neighbors one by one in the subsequent rounds once it is settled while the group of unsettled robots waits for 2φ rounds, where φ is the maximum degree observed by the group of unsettled robots till reaching the current node, in order to check the presence of any settled robot in its one-hop neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Similar to the algorithm described in the Section 2, the group of unsettled robots update the value of φ when they reach a new node u by max{φ, δu} as they have no prior knowledge of the maximum degree of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of unsettled robots maintains a counter ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist from the last encountered settled robot which stores the distance from the settled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The variable ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist can take values 0, 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of unsettled robots decides to do forward exploration or backtrack based on the value of ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist, the number of settled robots visiting this group during the waiting period and the ids of these visiting settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, the state of any robot can be explore or backtrack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots maintain another variable ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage which takes the value 2 when the D-2-D configuration is achieved and the termination stage has started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In order to terminate the algorithm, the settled robots maintain variables ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′, and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled whose description is given in the table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Whenever the group of unsettled robots moves from a node u containing a settled robot rj with the lowest id to one of its neighbors, say u(p) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' the unsettled group of robots have ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 at u(p), the state is explore and no other settled robot visits the group at u(p) except rj, then the parent pointer of u(p) is required 8 to be stored by the settled robot rj (rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent as defined in Table 1) and the group explores further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' From the settled robot’s point of view, it stores its parent pointer (rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent as defined in Table 1) along with the parent pointer of only one neighbor u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The idea is rj may store the parent pointer corresponding to another neighbor u(q), say, when q is the last port that was explored from the node where rj is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, if one or more settled robots with id lower than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id meet the group at u(p), then the group backtracks to the node u along with the settled robot rj and rj does not need to store the parent pointer of u(p) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' All the settled robots including rj which meet the group of unsettled robots will increment their value of count variable each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that, to avoid needless increments in the value of the count variable by the settled robots, when the settled robots encounter the group of unsettled robots at a node during their back and forth movement, they wait with the group until it leaves that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 as the group reaches a node say v after exiting u(p) where rj is settled, a robot settles there only if there are no neighboring settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this case, the newly settled robot keeps the parent pointer as the group moves forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, if at the node v, one or more settled robots are visiting, then the group backtracks as there are options to explore this node later through some other already settled robots with possibly lower id than ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For instance, if the unsettled group of robots at dist = 2 from the node u are witnessing their waiting period of 2φ rounds and during this waiting period, a robot rm visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group not only understands that no robot can settle at this node, but also, this node can be explored from the node that has the settled robot rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In other words, if the group of unsettled robots are at dist = 2 from a node with a settled robot and are in the explore state, and some settled robot rm visits the group during the waiting period then this implies a direct link to this node from rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This means the current node can be explored from the node where rm is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, in no circumstances, any settled robot, here rj, is keeping parent pointers of a node that is at least two hops away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' All the settled robots rj which meet the group of unsettled robots increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count each time they meet the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' These set of steps comprise the stage 1 of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After the last robot settles, it begins the next stage, namely, the termination stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The last settled robot starts from the root node and follows similar path as described in the above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this termination stage, the last robot rL acts as the group of unsettled robots in the stage 1 while the robot rj which originally settled during the Distance-2- Dispersion, set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 in the same order as it settled in the previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, the termination stage is a replica of stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Each time the act settled robot rj meets the last settled robot rL during its traversal, rj increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As and when the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count becomes equal to rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′, the robot rj terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this way the robots terminate and Distance-2-Dispersion with termination is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 9 Variables Descriptions ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent This variable contains the parent port of the node u where ri is settled in stage 1 or act settled in stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered The port through which robot ri enters the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Initially ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = −1 for all the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent The parent port of u(p) where p be the last port that was explored from the node where ri is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Initially ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = −1 for all the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist Each unsettled robot ri maintains the distance from the settled robot it last encountered during its traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' According to our algorithm, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For each settled robot, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0 if it is at the node where it is settled, else ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special A robot ri settled at some node u, say, updates ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 only when the group of unsettled robots is at u with ri and will move through one of the adjacent edges of u in the explore state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For other settled robots, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 and for any unsettled robot, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled Takes value 1 if ri is a settled robot in stage 1, else takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count If ri is a settled robot in stage 1, this variable counts the number of times ri meets the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage This variable can take values 1 or 2 where ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 1 indicates stage 1 of the algorithm whereas ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 indicates the stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled Takes value 1 if ri settles in the stage 2, else takes value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ In the stage 2, if ri is an act settled robot, this variable counts the number of times ri meets the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Table 1: Description of variables 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1 The Algorithm In this section, we describe our algorithm that can be run by the robots with O(log∆) memory per robot to achieve rooted D-2-D and terminate in 2∆(8m − 10 3n + 3) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Consider an arbitrary graph G and let k robots be initially placed on a single node, say u, of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Each robot ri maintains variables ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We have defined the variables in detail in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Apart from that, each unsettled robot maintains its state, which can be either explore or backtrack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In explore state, it does forward exploration by moving through a computed port number from the current node, whereas in backtrack state, it learns the parent pointer from a settled robot and backtracks through that port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Robots also main- tain a variable φ which is initialized to δ(u), where u is the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Each unsettle robot updates φ as and when they see some node with degree more than the cur- rent value of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, settled robots do not modify φ once they are settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that the each robot in the group of unsettled robots has the same value of φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' also φ ≤ ∆, the maximum degree of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' More specifically, when the group of unsettled robots reach a new node, say v, each robot in the group updates the value of φ with max{δ(v), φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Our algorithm works in two stages, stage 1 and stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In stage 1, robots achieve D-2-D and in stage 2, robots terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of unsettled robots run the algorithm in phases, where each phase consists of 2φ rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Since each unsettled robot agrees on their φ value, the group starts and ends each phase at the same round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Our algorithm starts by settling the minimum id unsettled robot, say r1, at the root node u at the end of phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r1 updates r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent ← −1, r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled ← 1 and r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The remaining unsettled robots update their state to explore and since they are present with the settled robot r1, the variable r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special is updated to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The unsettled robots update the port number as ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ, where δ is the degree of the node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The unsettled robots move along the incremented port number, which in this case will be port 0, to the neighboring node u(0), and update ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' With this, the phase 1 ends for the unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After reaching this node, the unsettled robots update φ and wait for 2φ rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As this is a rooted configuration and r1 is the first robot to settle(which is currently present along with the group of unsettled robots), no other settled robot visits this group during the wait of 2φ rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This implies that the current node can be further explored if there are unexplored edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', degree of u(0) is more than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r1 updates r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let p1 be the port through which robots entered u(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r1 updates r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent ← p1, return through port p1 to the root u and updates r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The unsettled robots update ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (p1 + 1)mod(δ(u(0))), where δ(u(0)) is the degree of node u(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The unsettled robots move through the updated value of portentered in the explore mode and update ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered ̸= r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, the unsettled robots move through the updated value of portentered in the backtrack mode and update ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This ends phase 2 for the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Basically, the group of unsettled robots move only at the end of each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After the group of unsettled robots reaches a new node at distance 2 apart from u, they update φ and wait for 2φ rounds while the settled robot r1 now has 11 r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0, and it continuously traverses through each of the ports of u one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' During this waiting period, if any settled robot rj visits the group of unsettled robots, they backtrack to the previous node using the ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, the settled robot rj which meets this group of unsettled robots, waits with the group of unsettled robots unless the group leaves that node and increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, if no settled robot visits this group then the minimum id robot from the group of unsettled robot settles at this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So each phase of unsettled robots corresponds to one edge traversal and each phase requires 2φ rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Settled robots do not bother with phases, they either wait with the group of unsettled robots or continue back and forth traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, the time complexity of stage 1 of our algorithm depends on how many phases the group of unsettled robots works before the last robot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', the largest id robot, say rL, settles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled and unsettled robots decide what to do based on the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' First we write what settled robots do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If a settled robot rj has the value special = 0, then it continues its visit to each of its neighbors one by one and keeps modifying ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As and when the settled robot meets the group of unsettled robots, it increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count and waits with the group unless it leaves that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Post that, the settled robot resumes its visit to each of its neighbors one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If a settled robot rj has the value special = 1, then it moves along with the group of unsettled robots in the next round to the neighboring node and waits along with the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot rj increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' During this waiting period, if any settled robot with id lower than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id visits then rj does not store the parent pointer for this node and return back to its original position after the group leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, if no settled robot with id smaller than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id visits the group of unsettled robots then the settled robot rj updates rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The pseudo-code for the settled robots is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now we see how the unsettled robots work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the group of unsettled robots reach a node at the end of some phase, then the decision of what to do at the end of the next phase is made based on the following cases: When ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 and state = explore at the beginning of a phase – If no settled robot with special = 0 visits the group of unsettled robots such that the visiting settled robot’s id is lesser than the id of rj with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1, then the group updates ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that whenever the group of unset- tled robots have ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 and state = explore, there is definitely a settled robot rj present with the group having rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent, then the group of unsettled robots backtracks to the previous node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered ̸= rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent then the group of unsettled robots leaves the current node via the up- dated port number in the explore state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 12 Algorithm 1: Algorithm for each settled robot rj 1 if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id is the largest among the group from which it settled then 2 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 3 else 4 if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 then 5 do the back and forth movement through all its ports and updates rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist value to 0 or 1 according to its position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 6 if the settled robot rj meets the group of unsettled robots then 7 wait with the group of unsettled robots till the group leaves that node 8 increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count 9 move to the original position where it was settled, if not already there, and resume the back and forth movement 10 else if rj meets any robot ri with ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 then 11 move to its original position where it was settled, if not already there, and wait until it meets the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 12 do not increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count 13 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 14 if the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 visits this node where rj is waiting then 15 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 16 if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 then 17 move along with the group of unsettled robots through updated value of ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and wait till the group leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 18 increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count 19 if no settled robot visits with id lower than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id then 20 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 21 move to the original position via rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 22 else 23 move to the original position via rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 24 update rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 – If at least one settled robot, with special = 0 and having id lesser than id of rj with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1, visits the group of unsettled robots, then the group backtracks via ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 and state = explore at the beginning of a phase – If no settled robot visits the group of unsettled robots in 2φ rounds, then the lowest id robot from the group settles on this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' – If one or more than one settled robot visits the group of unsettled robots, then the group backtracks via the port entered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, when the group of unsettled robots is in state = backtrack, the group of unsettled robots reaches a node which has been visited earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, the node which is visited in backtrack state has either a settled robot on it or any settled robot in its one hop neighbor has stored the virtual parent for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After backtracking to the current node, the group of unsettled robots decides to further explore or backtrack based on the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack at the beginning of a phase 13 – If all the ports are already explored i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ is equal to the rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parentpointer where rm is the settled robot on that node or rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent where rj is the settled robot which has stored the parent pointer for its neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Recall that this information can be exchanged during the waiting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this case, the group of unsettled robots backtracks through rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parentpointer or rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' – If the (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ is not equal to the ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parentpointer of the node then the unsettled group of robots change their state to explore and move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The pseudo-code for the unsettled robots is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Using this we achieve D-2-D configuration and stage 1 completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now the stage 2 begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robot with the largest id, say rL, is said to be the last settled robot and after it settles it sets rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In order to terminate the algorithm, the rL restarts similar traversal as described in stage 1 from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now two cases arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Either the robot rL settles at a node other than the root node (in this case it goes to the root node as described below) or rL settles at the root itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In any case, while at the root, rL sets rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' From root, rL starts mimicking the stage 1 again by acting as the group of unsettled robots, to help the remaining settled robots terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Finally, rL terminates after reaching the node where it settled at the end of stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Note that, if and when rL backtracks till the root, as some settled robots may meet rL on the path since they are continuing their back and forth movement, yet no settled robot increments their respective count variable since rL is not an unsettled robot anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the last robot rL settles at node ul other than the root – The robot rL backtracks through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It continues back- tracking through the parent pointer to reach the root node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After reaching the root node, the robot rL begins mimicking stage 1 as the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the remaining robots settle at the root and rL is the largest id robot among the group – The robot rL begins mimicking stage 1 as the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When a robot ri with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled = 1 meets rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 or any robot rj with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2, it understands stage 2 is under progres, updates ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2, goes back to its original position and waits for rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 to arrive there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When a robot rj other than rL meets rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 for the first time, following two cases are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If this meeting is done at the node where rj settled in phase 1, it understands it has to wait here till rL is here by acting as rj is also an unsettled robot which is with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Then rj updates rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 as if it becomes settle by the end of this phase and start following the algorithm of a settled robot of stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The only difference is that, now it increments rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ instead 14 Algorithm 2: Algorithm for each unsettled robot ri 1 initialise ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = −1, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = −1, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled = 0, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = −1, φ = 0, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 0, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 1 2 for phase = 0 do 3 the minimum id robot rj settles on the node after waiting for 2φ rounds 4 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = −1 5 set φ = δ(u) 6 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 7 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 8 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist=ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist + 1 9 for phase > 0 do 10 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0 then 11 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist + 1 12 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore then 13 wait for 2φ rounds 14 if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id is smaller than the settled robot already present with the group then 15 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 16 update φ = max{φ, δ(u)} 17 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 18 decrement ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist value to 0 19 else 20 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 21 update φ = max{φ, δ(u)} 22 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 23 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent then 24 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 25 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 26 else 27 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 28 increment ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist to 2 29 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore then 30 wait for 2φ rounds 31 update φ = max{φ, δ(u)} 32 if no settled robot visit the group of unsettled robots then 33 the robot ri with lowest id settles on that node and sets ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 34 each unsettled robot makes ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0 35 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = parent pointer of the settled robot at the current node then 36 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 37 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 38 else 39 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 40 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 41 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack then 42 wait for 2φ rounds 43 decrement ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist by 1 44 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 45 if the settled robot rj has rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = −1 then 46 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = 0 then 47 the unsettled robots settle at the root 48 else 49 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 50 move through ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 51 else 52 if portentered = parent or virtualparent then 53 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 54 move through portentered 55 else 56 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 57 move through portentered 58 increment ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist by 1 15 of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else if this meeting is done at a neighboring node of the node where rj settled in phase 1, it changes rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2, goes back to its original position and waits for rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 to arrive there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The above description shows that, any settled robot r in phase 2 increments their variable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ only after becoming act settled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', only after the corre- sponding round of stage 1 when r became settled and started incrementing its r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Since rL works as the unsettled robot by following the algorithm of unset- tled robots in stage 1 and its id is the largest, rL will settle in phase 2 again at the end and will terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' By that time, all settled robots count value match with their count′ value, and all terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' With this, below we provide the algorithm of stage 2 in a formal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 reaches the root u, it initializes the value of φ = δ(u), and begins the traversal in order to terminate the settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Similar to the algorithm described above for the unsettled group of robots in stage 1, every time rL reaches a node, say, v, it waits for 2φ rounds at that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also, it updates the value of φ with max{φ, δ(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After reaching a new node, decisions are made based on the following cases: When rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore – If any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id is smaller than the id of act settled robot already present with rL then rL backtracks via rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered – if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 but rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id > the id of act settled robot already present with rL or no act settled = 1 robot visits then rL is set to explore and it moves through the incremented value of portentered When rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore – If no settled robot with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 visits then this indicates that it is the original position where rL is settled and hence set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Terminate rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' – if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 then rL backtracks through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered – if there is a robot rj present at the node with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 but rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 0 then rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent is set to rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack – If the reached node is the root node and there are some ports to be explored i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ ̸= 0 then rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state is updated to explore, else rL and the remaining settled robots are terminated at this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' – If the reached node is other than the root node then rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered is compared with the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent or rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If all the 16 ports are explored then the state is changed to backtrack otherwise rL explores through the incremented value of rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The pseudo-code of the algorithm for the last settled robot rL is given as Algo- rithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The Algorithm 4 uses the Algorithm 3 as a subroutine, where Algorithm 3 is the pseudo-code of what the last settled robot rL does after reaching the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 0/1 proceeds based on the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If rj is a settled robot with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled = 1 but rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 0, when it meets any robot ri with ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 then it returns to its original position and waits there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It now becomes aware that the termination stage has started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, it does not increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 then it continues the back and forth movement through all its neighbors and increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ by 1 as and when it meets rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count becomes equal to rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′, it terminates at its original position where it was settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 then it moves with the robot rL to the neighboring node through the updated value of rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and waits with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It increments the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now two cases arise: – If no act settled robot visits with id lower than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id then rj sets rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered – Else move to the original settled position and set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 The pseudo-code of the algorithm for each settled robot rj with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 is given in the Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The pseudo-code of the Distance-2-Dispersion with Termination is given in the Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Figure 2 shows the run of stage 1 of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 Analysis of the Algorithm Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Tree Edge: An edge (u, v) is said to be a tree edge if the group of unsettled robots in stage 1 reaches v through (u, v) such that either the settled robot at node u (if exists) stores the parent pointer of the node v or the minimum id robot among the group of unsettled robots settles at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' A settled robot ri in stage 1 stores the parent pointer for its adjacent node u at some round t only if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 for any unsettled robot rj at u and no settled robot with id lower than ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id visits rj during the 2φ rounds i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', during the waiting period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' By the end of the Algorithm 6, there are no two robots that are settled at adjacent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 17 Algorithm 3: Help Termination(): Algorithm for the robot rL 1 set φ = δu, rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = 0 2 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 3 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist + 1 4 for phase > 0 do 5 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore then 6 wait for 2φ rounds 7 if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0, and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id < id of act settled robot already present with rL then 8 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 9 update φ= max{φ, δ(u)} 10 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 11 decrement rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist value to 0 12 if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0, but rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id > id of act settled robot already present with rL or no rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 robot visits then 13 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 14 update φ = max{φ, δ(u)} 15 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 16 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent then 17 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 18 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 19 else 20 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 21 increment rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist to 2 22 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 and rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore then 23 wait for 2φ rounds 24 update φ= max{φ, δ(u)} 25 if no settled robot with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 visits then 26 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 27 terminate rL 28 if any settled robot visits with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 then 29 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 30 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 31 decrement the value of rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist 32 if there is a robot rj present at the node with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2 then 33 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 0 34 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 35 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 36 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack then 37 wait for 2φ rounds 38 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = (rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered + 1)modδ 39 if the act settled robot rj has rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent = −1 then 40 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = 0 then 41 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 42 terminate rL and all the remaining settled robots at this node 43 else 44 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 45 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 46 else 47 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent or rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent then 48 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 49 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 50 else 51 rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = explore 52 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 53 increment rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist by 1 18 (a) (b) (c) (d) Figure 2: (a) The initial configuration with four robots at v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (b) One robot, say r, settles at v1 and updates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The remaining robots leave through port 0 and the settled robot moves along with the group to the neighboring node v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robot r updates the value r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' They wait for 2φ rounds and since no settled robot visit the group, they move further to node v3 and update ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2 while r moves back to v1 after storing 0 as the parent pointer of v2 in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent and updates r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of unsettled robots wait for 2φ rounds at v3 and due to the visit by the settled robot r they backtrack to the previous node v2 while r updates the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = 2 From v2, it explores v5 and since no settled robot visit v5 during the wait of 2φ rounds, one robot, say r′ settles there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group backtracks to v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r′ updates r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot r increments the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The group of unsettled robots further backtracks to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (c) Now the group of unsettled robots as well as r, move through port 1 and explore v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' From v3 it explores v2 while r goes back to v1 after storing 2 as the parent pointer of v3 in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As and when the settled robots r and r′ meets the group of unsettled robots at any node, they increment their value of count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' During the wait of 2φ rounds, r visits v2 and thus, the group backtracks to the node v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Then the group explores v4 and due similar reasons, the group backtracks to to v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After waiting 2φ rounds, it learns the parent pointer from r and backtracks to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (d) Now the group moves through port 2 to explore v4 and it further moves to explore v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Due to the visit of settled robot at v1, the group backtracks to v4 and then further backtracks to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As v1 is the root and tall the ports of v1 are explored, the unsettled robots settle here at v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' At the end of this stage 1, the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = 15 while r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 19 V5 C V1 0 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 1 1 2 0 2 0 V4 V2 0 1 V3V5 V1 0 1 1 0 2 0 04 V2 0 1 0305 V1 0 2 1 14 1 2 2 4 V2 0 1 V305 V1 0 2 1 1 2 2 04 V2 0 0 1 V3Algorithm 4: Algorithm for the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 to reach the root node 1 if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 and settled at a node other than the root node then 2 set rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage = 2, rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state = backtrack 3 move through rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent 4 while reached node is not the root node do 5 wait for 2φ rounds to get the parent port information 6 move through the parent pointer of that node 7 call Help Termination() 8 else if rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 and rL is settled at the root node then 9 call Help Termination() Algorithm 5: Algorithm for each robot rj with rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 during the termination stage 1 if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 then 2 do the back and forth movement through all its ports 3 if rj meets rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 then 4 wait with rL till it leaves that node 5 set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ = rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ + 1 6 move to its original position and then resume the back and forth movement 7 if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count == rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ then 8 terminate rj 9 else if rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 and rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1 then 10 move along with rL through the updated value of rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and wait till rL leaves 11 increment the value of rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ 12 if no act settled robot visits with id lower than rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id then 13 Set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent = rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 14 move to the original position via rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 15 else 16 move to the original position via rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered 17 update rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The Algorithm 1 for the settled robots in stage 1 guarantees that the settled robots show their presence by back-and-forth movement to their one-hop neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, when the group of unsettled robots visits a node and wait for 2φ rounds, then they encounter the settled robot, if present, in its one-hop neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' And according to the Algorithm 2 for unsettled robots in stage 1, no unsettled robot settles if some settled robot meets the unsettled robot in some node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also according to Algorithm 5, the settled robots in stage 2 settle at nodes where they get settled in stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This guarantees that no two adjacent nodes are occupied by the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Multiple robots can settle only at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the robots complete the traversal of the graph and do not find any node to settle satisfying the conditions of the D-2-D problem, they finally reach the root of the graph to continue Algorithm 2 and traverse through the root of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots can easily recognize the root as the parent pointer of the 20 Algorithm 6: D-2-D with Termination 1 if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settle = 0 then 2 call Algorithm 2 3 else if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settle = 1 and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 0 then 4 call Algorithm 1 5 else if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settle = 1 and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 then 6 call Algorithm 4 7 else if ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settle = 1 and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 then 8 call Algorithm 5 root is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After following the algorithm from the root and subsequently exploring through all outgoing edges, robots backtrack to root only if they don’t find nodes to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In this case, they settle at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, algorithm 2 leads multiple robots to settle only at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If multiple robots settle at the root in stage 1, then it is guaranteed that each node is visited by a group of unsettled robots at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us suppose there is a node u which is not visited at all but at least one of its one-hop neighbors, say v, is visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This implies, that every time the group reached v, either it backtracked from v or it explored all the ports except the port joining v with u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The latter case is not possible as the Algorithm 2 increments the value of portentered unless its value is equal to the value of the parent pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now without loss of generality let us consider the case when v is visited by the group from the node that contains the lowest id settled robot among the ids of the settled robots at one-hop neighbors of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The Algorithm 2 ensures that the group of unsettled robots backtracks from v only when all the ports of v are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' And hence u must be explored and this is a contradiction to the existence of such a node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' By the end of the Algorithm 6, multiple robots settled at the root implies no vacant node left such that none of its neighbors contains a settled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us suppose there is a vacant node u in the graph such that no settled robot is present in any of its one-hop neighbors in the stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 1 proves that node u is visited at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' According to our algorithm for unsettled robots in stage 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Algorithm 2, when the group visited u, each of the robots rj in the group must set rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' During the waiting period, there were no settled robots in the neighbors of u to visit u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence the minimum id robot must have settled there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This contradicts the presence of such a node in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If multiple robots settle in the root, it follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 that the nodes with settled robots form a maximal independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' D-2-D with termination can be run by the robots with O(log ∆) additional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The variables ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='state, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='stage, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='settled, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special re- quires 1 bit of memory while ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist requires 2 bits of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The variables ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='parent, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='portentered and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='virtualparent requires O(log ∆) bits of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The settled robot at a node v with δ(v) ≤ ∆ can meet the group of unsettled robots at at most (∆ + 1) nodes including node v and there can be at most O(∆2) asso- ciated edges with these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Since the group of unsettled robots visits any edge at most 4 times, the variable ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count can take maximum value that is in O(∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Similarly, in stage 2 the act settled robot at a node v with δ(v) ≤ ∆ can meet rL at (∆ + 1) nodes and thus, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count′ can take maximum value that is in O(∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Therefore, O(log∆) is the amount of memory needed by the robots to store the information relating to these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As a result, each robot only needs O(log ∆) bits of additional memory to run the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When the group of unsettled robots in stage 1 are in explore state and ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='dist = 1 then there is exactly one settled robot present along with the group which has ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It is easy to observe this from the description of ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='special variable of a settled robot as mentioned in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As no node except the root contains multiple settled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also the root contains multiple robots only when no robots are left to settle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' no robot is in the explore state in stage 1 anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Every tree edge in stage 1 is traversed exactly twice by the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Without loss of generality, according to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1, let u has a settled robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' the tree edge (u, v) has either a settled robot at v, or a settled robot at u that stores the parent pointer for node v during the exploration of edge (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This ensures that v is visited for the first time as we have its parent pointer stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, the edge (u, v) is traversed twice once in the explore state and the next in the backtrack state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As mentioned in Algorithm 1, the parent pointer of node v is saved by robot ru settled at node u only when no robot visits v with id < ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence the robots do not backtrack from v with the objective of exploring node v from another node with lower id robot settled on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This proves the edge (u, v) is traversed exactly two times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Every non-tree edge is traversed at most four times by the group of unsettled robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let (u, v) be a non-tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' According to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1, the robots back- track from node v and the parent pointer for v is not yet stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Till this round, the edge (u, v) has been traversed twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robots reach v from the smallest id settled robot in its neighborhood to explore v later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' At that time edge (v, u) is traversed again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence, every non-tree edge is traversed at most four times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The graph induced by the tree edges is connected and cycle free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Consider a rooted configuration on a graph G with root u such that degree of u is at least 2 and also k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' First we show that the tree edges form a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It is easy to see that first two tree edges form a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', eh be the first h tree edges and they form a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let there be still a group of unsettled robots that is doing the traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let eh = uw for some nodes u, w such that the tree edge was formed when the group of unsettled robot visited w from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If ww′ becomes a tree edge for some neighbor w′ of w then we are done, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', the (h + 1)th tree edge also remains in the same connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, if no more associated edge of w becomes a tree edge, the group backtracks from w via a tree edge and reaches a new node v, say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Again, either an adjacent edge of v becomes a new tree edge (in which case we are done), or it backtracks through another tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' And in this way the group continues to stay on a path consisting of tree edges until if finds a new tree edge, or it completes the exploration and all the robots of the group settles at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Whatever be the case, the tree edges form a connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now we prove that the induced graph is cycle free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us assume on contrary that there is a cycle consisting of the tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', up, u1 be the cycle consisting of the tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', assume that uiui+1 be the last tree edge due to which cycle is formed and group of unsettled robots moved from ui to ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now we have two cases: either there is a settled robot at ui+1 or there is no settled robot at ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In case there is a settled robot at ui+1, then the group of unsettled robots should have done a backtrack from ui+1 to ui and hence uiui+1 can not be a tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This is a contradiction to our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, let us assume there is no settled robot at ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1 implies there will be settled robots both at ui and ui+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now, ui+1 is at one hop distance from these two settled robots and the exploration is being done from ui to ui+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Either of the two settled robots at ui and ui+2 have smaller id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If the settled robot at ui+2 has smaller id then the robots will backtrack from ui+1 to ui and thus uiui+1 will not be a tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' However, if the settled robot at ui has smaller id then while exploring the node ui+2 and traversing from ui+2 to ui+1, the group of unsettled robots must have backtracked due to presence of a smaller id settled robot at ui thus forming ui+2ui+1 as the non tree edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, we see that uiui+1 and ui+2ui+1 cannot be tree edges simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence, our assumption of the presence of a cycle consisting of all the tree edges is wrong and the graph induced by the tree edges is connected and cycle free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' By the time stage 2 finishes, each robot terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Since the robot rL with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='terminate = 1 replicates the group of unsettled robots in stage 1 and all the robots with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='act settled = 1 replicates the settled robots in stage 1, so, the number of times each settled robot meets with the group of unsettled robots in stage 1 is same as the number of times each act settled robot meets with rL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1, the stage 2 is replay of stage 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So the correctness of stage 1 implies the correctness of stage 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' And hence for each settled robot ri except rL, ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count = ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='count, and terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Finally, rL settles at the node where it settled at the end of stage 1 and terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 23 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The Algorithm 6 achieves D-2-D with termination in 2∆(8m−3n+ 3) rounds on arbitrary rooted graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It is clear from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='5 that every edge is traversed at most 4 times except the tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Also from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='6, there can be at most (n − 1) tree edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So the total number of edge traversal is no more than 4(m − (n − 1)) + 2(n − 1) = 4m − 2n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After each edge traversal, the robots wait for 2φ rounds and φ ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So at most 2∆(4m − 2n + 2) rounds are required for all the robots to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus Stage 1 is completed within 2∆(4m − 2n + 2) many rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' After the last robot settles, it may take at most 2∆(n − 1) rounds to reach the root node in the worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now, the remaining part of stage 2 is replica of the stage 1 of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, it takes 2∆(8m − 3n + 3) many rounds in order to achieve D-2-D with termination 4 Lower Bound In this section we discuss the lower bound on number of rounds of D-2-D problem considering robots do not have more than O(log ∆) additional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We start by defining view of a node to a robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' View: View of a node v to a robot is the information of whether there is a settled robot at any of its one hop neighbor or not, including v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Next we prove the theorem by constructing a class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The idea is that, each graph in the class is a regular graph of degree n−1 and has 2n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We start with two robots, one of which settles first and the other looks for a node to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The graphs are such that, unless the unsettled robot reaches two particular nodes, it will not be able to differentiate the graph with a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, before reaching one of those nodes, if it decides to settle, that will lead to a wrong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We show that, with limited memory, finding one of those nodes requires at least Ω(m∆) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The lower bound on number of rounds of D-2-D problem on arbi- trary graphs is Ω(m∆) considering robots have no more than O(log ∆) additional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We will prove this using a class of graphs where we show that there will be at least one graph for which the robots require at least ∆m 12 many rounds to complete D-2-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us consider two cliques of n vertices but with one edge missing from each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', vn be the vertices of the first clique Q1 and u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', un be the vertices of the second clique Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let v1v2 be the missing edge from the first clique and u1u2 be missing from the second clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We join v1 with u1 and v2 with u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now, the graph G has 2n nodes with ∆ = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Considering all possible different port-numbering of this graph gives us a graph class G which has cardinality equal to [(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=']2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let two robots r1 and r2 are initially present at vj where j ̸= 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us assume that there exists an algorithm A which solves D-2-D in time less than m∆ 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let r1 settles first and at node w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We can claim that 24 there will be at least |G | 2 graphs where, w /∈ {v1, v2, u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' let w = vi be some vertex of Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let us denote |G | 2 by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' As the robots have O(log ∆) memory, they can remember only a constant many port numbers at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We provide r2 more power by letting it know that there is a node to settle within two hop distance of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' The robot r2 aims to explore all the ∆(∆ − 1) many two hop neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' There are enough graphs(in particular, N 4 ) wherein the robot r2 needs to explore at least ∆(∆−1) 2 many vertices before exploring u1 or u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Unless it reaches u1 or u2 and has the view, r2 can not distinguish any graph of our graph class from a clique of n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Let the sequence in which the nodes are explored is as follows {vi1, vi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', vi ∆(∆−1) 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' When r2 reaches vi1, it needs to know the view of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' If vi1 is reached from vi directly, then getting the view takes only one round as r2 under- stands it is one hop away from vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Else, if vi1 is not reached directly from vi, then it is easy to see that, in at least half of the graphs, r2 needs at least ∆ 2 rounds to get the view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' So, there exists enough instances(in particular at least N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2) where r2 requires ∆ 2 rounds to find the view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Similarly, after reaching vi2 there exists at least N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='22 many graphs where ∆ 2 many rounds will be required to find the view of that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In similar fashion, at vi ∆(∆−1) 2 there exists at least N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 ∆(∆−1) 2 many graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Now N 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='2 ∆(∆−1) 2 is a function of n and the value becomes more than 1 for all n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Hence, there is at least one graph where robot r2 needs to spend at least ∆(∆−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ∆ 2 rounds to settle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' For n ≥ 3, ∆ ≥ M 3 where M = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Thus, ∆(∆−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ∆ 2 ≥ M 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' (∆−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ∆ 2 = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' ∆−1 6 ≥ m∆ 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This proves there is at least one such instance in the class G where the robot r2 requires m∆ 12 many rounds to complete D-2-D, else both r1 and r2 settles either on Q1 or on Q2 and this leads to wrong D-2-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 5 Conclusion and Future Work We propose a variant of the dispersion problem and provide an algorithm that solves it for the rooted initial configuration with O(log ∆) additional memory per robot and in 2∆(8m−3n+3) synchronous rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' We also provide a Ω(m∆) lower bound of the problem on number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In some cases, we guarantee forming a maximal independent set by the robots which can be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' It will be interesting to see how to solve the problem for arbitrary initial configuration of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' References [1] Ankush Agarwalla, John Augustine, William K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Moses Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', Sankar Madhav K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In ICDCN, pages 218–227, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [12] Ajay D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Kshemkalyani, Anisur Rahaman Molla, and Gokarna Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Dis- persion of mobile robots using global communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Parallel Distributed Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', 161:100–117, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [13] Ajay D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Kshemkalyani and Gokarna Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Near-optimal dispersion on ar- bitrary anonymous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In OPODIS, pages 8:1–8:19, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [14] Anisur Rahaman Molla and William K.' 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on an anonymous ring in the presence of weak byzantine robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In ALGOSENSORS, volume 12503, pages 154–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [16] Anisur Rahaman Molla, Kaushik Mondal, and William K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Moses Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Byzantine dispersion on graphs.' metadata={'source': 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Pattanayak, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Mandal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Filling MIS vertices by myopic luminous robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' CoRR, abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content='04885, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 26 [19] Masahiro Shibata, Toshiya Mega, Fukuhito Ooshita, Hirotsugu Kakugawa, and Toshimitsu Masuzawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Uniform deployment of mobile agents in asynchronous rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Parallel Distributed Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=', 119:92–106, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' [20] Takahiro Shintaku, Yuichi Sudo, Hirotsugu Kakugawa, and Toshimitsu Ma- suzawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' Efficient dispersion of mobile agents without global knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' In SSS, volume 12514, pages 280–294, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfLgwd/content/2301.04938v1.pdf'} diff --git a/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/2301.05165v1.pdf.txt b/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/2301.05165v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0111a8937c32aea9ee99050b0da52fa0878d8c01 --- /dev/null +++ b/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/2301.05165v1.pdf.txt @@ -0,0 +1,844 @@ +Optimal filtering techniques for the adaptive optics system of the LBT +G. Agapito, F. Quir´os-Pacheco, P. Tesi, A. Riccardi, S. Esposito +Abstract— In this paper we will discuss the application of +optimal filtering techniques for the adaptive optics system of +the LBT telescope. We have studied the application of both +Kalman and H∞ filters to estimate the temporal evolution of +the phase perturbations due to the atmospheric turbulence and +the telescope vibrations on tip/tilt modes. We will focus on the +H∞ filter and on its advantages and disadvantages over the +Kalman filter. +I. INTRODUCTION +The +Large +Binocular +Telescope +(LBT) +is +an +opti- +cal/infrared telescope using two 8.4m diameter primary mir- +rors. By having both primary mirrors on the same mechanical +mount, LBT will be able to achieve the diffraction-limited +image sharpness of a 22.8m diameter aperture. As in any +large ground-based telescope, the diffraction limit can only +be obtained with the assistance of adaptive optics (AO), +which is a technique aimed at reducing the effects of +wavefront distortion due to atmospheric turbulence [12]. +LBT will be equipped soon1 with two AO systems, one for +each arm of the telescope. Each AO unit (fig.1) comprises +a pyramid wavefront sensor (WFS), an adaptive secondary +mirror (ASM), and a real-time computer (RTC). The pyramid +wavefront sensor delivers a signal that is proportional, as +a first-order approximation, to the first derivative of the +incoming wavefront, sampled with a maximum of 30 × 30 +subapertures [14]. The ASM is a deformable mirror with +672 voice-coil (electro-magnetic force) actuators, distributed +in concentric rings, to change the shape of the 1.6mm-thick +and 911mm-diameter Zerodur shell [10]. +Large telescopes suffer from structure vibrations that can +reduce the AO performance [2]. Recent theoretical studies +and preliminary laboratory validations have shown that op- +timal control techniques can be used to reduce the impact +of these vibrations [8], [9]. We will present in this paper +an analysis of a mixed-control strategy for the LBT based +on both optimal filtering and classical control techniques, +aimed at reducing the impact of telescope vibrations without +burdening the RTC with heavy computations. In section II +G. Agapito, F. Quir´os-Pacheco, A. Riccardi, S. Esposito are with Osser- +vatorio Astrofisico di Arcetri, Largo E. Fermi 5, Firenze, Italy +P. Tesi is with Universit`a degli Studi di Firenze, Facolt`a di Ingegneria, +via Santa Marta 3, Firenze, Italy +G. Agapito: E-mail:agapito@arcetri.astro.it +F. Quir´os-Pacheco: E-mail:fquiros@arcetri.astro.it +1at the beginning of 2009 the first LBT-AO system will be commissioned +to the telescope. +Fig. 1. +Illustration of the optical configuration of one-arm of the LBT, +including the AO system components: the wavefront sensor (WFS), the +adaptive secondary mirror (ASM), and the real-time computer (RTC). +we will present the general control strategy for the LBT-AO +system. Section III describes the models required to design +the filter-based controllers. We have compared the perfor- +mance of these controllers based on numerical simulations. +These results will be presented in section V. +II. GENERAL CONTROL STRATEGY +The AO control diagram for the LBT is illustrated in +figure 2. The AO controller receives the WFS measurements +y(k) and computes the commands vector u(k) to drive the +actuators of the ASM. It is important to mention that the +ASM has non-negligible dynamics and, to compensate for +wavefront distortions, it must take the desired shape with +good accuracy and within a short settling time. For this +reason, it was chosen to control the ASM with a dedicated +control loop. The ASM control loop relies on the position +feedback provided by a set of capacitive sensors placed at the +back of the mirror shell. The design of the ASM controller +is based on a proportional-derivative (PD) position feedback +plus a feedforward signal that is proportional to the desired +position [11]. Summing up, there are two control systems +involved: +• A global AO control system (working @1kHz) whose +arXiv:2301.05165v1 [astro-ph.IM] 12 Jan 2023 + +Light beam +ASM * +AO RTC +WFS +Tertiary +mirror +Primary +mirrorFig. 2. +LBT-AO system control loop scheme. +goal is to determine the commands to the ASM that +corrects for a residual wavefront distortion; +• A local ASM control system (working @72kHz) with +the goal of shaping the mirror in the time of one AO +loop step (< 1ms). +The main subject of this paper regards the controller +for the global AO loop. We will follow the modal control +approach widely used in the analysis of AO systems [12]. +The controllers implemented in the current generation of AO +systems are based on classical (modal) integrators [4]. These +controllers have provided good performance on atmospheric +turbulence correction, but they have been unable to attenuate +substantially the effects of telescope structure vibrations. In +the LBT case, the swinging arm supporting the ASM has +resonance frequencies in the band between 15 and 30Hz [3]. +These vibrations affect mostly the tip/tilt modes2. As we will +discuss in section V, the vibration attenuation performed by +classical controllers is not enough to meet the expected AO +performance. For this reason it has been chosen to control +tip/tilt modes with a filter-based control. We will review in +section III the models of the AO system and of the input +signals (atmospheric turbulence, telescope vibrations, mea- +surement noise, etc.) required to define a system state vector +and estimate its evolution with an optimal filtering technique +such as Kalman or H∞ one (see Agapito et. al. 2008 [1] for +further considerations). +Finally, let us emphasize that the AO controller studied +in this work is a mixed controller (see fig. 2): tip/tilt modes +are controlled by a Kalman or H∞ filter-based controller +whereas the other modes are controlled with a simple in- +tegrator controller. The modal basis we chose was created +from Karhunen-Lo`eve modes [12] defined in the LBT pupil, +2Tip and tilt modal coefficients quantify the displacements of the image +in the two orthogonal directions. +projected onto the ASM influence functions and then re- +orthonormalized. A total of 672 modes (corresponding to the +total number of ASM actuators and hence, the total number +of degrees of freedom) were computed in this way. Finally, +tip/tilt modes were projected out from all modes in order to +decouple the control of tip/tilt and the rest of the modes for +the mixed-control strategy implementation. +III. AO SYSTEM MODEL +A. WFS and ASM models +The pyramid WFS model is described by: +y(k) = DΦres(k − 1) + w(k) +(1) +where y(k) is the measurement vector, w(k) is the mea- +surement noise vector —y(k) and w(k) ∈ Rq×1 where q +is the number of measurements—, D is the WFS response +matrix, and Φres(k) stands for the residual phase after ASM +correction computed as Φres(k) = Φtot(k)−Φcor(k), where +Φcor(k) is the phase correction applied by the ASM and +Φtot(k) = Φtur(k) + Φvib(k), i.e. the sum of the phase +distortions introduced by the turbulence Φtur(k) and the +telescope vibrations Φvib(k). All phase variables are modal +coefficient vectors Φ(k) ∈ Rn×1 where n is the number of +coefficients. +The ASM model can be expressed by: +Φcor(k − 1) = Nu(k − 2) +(2) +where N is the ASM influence matrix, and u(k) is the com- +mand vector for the actuators of the ASM —u(k) ∈ Rm×1, +where m is the number of actuators. Note that this equation +does not take into account for the mirror dynamics. However, +the AO command vector u(k) becomes the reference to the +ASM control loop and, as we mentioned above, this loop +guarantees that the ASM takes the desired shape. + +AOcontrollet +turbulcnccand +oa +Qtur (k) +vib +(k +Filter based +ASMcontrolloop +controller +tip&tilt +Modal +u(k) +D +(k) +Frommodes +ASM +reconstruction +tocommands +other +modes +Integrator +controller +ASM +Control +Pyramid +y(k) +WFS +w(k)Fig. 3. +Classical control strategy scheme. +B. Turbulence and vibration models +The atmospheric turbulence evolution can be described by: +Φtur(k + 1) = f(Φtur(k), Φtur(k − 1), . . .) + v(k) +(3) +where v(k) is the model’s white noise. We have chosen +to approximate this equation with an Auto-Regressive (AR) +first-order model [6]: +Φtur(k + 1) = AtΦtur(k) + vt(k) +(4) +where +At +is +a +diagonal +matrix +calculated +as +in +Le Roux et. al. 2004 [13], whose diagonal elements +are e−2π0.3ηV/f (η radial order, V wind speed, f sampling +frequency), and vt(k) is the model’s white noise calculated +from the Noll matrix [7]. +The vibrations model can be expressed as: +Φvib(k + 1) = A1Φvib(k) − A2Φvib(k − 1) + vv(k) +(5) +where A1 and A2 are two diagonal matrices whose diagonal +elements depend upon vibration frequency and damping +constant, and vv(k) is a white noise vector whose variance +depends upon input force power [8]. Φvib(k) and vv(k) have +p non-zero elements corresponding to the modes affected by +vibrations. +C. Classical control strategy +The classical control strategy (see figure 3) is based on a +reconstruction matrix R and on a simple integrator: +u(k) = u(k − 1) + g∆u(k − 2) +(6) +where g is the integrator gain (equal for all modes), and the +command increment is computed as: +∆u(k) = Ry(k) . +(7) +The +reconstruction +matrix +R +is +computed +as +R += +(M ′ +intMint)−1M ′ +int, that is, the generalized inverse of the +interaction matrix Mint = DN, measured experimentally +during the AO system calibration. This is the control applied +to all modes except for tip/tilt in the mixed-control strategy. +We should note that the gain g can be optimized for each +mode, as in the case of the optimized modal gain integrator +(OMGI) [4] controller. In this work we did not implement +it, but this will be considered as a future improvement. +Fig. 4. +Filter-based control strategy scheme (K is the filter asymptotic gain +matrix). +D. Filter-based control strategy +The control based on the Kalman or the H∞ filter (see +figure 4) generates the command vector from the predicted +state vector. We have defined the following state vector: +x(k) = +� +����� +Φvib(k) +Φvib(k − 1) +Φtur(k) +Φtot(k − 1) +u(k − 2) +� +����� +(8) +comprising all the variables required to estimate the total +phase vector Φtot(k + 1). The dimension of the state vector +is (2p+2n+m)×1. It turns out that the command vector u(k) +is computed by projecting ˆΦtot(k + 1) onto the command +space: +u(k) += (N ′N)−1N ′F ˆx(k + 1) += (N ′N)−1N ′ ˆΦtot(k + 1) , +(9) +where F = +� I +0 +I +0 +0 � +and I is the identity matrix. +For the Kalman filter, the state model is expressed as: +x(k + 1) += +� +����� +A1 +A2 +0 +0 +0 +I +0 +0 +0 +0 +0 +0 +At +0 +0 +I +0 +I +0 +0 +0 +0 +0 +0 +0 +� +����� +� +�� +� +A +x(k)+ +� +����� +0 +0 +0 +0 +I +� +����� +� �� � +B +u(k − 1) + v(k) , +(10) +where v(k) = +� +vv(k) +0 +vt(k) +0 +0 +� +. Finally, the +measurement equation is expressed as: +y(k) += D +� 0 +0 +0 +I +−N � +� +�� +� +C +x(k) + w(k) . +(11) + +y(k) +u(k) +R +9父(k) +A +F +u(k) +K +父(k+1) +y(k) +BOn the other hand, the state model for the H∞ filter is +expressed as: +� +� +x(k + 1) +ˆz(k) − z(k) +y(k) +� +� = +� +��������� +A1 +A2 +0 +0 +0 +I +0 +0 +0 +I +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +At +0 +0 +0 +I +0 +0 +I +0 +I +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−I +0 +−I +0 +0 +0 +0 +0 +I +0 +0 +0 +D +−DN +0 +0 +I +0 +� +��������� +� +� +x(k) +µ(k) +ˆz(k) +� +� ++ +� +��������� +0 +0 +0 +0 +I +0 +0 +� +��������� +u(k − 1) , +(12) +with x(0) = 0. Also, z(k) = Φtot(k + 1), ˆz(k) is an +estimate of z(k), and µ(k) is the disturbances vector µ(k) = +� vv(k) +vt(k) +w(k) �′. +IV. FILTERS +We decided to implement the Kalman filter because it is +the best linear state estimator and the H∞ filter because it +is capable of dealing with plant errors and unknown distur- +bances. Kalman and H∞ filters have different objectives: +• the Kalman filter’s aim is to minimize either the vari- +ance of the final state estimation error: +J1 = ε [(ˆx(N) − x(N))′(ˆx(N) − x(N))] , +(13) +or to minimize the average RMS power of the estima- +tion error [5]: +J2 = ε +� +� 1 +N +N +� +k=0 +(ˆx(k) − x(k))′ (ˆx(k) − x(k)) +� +� +1 +2 +(14) +where ε[·] denotes the expected value, and x(N) denotes +the final state; +• the H∞ filter’s aim is to ensure that the energy gain +from the disturbances to the estimation error is less than +a prespecified level γ2 [5]: +∥ˆz − z∥2 +2,[0,N] − γ2∥µ∥2 +2,[0,N] ≤ −ϵ∥µ∥2 +2,[0,N] +(15) +where ϵ > 0, µ ∈ l2[0, N] is the disturbances vector3, +3The space l2[0, N] is defined as: +l2[0, N] = � +f : f(k) = 0 ∀ k /∈ [0, N], ∥f∥2,[0,N] < ∞� +, +z = Lx, and ˆz = Fy is a z estimate (F must be casual +and linear). +Kalman and H∞ filters use different problem descriptions +too: +• for Kalman filter the signal generating system is as- +sumed to be a state-space system driven by a white noise +process with known statistical properties. The observed +output is also corrupted by a white noise process with +known statistical properties; +• for H∞ filter the system has unknown disturbances of +finite energy that drive the signal generating system and +corrupt the observations. +V. SIMULATIONS AND DATA ANALYSIS +A. Preliminary considerations +Telescope structure vibrations may exhibit large ampli- +tudes, in particular on tip/tilt modes, and they depend on +many factors such as telescope orientation, telescope tracking +errors, and wind shaking. +The LBT relies on a set of accelerometers placed on the +structure supporting the ASM to characterize the vibrations +(frequency and amplitude) affecting the AO system. In this +work we did not consider an adaptive controller, so the +vibration’s parameters will be previously calibrated with the +accelerometers and used to build the Kalman (or H∞) filter. +For these reasons it is important to study the robustness of +the controllers with respect to errors in the vibration’s model. +We will first compare the performance of the three con- +trollers (classical, mixed-Kalman, and mixed-H∞) under +the presence of only atmospheric turbulence (sec. V-B). +Then, we will consider the presence of a telescope vibration +affecting tip-tilt modes (sec. V-C). Finally, we will study the +robustness of the mixed-controllers with respect to changes +on the vibration frequency (sec. V-D). +All the simulations were made on an end-to-end simulator +of the LBT-AO system. Table I presents a summary of the +simulation parameters. +B. Performance under the presence of turbulence +First, let us consider that there are no vibrations. In this +case, mixed-Kalman controller gives a SR4 of 80.7%, the +mixed-H∞ controller a SR of 80.4%, and the classical con- +troller a SR of 84.1% (Table II). Note that the performance +where ∥ · ∥2,[0,N] is the finite-horizon 2-norm, defined as: +∥f∥2,[0,N] = +� N +� +k=0 +f′(k)f(k) +� 1 +2 +, +where f = {f(k)}∞ +−∞. +4To measure the performance of an AO system we use the Strehl Ratio +(SR). It is the ratio of the observed peak intensity at the detection plane +compared to the theoretical maximum peak intensity of a diffraction-limited +image. + +Telescope +Effective diameter (D) +8.22m +Central obstruction +0.11D +Pyramid WFS +Sensing wavelength (λ) +0.75µm +Tilt modulation radius +4.0 λ +D +Number of subapertures +30 × 30 +Number of photons per integration time per subaperture +50 +Number of electrons per pixel of readout noise +8 +ASM +Number of modes +672 +Turbulence +Seeing +0.8 (@ 0.5µm) +Outer scale (L0) +22m +Wind speed +20m/s +Loop parameters +Sampling frequency +800Hz +Total delay +2 frames +TABLE I +SUMMARY TABLE OF SIMULATION PARAMETERS. +%SR @ 2.2µm +vibration +Classical +mixed-Kalman +mixed-H∞ +No +84.1 +80.7 +80.4 +Yes +30.9 +80.4 +80.2 +TABLE II +SIMULATION RESULTS: PERFORMANCE OF THE LBT-AO SYSTEM WITH +AND WITHOUT TELESCOPE VIBRATIONS. +of the mixed controllers is slightly lower with respect to the +classical one because the AR1 dynamic model of the turbu- +lence is a simple one; it is just a first-order approximation of +the Taylor’s hypothesis model of the turbulence’s temporal +evolution [13]. +C. Performance under the presence of turbulence and vibra- +tions +Let us now consider the case where there is a telescope vi- +bration affecting tip/tilt modes with an amplitude of 80 milli- +arcseconds at a frequency of 20Hz. Under these conditions, +mixed-Kalman controller provides a SR of 80.4%, and +mixed-H∞ controller a SR of 80.2%. Their performances +are very similar to the ones obtained in the previous case. +On the other hand, the classical controller has a very different +performance; the SR has been reduced to 30.9% under the +presence of this vibration (Table II). We should note that +this result was obtained by increasing the integrator’s gain in +order to increase the attenuation at the vibration’s frequency. +Of course, the gain cannot be increased arbitrarily due to +stability constraints. Therefore, the AO performance with the +classical controller will remain limited by the presence of +telescope vibrations. +D. Robustness study +In order to test the robustness of the controllers based on +the Kalman and the H∞ filters, we introduced an error on +the value of the vibration’s frequency in the state model, +whereas the actual vibration’s frequency was left equal to +20Hz. From figure 5 (and table III) we can see that the two +filters have very similar performance when the error on the +frequency (Hz) +%SR @ 2.2µm +model +error +mixed-Kalman +mixed-H∞ +16.5 +-3.5 +22.4 +30.9 +17 +-3 +25.5 +36.4 +17.5 +-2.5 +30.6 +43.7 +18 +-2 +39.1 +52.7 +18.5 +-1.5 +51.0 +62.8 +19 +-1 +65 +71.3 +19.5 +-0.5 +76.8 +78.4 +20 +0 +80.4 +80.2 +20.5 +0.5 +77.8 +78.4 +21 +1 +70.1 +74.1 +21.5 +1.5 +61 +67.6 +22 +2 +52.8 +60.7 +22.5 +2.5 +46.1 +54.5 +23 +3 +40.9 +48.9 +23.5 +3.5 +36.9 +44.6 +TABLE III +STREHL RATIO VALUES SHOWN IN FIGURE 5. +frequency is less than |0.5Hz|. When the error is greater +than |1Hz| the performance of the mixed-H∞ is ≈ 10% in +SR better than the mixed-Kalman controller. Note that the +SR of the classical controller with this vibration is lower +than the mixed controllers almost for every considered error +values. +These simulation results can also be explained by looking +at the corresponding sensitivity functions. Figure 6 represents +the maximal singular values of the transfer functions between +disturbances and estimation error5. From this figure we can +see that the Kalman filter estimation sensitivity functions in +correspondence of the vibration frequency have a peak. This +means that the Kalman filter is more sensitive to disturbances +around this frequency, and that model errors around this +frequency will have a greater influence on the estimation. +Instead, the H∞ filter is characterized by flatter sensitivity +functions. Hence, this filter should be more robust to errors +on the vibration’s frequency value, as has been shown with +numerical simulations above. +5We trace this graph and not all the sensitivity functions for a simpler +and better comprehension - the sensitivity functions are n × m. + +Fig. 5. +Robustness study: performance of the mixed controllers under the +presence of model errors regarding the vibration’s frequency. +Fig. 6. +Singular values of the disturbances - esitmation error transfer +function for Kalman and H∞ filters. +VI. CONCLUSIONS AND FURTHER WORK +We have presented in this work a mixed-control strategy +combining classical and filter-based techniques for the LBT- +AO system. We have shown with numerical simulations +that the mixed controllers are able to effectively eliminate +the effects of telescope’s structure vibrations on the AO +performance. In order to achieve this, it is crucial to charac- +terize accurately the vibration parameters, in particular the +vibration’s frequency value. We have verified that the H∞ +filter is more robust than the Kalman filter with respect +to uncertainties on the vibration’s frequency value. For the +particular parameters simulated in this work, an absolute loss +of 10% of SR at 2.2µm is expected in the presence of a +frequency error of ±1.2Hz and ±0.9Hz in the vibration’s +model for the H∞ and the Kalman filter respectively. +We should note that more than one vibration frequencies +can be taken into account straightforwardly by extending the +model and the state vector. As a next step, we will implement +the mixed-control strategy in a test bench based on the +real-time computer of the LBT-AO system. We should note +that the mixed-control strategy can be implemented without +changes on the existing hardware and firmware. +REFERENCES +[1] G. Agapito, F. Quiros-Pacheco, P. Tesi, S. Esposito, and M. Xompero, +“Optimal control techniques for the adaptive optics system of the +LBT,” Proceedings of SPIE, vol. 7015, no. 123, 2008. +[2] Y. Cl´enet, M. Kasper, N. Ageorges, C. Lidman, T. Fusco, O. P. Marco, +M. Hartung, D. Mouillet, B. Koehler, G. Rousset, and N. Hubin, +“NAOS performances: impact of the telescope vibrations and possible +origins,” SF2A, p. p179, 2004. +[3] D. Gallieni, “F/15 Adaptive secondary mechanical Design,” LBT +Project technical Report, Tech. Rep. 640a005 F, 22 August 2007. +[4] E. Gendron and P. Lena, “Astronomical adaptive optics. 1. Modal +control optimization,” Astronomy and Astrophysics, vol. 291, no. 1, +pp. 337–347, 1994. +[5] M. Green and D. J. Limebeer, Linear Robust Control. +Prentice Hall, +1994. +[6] L. Ljung, System Identification - Theory For the User, 2nd ed. Upper +Saddle River, N.J.: PTR Prentice Hall, 1999. +[7] R. J. Noll, “Zernike polynomials and atmospheric turbulence,” Opt. +Soc. Am., vol. 66, pp. 207–211, 1976. +[8] C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, and T. Fusco, “First +laboratory validation of vibration filtering with LQG control law for +Adaptive Optics,” Optics Express, 2008. +[9] C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcs´ar, H.-F. Raynaud, +T. Fusco, and G. Rousset, “Kalman filter based control for adaptive +optics,” Proceedings of SPIE, vol. 5490, pp. 1414–1425, 2004. +[10] A. Riccardi, M. Xompero, D. Zanotti, L. Busoni, C. D. Vecchio, +P. Salinari, P. Ranfagni, G. B. Zappellini, R. Biasi, M. Andrighettoni, +D. Gallieni, E. Anaclerio, H. M. Martin, and S. M. Miller, “Adap- +tive secondary mirror for the Large Binocular Telescope: results of +acceptance laboratory test,” Proceedings of SPIE, vol. 7015, no. 37, +2008. +[11] A. Riccardi, G. Brusa, P. Salinari, D. Gallieni, R. Biasi, M. Andrighet- +toni, and H. M. Martin, “Adaptive secondary mirrors for the Large +Binocular Telescope,” Proceedings of SPIE, vol. 4839, pp. 721–732, +2003. +[12] F. Roddier, Adaptive optics in astronomy. +Cambridge, U.K: Cam- +bridge University Press, 1999. +[13] B. L. Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, +and T. Fusco, “Optimal control law for classical and multiconjugate +adaptive optics,” JOSAA, vol. 21, no. 7, pp. 1261–1276, 2004. +[14] A. Tozzi, P.Stefanini, E. Pinna, and S.Esposito, “The double pyramid +wavefront sensor for LBT,” Proceedings of SPIE, vol. 7015, no. 190, +2008. + +90 +80 - +70 - +60 - +(%） +Strehi ratio +50 +-Kalman +Hinfinity +integrator +40 +30 +20 +10 +-3,5 +-3 +-2.5 +-2 +-1,5 +-0'.5 +0 +0,5 +1'5 +2 +2,5 +3 +3'5 +error (Hz)SingularValues +10 +10 +Kalman +Ho +12 +10° +10 +102 +10° +Freguency (rad/sec) \ No newline at end of file diff --git a/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/load_file.txt b/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77d6cd4a4a8bb45cc41d53802fffaa6b95fb9c9c --- /dev/null +++ b/TNE4T4oBgHgl3EQfmA2x/content/tmp_files/load_file.txt @@ -0,0 +1,387 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf,len=386 +page_content='Optimal filtering techniques for the adaptive optics system of the LBT G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Agapito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Quir´os-Pacheco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Tesi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Riccardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Esposito Abstract— In this paper we will discuss the application of optimal filtering techniques for the adaptive optics system of the LBT telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have studied the application of both Kalman and H∞ filters to estimate the temporal evolution of the phase perturbations due to the atmospheric turbulence and the telescope vibrations on tip/tilt modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We will focus on the H∞ filter and on its advantages and disadvantages over the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' INTRODUCTION The Large Binocular Telescope (LBT) is an opti- cal/infrared telescope using two 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4m diameter primary mir- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' By having both primary mirrors on the same mechanical mount, LBT will be able to achieve the diffraction-limited image sharpness of a 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8m diameter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' As in any large ground-based telescope, the diffraction limit can only be obtained with the assistance of adaptive optics (AO), which is a technique aimed at reducing the effects of wavefront distortion due to atmospheric turbulence [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' LBT will be equipped soon1 with two AO systems, one for each arm of the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Each AO unit (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1) comprises a pyramid wavefront sensor (WFS), an adaptive secondary mirror (ASM), and a real-time computer (RTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The pyramid wavefront sensor delivers a signal that is proportional, as a first-order approximation, to the first derivative of the incoming wavefront, sampled with a maximum of 30 × 30 subapertures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The ASM is a deformable mirror with 672 voice-coil (electro-magnetic force) actuators, distributed in concentric rings, to change the shape of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='6mm-thick and 911mm-diameter Zerodur shell [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Large telescopes suffer from structure vibrations that can reduce the AO performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Recent theoretical studies and preliminary laboratory validations have shown that op- timal control techniques can be used to reduce the impact of these vibrations [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We will present in this paper an analysis of a mixed-control strategy for the LBT based on both optimal filtering and classical control techniques, aimed at reducing the impact of telescope vibrations without burdening the RTC with heavy computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In section II G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Agapito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Quir´os-Pacheco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Riccardi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Esposito are with Osser- vatorio Astrofisico di Arcetri, Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Fermi 5, Firenze, Italy P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Tesi is with Universit`a degli Studi di Firenze, Facolt`a di Ingegneria, via Santa Marta 3, Firenze, Italy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Agapito: E-mail:agapito@arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='it F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Quir´os-Pacheco: E-mail:fquiros@arcetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='it 1at the beginning of 2009 the first LBT-AO system will be commissioned to the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Illustration of the optical configuration of one-arm of the LBT, including the AO system components: the wavefront sensor (WFS), the adaptive secondary mirror (ASM), and the real-time computer (RTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' we will present the general control strategy for the LBT-AO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Section III describes the models required to design the filter-based controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have compared the perfor- mance of these controllers based on numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' These results will be presented in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' GENERAL CONTROL STRATEGY The AO control diagram for the LBT is illustrated in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The AO controller receives the WFS measurements y(k) and computes the commands vector u(k) to drive the actuators of the ASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' It is important to mention that the ASM has non-negligible dynamics and, to compensate for wavefront distortions, it must take the desired shape with good accuracy and within a short settling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' For this reason, it was chosen to control the ASM with a dedicated control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The ASM control loop relies on the position feedback provided by a set of capacitive sensors placed at the back of the mirror shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The design of the ASM controller is based on a proportional-derivative (PD) position feedback plus a feedforward signal that is proportional to the desired position [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Summing up, there are two control systems involved: A global AO control system (working @1kHz) whose arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='05165v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='IM] 12 Jan 2023 Light beam ASM * AO RTC WFS Tertiary mirror Primary mirrorFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' LBT-AO system control loop scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' goal is to determine the commands to the ASM that corrects for a residual wavefront distortion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' A local ASM control system (working @72kHz) with the goal of shaping the mirror in the time of one AO loop step (< 1ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The main subject of this paper regards the controller for the global AO loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We will follow the modal control approach widely used in the analysis of AO systems [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The controllers implemented in the current generation of AO systems are based on classical (modal) integrators [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' These controllers have provided good performance on atmospheric turbulence correction, but they have been unable to attenuate substantially the effects of telescope structure vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In the LBT case, the swinging arm supporting the ASM has resonance frequencies in the band between 15 and 30Hz [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' These vibrations affect mostly the tip/tilt modes2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' As we will discuss in section V, the vibration attenuation performed by classical controllers is not enough to meet the expected AO performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' For this reason it has been chosen to control tip/tilt modes with a filter-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We will review in section III the models of the AO system and of the input signals (atmospheric turbulence, telescope vibrations, mea- surement noise, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=') required to define a system state vector and estimate its evolution with an optimal filtering technique such as Kalman or H∞ one (see Agapito et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 2008 [1] for further considerations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Finally, let us emphasize that the AO controller studied in this work is a mixed controller (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 2): tip/tilt modes are controlled by a Kalman or H∞ filter-based controller whereas the other modes are controlled with a simple in- tegrator controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The modal basis we chose was created from Karhunen-Lo`eve modes [12] defined in the LBT pupil, 2Tip and tilt modal coefficients quantify the displacements of the image in the two orthogonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' projected onto the ASM influence functions and then re- orthonormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' A total of 672 modes (corresponding to the total number of ASM actuators and hence, the total number of degrees of freedom) were computed in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Finally, tip/tilt modes were projected out from all modes in order to decouple the control of tip/tilt and the rest of the modes for the mixed-control strategy implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' AO SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' WFS and ASM models The pyramid WFS model is described by: y(k) = DΦres(k − 1) + w(k) (1) where y(k) is the measurement vector, w(k) is the mea- surement noise vector —y(k) and w(k) ∈ Rq×1 where q is the number of measurements—, D is the WFS response matrix, and Φres(k) stands for the residual phase after ASM correction computed as Φres(k) = Φtot(k)−Φcor(k), where Φcor(k) is the phase correction applied by the ASM and Φtot(k) = Φtur(k) + Φvib(k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' the sum of the phase distortions introduced by the turbulence Φtur(k) and the telescope vibrations Φvib(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' All phase variables are modal coefficient vectors Φ(k) ∈ Rn×1 where n is the number of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The ASM model can be expressed by: Φcor(k − 1) = Nu(k − 2) (2) where N is the ASM influence matrix, and u(k) is the com- mand vector for the actuators of the ASM —u(k) ∈ Rm×1, where m is the number of actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Note that this equation does not take into account for the mirror dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' However, the AO command vector u(k) becomes the reference to the ASM control loop and, as we mentioned above, this loop guarantees that the ASM takes the desired shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' AOcontrollet turbulcnccand oa Qtur (k) +vib (k Filter based ASMcontrolloop controller tip&tilt Modal u(k) D (k) Frommodes ASM reconstruction tocommands other modes Integrator controller ASM Control Pyramid y(k) WFS w(k)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Classical control strategy scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Turbulence and vibration models The atmospheric turbulence evolution can be described by: Φtur(k + 1) = f(Φtur(k), Φtur(k − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=') + v(k) (3) where v(k) is the model’s white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have chosen to approximate this equation with an Auto-Regressive (AR) first-order model [6]: Φtur(k + 1) = AtΦtur(k) + vt(k) (4) where At is a diagonal matrix calculated as in Le Roux et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 2004 [13], whose diagonal elements are e−2π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='3ηV/f (η radial order, V wind speed, f sampling frequency), and vt(k) is the model’s white noise calculated from the Noll matrix [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The vibrations model can be expressed as: Φvib(k + 1) = A1Φvib(k) − A2Φvib(k − 1) + vv(k) (5) where A1 and A2 are two diagonal matrices whose diagonal elements depend upon vibration frequency and damping constant, and vv(k) is a white noise vector whose variance depends upon input force power [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Φvib(k) and vv(k) have p non-zero elements corresponding to the modes affected by vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Classical control strategy The classical control strategy (see figure 3) is based on a reconstruction matrix R and on a simple integrator: u(k) = u(k − 1) + g∆u(k − 2) (6) where g is the integrator gain (equal for all modes), and the command increment is computed as: ∆u(k) = Ry(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' (7) The reconstruction matrix R is computed as R = (M ′ intMint)−1M ′ int, that is, the generalized inverse of the interaction matrix Mint = DN, measured experimentally during the AO system calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' This is the control applied to all modes except for tip/tilt in the mixed-control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We should note that the gain g can be optimized for each mode, as in the case of the optimized modal gain integrator (OMGI) [4] controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In this work we did not implement it, but this will be considered as a future improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Filter-based control strategy scheme (K is the filter asymptotic gain matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Filter-based control strategy The control based on the Kalman or the H∞ filter (see figure 4) generates the command vector from the predicted state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have defined the following state vector: x(k) = � ����� Φvib(k) Φvib(k − 1) Φtur(k) Φtot(k − 1) u(k − 2) � ����� (8) comprising all the variables required to estimate the total phase vector Φtot(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The dimension of the state vector is (2p+2n+m)×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' It turns out that the command vector u(k) is computed by projecting ˆΦtot(k + 1) onto the command space: u(k) = (N ′N)−1N ′F ˆx(k + 1) = (N ′N)−1N ′ ˆΦtot(k + 1) , (9) where F = � I 0 I 0 0 � and I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' For the Kalman filter, the state model is expressed as: x(k + 1) = � ����� A1 A2 0 0 0 I 0 0 0 0 0 0 At 0 0 I 0 I 0 0 0 0 0 0 0 � ����� � �� � A x(k)+ � ����� 0 0 0 0 I � ����� � �� � B u(k − 1) + v(k) , (10) where v(k) = � vv(k) 0 vt(k) 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Finally, the measurement equation is expressed as: y(k) = D � 0 0 0 I −N � � �� � C x(k) + w(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' (11) y(k) u(k) R 9父(k) A F u(k) K 父(k+1) y(k) BOn the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' the state model for the H∞ filter is expressed as: � � x(k + 1) ˆz(k) − z(k) y(k) � � = � ��������� A1 A2 0 0 0 I 0 0 0 I 0 0 0 0 0 0 0 0 0 0 At 0 0 0 I 0 0 I 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −I 0 −I 0 0 0 0 0 I 0 0 0 D −DN 0 0 I 0 � ��������� � � x(k) µ(k) ˆz(k) � � + � ��������� 0 0 0 0 I 0 0 � ��������� u(k − 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' (12) with x(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Also, z(k) = Φtot(k + 1), ˆz(k) is an estimate of z(k), and µ(k) is the disturbances vector µ(k) = � vv(k) vt(k) w(k) �′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' FILTERS We decided to implement the Kalman filter because it is the best linear state estimator and the H∞ filter because it is capable of dealing with plant errors and unknown distur- bances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Kalman and H∞ filters have different objectives: the Kalman filter’s aim is to minimize either the vari- ance of the final state estimation error: J1 = ε [(ˆx(N) − x(N))′(ˆx(N) − x(N))] , (13) or to minimize the average RMS power of the estima- tion error [5]: J2 = ε � � 1 N N � k=0 (ˆx(k) − x(k))′ (ˆx(k) − x(k)) � � 1 2 (14) where ε[·] denotes the expected value, and x(N) denotes the final state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' the H∞ filter’s aim is to ensure that the energy gain from the disturbances to the estimation error is less than a prespecified level γ2 [5]: ∥ˆz − z∥2 2,[0,N] − γ2∥µ∥2 2,[0,N] ≤ −ϵ∥µ∥2 2,[0,N] (15) where ϵ > 0, µ ∈ l2[0, N] is the disturbances vector3, 3The space l2[0, N] is defined as: l2[0, N] = � f : f(k) = 0 ∀ k /∈ [0, N], ∥f∥2,[0,N] < ∞� , z = Lx, and ˆz = Fy is a z estimate (F must be casual and linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Kalman and H∞ filters use different problem descriptions too: for Kalman filter the signal generating system is as- sumed to be a state-space system driven by a white noise process with known statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The observed output is also corrupted by a white noise process with known statistical properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' for H∞ filter the system has unknown disturbances of finite energy that drive the signal generating system and corrupt the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' SIMULATIONS AND DATA ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Preliminary considerations Telescope structure vibrations may exhibit large ampli- tudes, in particular on tip/tilt modes, and they depend on many factors such as telescope orientation, telescope tracking errors, and wind shaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' The LBT relies on a set of accelerometers placed on the structure supporting the ASM to characterize the vibrations (frequency and amplitude) affecting the AO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In this work we did not consider an adaptive controller, so the vibration’s parameters will be previously calibrated with the accelerometers and used to build the Kalman (or H∞) filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' For these reasons it is important to study the robustness of the controllers with respect to errors in the vibration’s model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We will first compare the performance of the three con- trollers (classical, mixed-Kalman, and mixed-H∞) under the presence of only atmospheric turbulence (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' V-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Then, we will consider the presence of a telescope vibration affecting tip-tilt modes (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' V-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Finally, we will study the robustness of the mixed-controllers with respect to changes on the vibration frequency (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' V-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' All the simulations were made on an end-to-end simulator of the LBT-AO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Table I presents a summary of the simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Performance under the presence of turbulence First, let us consider that there are no vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In this case, mixed-Kalman controller gives a SR4 of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='7%, the mixed-H∞ controller a SR of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4%, and the classical con- troller a SR of 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1% (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Note that the performance where ∥ · ∥2,[0,N] is the finite-horizon 2-norm, defined as: ∥f∥2,[0,N] = � N � k=0 f′(k)f(k) � 1 2 , where f = {f(k)}∞ −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 4To measure the performance of an AO system we use the Strehl Ratio (SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' It is the ratio of the observed peak intensity at the detection plane compared to the theoretical maximum peak intensity of a diffraction-limited image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Telescope Effective diameter (D) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='22m Central obstruction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='11D Pyramid WFS Sensing wavelength (λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='75µm Tilt modulation radius 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='0 λ D Number of subapertures 30 × 30 Number of photons per integration time per subaperture 50 Number of electrons per pixel of readout noise 8 ASM Number of modes 672 Turbulence Seeing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8 (@ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5µm) Outer scale (L0) 22m Wind speed 20m/s Loop parameters Sampling frequency 800Hz Total delay 2 frames TABLE I SUMMARY TABLE OF SIMULATION PARAMETERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' %SR @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2µm vibration Classical mixed-Kalman mixed-H∞ No 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 Yes 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2 TABLE II SIMULATION RESULTS: PERFORMANCE OF THE LBT-AO SYSTEM WITH AND WITHOUT TELESCOPE VIBRATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' of the mixed controllers is slightly lower with respect to the classical one because the AR1 dynamic model of the turbu- lence is a simple one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' it is just a first-order approximation of the Taylor’s hypothesis model of the turbulence’s temporal evolution [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Performance under the presence of turbulence and vibra- tions Let us now consider the case where there is a telescope vi- bration affecting tip/tilt modes with an amplitude of 80 milli- arcseconds at a frequency of 20Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Under these conditions, mixed-Kalman controller provides a SR of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4%, and mixed-H∞ controller a SR of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Their performances are very similar to the ones obtained in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' On the other hand, the classical controller has a very different performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' the SR has been reduced to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9% under the presence of this vibration (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We should note that this result was obtained by increasing the integrator’s gain in order to increase the attenuation at the vibration’s frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Of course, the gain cannot be increased arbitrarily due to stability constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Therefore, the AO performance with the classical controller will remain limited by the presence of telescope vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Robustness study In order to test the robustness of the controllers based on the Kalman and the H∞ filters, we introduced an error on the value of the vibration’s frequency in the state model, whereas the actual vibration’s frequency was left equal to 20Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' From figure 5 (and table III) we can see that the two filters have very similar performance when the error on the frequency (Hz) %SR @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2µm model error mixed-Kalman mixed-H∞ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9 17 3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='7 18 2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8 19 1 65 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 20 0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='4 21 1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 61 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='6 22 2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 23 3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='6 TABLE III STREHL RATIO VALUES SHOWN IN FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' frequency is less than |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='5Hz|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' When the error is greater than |1Hz| the performance of the mixed-H∞ is ≈ 10% in SR better than the mixed-Kalman controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Note that the SR of the classical controller with this vibration is lower than the mixed controllers almost for every considered error values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' These simulation results can also be explained by looking at the corresponding sensitivity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Figure 6 represents the maximal singular values of the transfer functions between disturbances and estimation error5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' From this figure we can see that the Kalman filter estimation sensitivity functions in correspondence of the vibration frequency have a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' This means that the Kalman filter is more sensitive to disturbances around this frequency, and that model errors around this frequency will have a greater influence on the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Instead, the H∞ filter is characterized by flatter sensitivity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Hence, this filter should be more robust to errors on the vibration’s frequency value, as has been shown with numerical simulations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 5We trace this graph and not all the sensitivity functions for a simpler and better comprehension - the sensitivity functions are n × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Robustness study: performance of the mixed controllers under the presence of model errors regarding the vibration’s frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' Singular values of the disturbances - esitmation error transfer function for Kalman and H∞ filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' CONCLUSIONS AND FURTHER WORK We have presented in this work a mixed-control strategy combining classical and filter-based techniques for the LBT- AO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have shown with numerical simulations that the mixed controllers are able to effectively eliminate the effects of telescope’s structure vibrations on the AO performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' In order to achieve this, it is crucial to charac- terize accurately the vibration parameters, in particular the vibration’s frequency value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We have verified that the H∞ filter is more robust than the Kalman filter with respect to uncertainties on the vibration’s frequency value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' For the particular parameters simulated in this work, an absolute loss of 10% of SR at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2µm is expected in the presence of a frequency error of ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='2Hz and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content='9Hz in the vibration’s model for the H∞ and the Kalman filter respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We should note that more than one vibration frequencies can be taken into account straightforwardly by extending the model and the state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' As a next step, we will implement the mixed-control strategy in a test bench based on the real-time computer of the LBT-AO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' We should note that the mixed-control strategy can be implemented without changes on the existing hardware and firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' REFERENCES [1] G.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 190, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content=' 90 80 - 70 - 60 - (%） Strehi ratio 50 Kalman Hinfinity integrator 40 30 20 10 3,5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content="5 2 1,5 0'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} +page_content="5 0 0,5 1'5 2 2,5 3 3'5 error (Hz)SingularValues 10 10 Kalman Ho 12 10° 10 102 10° Freguency (rad/sec)" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfmA2x/content/2301.05165v1.pdf'} diff --git a/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/2301.04948v1.pdf.txt b/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/2301.04948v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f96c7a9f33601f7cd65889f50ae17c00ec9af1a --- /dev/null +++ b/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/2301.04948v1.pdf.txt @@ -0,0 +1,1207 @@ +DISCRIMINATION AND CERTIFICATION OF UNKNOWN +QUANTUM MEASUREMENTS +ALEKSANDRA KRAWIEC1,∗, �LUKASZ PAWELA1, AND ZBIGNIEW PUCHA�LA1 +Abstract. We study the discrimination of von Neumann measurement in the scenario +when we are given a reference measurement and some other measurement. The aim of +the discrimination is to determine whether the other measurement is the same as the +first one. We consider the cases when the reference measurement is given without the +classical description and when its classical description is known. Both cases are studied +in the symmetric and asymmetric discrimination setups. Moreover, we provide optimal +certification schemes enabling us to certify a known quantum measurement against the +unknown one. +1. Introduction +The need for appropriate certification tools is one of the barriers to the development +of large-scale quantum technologies. [1] In this work, we propose tests that verify if a +given device corresponds to its classical description or the reference device. +But why should we care about the discrimination of devices which description we do +not know? A lot is known about discrimination of quantum states, channels and mea- +surements, which description we do know. In the standard discrimination problem, there +are two quantum objects, and one of them is secretly chosen. The goal of discrimination +is to decide which of the objects was chosen. These objects can be quantum states but +also quantum channels and measurements. However, what if we were given a reference +quantum measurement or channel instead of its classical description? Then we may want +to discriminate them regardless of their classical descriptions. Therefore, we arrive at +the new problem of discrimination of unknown objects. +Discrimination of known quantum channels was mainly studied for certain classes of +channels like unitary channels [2–4]. Advantage of using entangled states for minimum- +error discrimination of quantum channels was studied in [5, 6]. +General conditions +when quantum channels can be discriminated in the minimum error, unambiguous and +asymmetric scenarios were derived in [7], [8] and [9] respectively. Another formalism +used for for studying discrimination of quantum channels is based on process POVM +(PPOVM) [10]. It was applied to discrimination of unitary channels in [11,12]. +Discrimination of unknown unitary channels was first studied in the work [13] in both +minimum-error and unambiguous setups. The authors calculated that the probability +of successful minimum-error discrimination between two random qubit unitary channels +1 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. +Ba�ltycka 5, 44-100 Gliwice, Poland +E-mail address: akrawiec@iitis.pl. +Date: January 13, 2023. +1 +arXiv:2301.04948v1 [quant-ph] 12 Jan 2023 + +2 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +equals 7/8 and they made use of the input state |ψ−⟩ = +1 +√ +2 (|01⟩ − |10⟩). The authors +of [14] proved that the probability 7/8 is optimal in the sense that it cannot be improved +by the use of any (even adaptive) discrimination strategy for the qubit case. Recent +results concerning discrimination of unknown unitary channels can be found in [15]. +Minimum error discrimination of quantum measurements was studied in single-shot [16] +and multiple-shot [17] regimes. Asymmetric discrimination of von Neumann measure- +ments was studied in [18] The advantage of using entangled stated for single-shot dis- +crimination between qubit measurements was experimentally shown in [19]. Application +of process POVMs for discrimination of quantum measurements can be found in [20,21] +In this work we study discrimination of unknown von Neumann measurements in +symmetric and asymmetric scenarios. +We begin with preliminaries in Section 2 and +detailed setups for symmetric and asymmetric discrimination of quantum measurements +will be presented therein. Next, we will study the problem when one of the measurements +is given without classical description and we want to verify if the other measurement is +a copy of the same measurements or it is some other one. This problem will be studied +in Section 3. Later, we will assume that one copy of a measurement is given with its +classical description and we want to know whether the other measurement is a copy of +the same measurement. This problem will be studied in Section 4. We will conclude in +Section 5. +2. Preliminaries +Let X, Y and Z be Hilbert spaces where dim(X) = dim(Y) = d, dim(Z) = d2. Let +L(X) be a set of linear operators acting from X to X. +Let U(X) denote the set of +unitary operators. Let D(X) denote the set of quantum states, C(X) denote the set of +quantum channels and T (X) denote the set of quantum operations. For U ∈ U(X), +a unitary channel will be denoted ΦU(·) := U · U †. +We will also utilize two special +quantum channels. The first one is the depolarizing channel, which transforms every +quantum state into the maximally mixed state. Formally, it is defined for X ∈ L(X) as +Φ∗(X) := Tr(X) +1l +dim(X). +The second one is the dephasing channel defined as +∆(X) := +� +i +|i⟩⟨i|X|i⟩⟨i|. +A quantum measurement is defined as a collection of positive semidefinite operator +P = {E1, . . . , Em} which satisfy �m +i=1 = 1l, where 1l is the identity operator. Operators +Ei are called effects. When a quantum state ρ is measured by the measurement P, then +we obtain a label i with probability p(i) = tr (Eiρ) and the state ρ ceases to exist. We +will be particularly interested in von Neumann measurements, which effects are of the +form PU = {|u1⟩⟨u1|, . . . , |ud⟩⟨ud|}, where |ui⟩ = U|i⟩ is the i-th column of the unitary +matrix U. Every quantum measurement can be associated with a quantum channel +(1) +P(ρ) = +� +i +|i⟩⟨i| tr(Eiρ), + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +3 +which outputs a diagonal matrix where i-th entry on the diagonal corresponds to the +probability of obtaining i-th label. +The Choi-Jamio�lkowski representation of quantum operation Ψ ∈ T (X) is defined as +J (Ψ) := (Ψ ⊗ 1lX ) (|1l⟩⟩⟨⟨1l|), where 1lX is the identity channel on the space L(X) and +|X⟩⟩ denotes the (lexicographical) vectorization of the operator X. +The diamond norm of a quantum operation Ψ ∈ T (X) is defined as +(2) +∥Ψ∥⋄ := +max +X:∥X∥1=1 ∥(Ψ ⊗ 1lX ) (X)∥1 , +where 1lX is, as previously, the identity channel on the space L(X). We will often use +the bounds on the diamond norm [22,23] +(3) +1 +d∥J(Ψ)∥1 ≤ ∥Ψ∥⋄ ≤ ∥ Tr1 |J(Ψ)|∥. +In this work we will focus on two approaches to discrimination of quantum measure- +ments, which are symmetric and asymmetric discrimination. +2.1. Symmetric discrimination. The goal of symmetric discrimination is to maximize +the probability of correct discrimination. It is also known as minimum-error discrimi- +nation. The schematic representation of symmetric discrimination of quantum measure- +ments is depicted in Figure 1. +X +P0 +• +Y +P? +• +Z +Ω +decision +� +� +� +� +� +� +� +� +� +|ψ⟩ +Figure 1. Entanglement-assisted discrimination of von Neumann mea- +surements +There are two black boxes. In the first black box there is a measurement P0. In the +second box there is a measurement P?, which can either the same measurement P0, or +some other measurement, P1. In other words P? ∈ {P0, P1}. As the input state to the +discrimination procedure we take a state |ψ⟩ ∈ X ⊗ Y ⊗ Z and we will write ψ := |ψ⟩⟨ψ| +for the sake of simplicity. The measurement in the first black box acts on the register +X and the second black box acts on the register Y. Basing on the outcomes of both +measurements in the black boxes, we prepare a final measurement on the register Z. +Having the output of the final register, we make a decision whether P? = P0 or P? = P1. +To calculate the probability of the successful discrimination between quantum mea- +surements, we will make use of the Holevo-Helstrom theorem. It states that the op- +timal probability of successful discrimination between any quantum channels Ψ0 and +Ψ1 ∈ C(X) is upper-bounded by +(4) +psucc ≤ 1 +2 + 1 +4 ∥Ψ0 − Ψ1∥⋄ +and this bound can be saturated. This optimal probability of successful discrimination +will be denoted pH +succ := 1 +2 + 1 +4 ∥Ψ0 − Ψ1∥⋄. + +4 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +2.2. Asymmetric discrimination. Asymmetric discrimination is based on hypothesis +testing. +The null hypothesis H0 corresponds to the situation when P? = P0. +The +converse situation, P? = P1 corresponds to alternative hypothesis H1. +The scheme +of asymmetric discrimination is as follows. +We begin with preparing an input state +|ψ⟩ ∈ L(X ⊗ Y ⊗ Z) and apply P0 and P? on registers X and Y respectively. Therefore, +in the case when P? = P0, we obtain as the output (P0 ⊗ P0 ⊗ 1l) (ψ) and if P? = P1, +then the output state yields (P0 ⊗ P1 ⊗ 1l) (ψ). Having the output states, we prepare a +binary measurement {Ω, 1l − Ω}, where the effect Ω accepts the null hypothesis and the +effect 1l − Ω accepts the alternative hypothesis. +The type I error (false positive) happens when we reject the correct null hypothesis. +When the input state ψ and measurement Ω are fixed, then the probability of making +the type I error is given by the expression +(5) +p(ψ,Ω) +I +:= Tr ((1l − Ω) (P0 ⊗ P0 ⊗ 1l) (ψ)) = 1 − Tr (Ω (P0 ⊗ P0 ⊗ 1l) (ψ)) . +The optimized probability of the type I error yields +(6) +pI := min +ψ,Ω p(ψ,Ω) +I +The probability of making the type II error (also known as false negative) for fixed input +state and measurement equals +(7) +p(ψ,Ω) +II += Tr (Ω (P0 ⊗ P1 ⊗ 1l) (ψ)) +and corresponds to the situation when we accept the null hypothesis when the alternative +one was correct. The optimized probability of making the type II error yields +(8) +pII := min +ψ,Ω p(ψ,Ω) +II +. +For both symmetric and asymmetric schemes we will study two cases. First we will +assume that both measurements are unknown. Later, we will assume that we know the +description of the reference measurement and the other measurement is unknown. We +will be also interested whether the additional register is necessary for optimal discrimi- +nation. The summary of results is presented in the following table. +pH +succ +pH +err +pI +pII +additional register +both unknown +1 +2 + 1 +2d +1 +2 − 1 +2d +0 +1 − 1 +d +no +one fixed +1 − 1 +2d +1 +2d +0 +1 +d +yes +Table 1. Summary of for symmetric and asymmetric discrimination of +unknown von Neumann measurements +3. Discrimination of both unknown von Neumann measurements +In this section we will study a situation when we are given a von Neumann measure- +ment P0 but no classical description of it. This measurement will be our reference. We +also have another von Neumann measurement P1, which can be the same as the reference +one, but it does not have to. In this section we will study the problem how to verify + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +5 +whether the second measurement is the same as the first one or not. Similar problem of +discrimination of both unknown unitary channels was recently studied in [15]. +3.1. Symmetric discrimination. We will be calculating the success probability for the +discrimination of von Neumann measurements in the scenario depicted in Fig. 1. There- +fore we will be actually discriminating between P0⊗P0 and P0⊗P1 in the entanglement- +assisted scenario. Thus, in order to use Holevo-Helstrom theorem we will need to cal- +culate the value of the diamond norm. As we do not have classical description of either +P0 or P1, we will assume that both measurement are Haar-random, that is we will be +discriminating between +� +PU ⊗PUdU and +� +PU ⊗PV dUdV . The probability of successful +discrimination is formulated as the following theorem. +Theorem 1. Let P0 be a reference von Neumann measurement of dimension d given +without classical description. +Let P1 be another von Neumann measurement of the +same dimension, also given without classical description. The optimal probability of +correct verification if P1 is the same as the reference channel in the scheme described in +Subsection 2.1 equals +(9) +pH +succ = 1 +2 + 1 +2d. +Remark 1. The above theorem is a direct application of Holevo-Helstrom Theorem (see +Eq. (4)) for discrimination between channels +� +PU ⊗ PUdU and +� +PU ⊗ PV dUdV , that +is +(10) +pH +succ = 1 +2 + 1 +4 +���� +� +PU ⊗ PUdU − +� +PU ⊗ PV dUdV +���� +⋄ += 1 +2 + 1 +2d. +Proof. Let U ∈ U(X), V ∈ U(Y) be unitary operators and dim(X) = dim(Y) = d. The +probability of successful discrimination is given by the Holevo-Helstrom theorem. To +calculate this probability (Eq. (4)), we need to calculate the diamond norm distance +between the averaged channels +(11) +���� +� +PU ⊗ PUdU − +� +PU ⊗ PV dUdV +���� +⋄ +. +As the von Neumann measurement PU can be seen as ∆ΦU†, where ∆ is a dephasing +channel defined in Eq. (2), we will actually be discriminating between +(12) +� +(∆ ⊗ ∆)(ΦU† ⊗ ΦU†)dU +and +� +(∆ ⊗ ∆)(ΦU† ⊗ ΦV †)dUdV. +Using [24,25] we calculate the Choi-Jamio�lkowski representations of averaged unitary +channels +J +�� +ΦU ⊗ ΦUdU +� += +1 +d2 − 1 (1l ⊗ 1l + S ⊗ S) − +1 +d(d2 − 1) (S ⊗ 1l + 1l ⊗ S) , +J +�� +ΦU ⊗ ΦV dUdV +� += 1 +d2 1l ⊗ 1l, +(13) +where, unless said otherwise, S is the Swap matrix of dimension d2 and identity matrices +1l-s are also of dimension d2. + +6 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +Using the above, we can calculate the Choi-Jamio�lkowski representations of the aver- +aged measurements, that is +(14) +J +�� +PU ⊗ PUdU +� += +1 +d2 − 1 +� +1l ⊗ +� +1l − 1 +dS +� ++ T ⊗ +� +S − 1 +d1l +�� +where T := ∆(S), and +(15) +J +�� +PU ⊗ PV dUdV +� += 1 +d2 1l ⊗ 1l. +For later convenience, we introduce J as a difference of Choi matrices of both ran- +domized measurements, that is +J := J +�� +PU ⊗ PUdU +� +− J +�� +PU ⊗ PV dUdV +� += +1 +d2 − 1 +� +1l ⊗ +� 1 +d2 1l − 1 +dS +� ++ T ⊗ +� +S − 1 +d1l +�� +. +(16) +The remaining part of the proof goes as follows. We will first calculate the upper +bound on the diamond norm ∥ +� +PU ⊗ PUdU − +� +PU ⊗ PV dUdV ∥⋄ ≤ ∥TrX,Y |J|∥ from +Eq. (3). Later, we will show that this inequality is saturated by Proposition 3 in [22]. +Now we will focus on the upper bound. To calculate the upper bound we first need +to find |J| = +√ +J†J. From Lemma 1 in Appendix A, taking W := (2T − 1l) ⊗ S it holds +that (WJ)2 = J2, and this gives a polar decomposition of J. +To calculate the upper bound for the diamond norm from Eq. (3) we need to calculate +∥TrX,Y |J|∥ = ∥TrX,Y WJ∥. Hence we calculate +TrX,Y(WJ) = +1 +d2 − 1 TrX,Y +�1 +d1l ⊗ 1l − 1 +d2 1l ⊗ S + d − 2 +d +T ⊗ 1l − d − 2 +d2 T ⊗ S +� += +1 +d2 − 1 +�d2 +d 1l − d2 +d2 S + d(d − 2) +d +1l − d(d − 2) +d2 +S +� += +1 +d2 − 1 +� +(2d − 2)1l − 2d − 2 +d +S +� += +2 +d + 1 +� +1l − 1 +dS +� +(17) +and eventually we have +(18) +∥TrX,Y |J|∥ = +���� +2 +d + 1 +� +1l − 1 +dS +����� = +2 +d + 1 +����1l − 1 +dS +���� = 2 +d. +Now we proceed to proving that the upper bound is saturated. By Proposition 3 +in [22] we need to check whether there exist a vector |a⟩ and a unitary matrix W such +that +(i) ⟨a| TrX,Y +√ +J†J|a⟩ = +���TrX,Y +√ +J†J +��� +(ii) (1l ⊗ |a⟩⟨a|) W = W (1l ⊗ |a⟩⟨a|) +(iii) W is the angular part of some polar decomposition of J (i.e. J = WP for some +positive semidefinite P) +As the matrix W we take W := (2T − 1l) ⊗ S and as the vector |a⟩ we take some vector +1 +√ +2 (|ij⟩ − |ji⟩) ∈ Z, where i > j and dim(Z) = d2. The condition (ii) translates to + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +7 +(1l ⊗ |a⟩⟨a|) S ⊗S = S ⊗S (1l ⊗ |a⟩⟨a|) hence it suffices to note that |a⟩⟨a|S = S|a⟩⟨a|. The +condition (iii) follows directly. +Therefore +(19) +���� +� +PU ⊗ PUdU − +� +PU ⊗ PV dUdV +���� +⋄ += 2 +d +and eventually +(20) +pH +succ = 1 +2 + 1 +2d. +□ +3.2. Asymmetric discrimination. In the asymmetric discrimination we will consider +two types of errors separately. We would like to verify whether measurements in both +black boxes are the same (which corresponds to H0 hypothesis) or they are different +(which corresponds to H1 hypothesis). Formally, when the measurement in the first +black box, P0, is unknown, we say that P0 = +� +PUdU. The measurement in the second +black box can be either the same as in the first black box (P? = P0) or it can be some +other measurement, that is P? = +� +PV dV . When performing asymmetric discrimina- +tion, we prepare an input state |ψ⟩ ∈ X ⊗ Y ⊗ Z. If in both black boxes there were +the same measurements, then the output state yields ρ(ψ) +0 += +� +(PU ⊗ PU ⊗ 1lZ) (ψ)dU. +If the measurements in the black boxes were different, when the output state is ρ(ψ) +1 += +� +(PU ⊗ PV ⊗ 1lZ) (ψ)dUdV. Next, we measure the output state by a binary measure- +ment {Ω, 1l − Ω}. We will focus on the case when he type I error cannot occur. The +optimal probability of the type II error is formulated as the following theorem. +Theorem 2. Let P0 be a reference von Neumann measurement of dimension d given +without classical description. Let P1 be another von Neumann measurement of the same +dimension, also given without classical description. +Consider the hypotheses testing +problem described in Subsection 2.2. Let H0 hypothesis state that P? = P0 and let the +alternative H1 hypothesis state that P? = P1. If no false positive error can occur, then +the optimal probability of false negative error yields +(21) +pII = 1 − 1 +d. +Moreover, no additional register is needed to obtain this value. +Proof. As the input state to the discrimination procedure we take some state |ψ⟩ ∈ X ⊗Y. +Note that we assumed that this state is only on two registers. In this proof we will +calculate the probability of the type II error assuming that the register Z is trivial. +Later, we will prove that this gives the optimal probability and the additional register +is not needed. +If both measurements are the same, then the output state will be +(22) +ρ(ψ) +0 += +� +(PU ⊗ PU) (ψ)dU. +If the measurement in the black boxes are different, then the output state will be +(23) +ρ(ψ) +1 += +� +(PU ⊗ PV ) (ψ)dUdV. + +8 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +We begin with calculating +� +(PU ⊗ PU) (ψ)dU by the use of formula for recovering +the action of a quantum channel given its Choi matrix. Using the formula for the Choi +matrix from Eq. (14) and using the notation T := ∆(S) we calculate +ρ(ψ) +0 += TrZ +� +J +�� +PU ⊗ PUdU +� � +1l ⊗ ψ⊤�� += +1 +d(d2 − 1) +�� +d − tr +� +Sψ⊤�� +1l + +� +d tr +� +Sψ⊤� +− 1 +� +T +� +. +(24) +Let us take the input state to be antisymmetric, that is it satisfies tr +� +Sψ⊤� += −1. +We calculate +ρ(ψ) +0 += +1 +d(d2 − 1) ((d + 1) 1l − (d + 1) T) = +1 +d(d − 1) (1l − T) . +(25) +By similar calculation, using the antisymmetric input state we have +ρ(ψ) +1 += TrZ +� +J +�� +PU ⊗ PV dU +� � +1l ⊗ ψ⊤�� += TrZ +�� 1 +d2 1l ⊗ 1l +� � +1l ⊗ ψ⊤�� += 1 +d2 TrZ +� +1l ⊗ ψ⊤� += 1 +d2 1l. +(26) +As the measurement effect we take Ω := 1l − T. Hence +p(ψ,Ω) +I += 1 − tr +� +Ωρ(ψ) +0 +� += 1 − +1 +d(d − 1) tr ((1l − T) (1l − T)) = 0, +(27) +and +p(ψ,Ω) +II += tr +� +Ωρ(ψ) +1 +� += 1 +d2 tr (1l − T) = d(d − 1) +d2 += 1 − 1 +d. +(28) +From Appendix B we know that the probability of erroneous discrimination is the +symmetric scheme (which equals 1 − pH +succ) is never bigger than the arithmetic mean of +probabilities of the type I and type II errors. As +(29) +1 +2 +� +p(ψ,Ω) +I ++ p(ψ,Ω) +II +� += 1 +2 − 1 +2d, +then we conclude that our value of p(ψ,Ω) +II += 1 − 1 +d is optimal and hence pII = p(ψ,Ω) +II +. +Finally, note the optimal value pII can be achieved for the input state |ψ⟩ ∈ X ⊗ Y, +that is when the register Z is trivial. Hence, the additional register is not needed for +asymmetric discrimination in this case. +□ +4. Discrimination between a fixed and unknown von Neumann +measurements +In this section we assume that instead of the unknown reference measurement from +the previous section, we are given P0 as a fixed von Neumann measurement PU. We +will begin with studying symmetric discrimination and later proceed to studying the +asymmetric discrimination scheme. + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +9 +4.1. Symmetric discrimination. Now we focus on the situation when we want to dis- +tinguish between a fixed von Neumann measurement PU and a Haar-random measure- +ment +� +PV dV . The probability of successful discrimination is formulated as a theorem. +Theorem 3. Let P0 = PU be a reference von Neumann measurement of dimension d. +Let P1 be another von Neumann measurement of the same dimension, but given without +classical description. The optimal probability of correct verification whether P1 = P0 or +P1 ̸= P0 in the scheme described in Subsection 2.1 equals +(30) +pH +succ = 1 − 1 +2d. +Proof. Without loss of generality we can take U = 1l. +To calculate the bound from +Holevo-Helstrom theorem (4), we want to calculate the diamond norm distance between +quantum measurements +(31) +����P1l ⊗ P1l − P1l ⊗ +� +PV dV +���� +⋄ +. +Using properties of the diamond norm [23] we calculate +����P1l ⊗ P1l − P1l ⊗ +� +PV dV +���� +⋄ += +����P1l ⊗ +� +P1l − +� +PV dV +����� +⋄ += ∥P1l∥⋄ +����P1l − +� +PV dV +���� +⋄ += +����P1l − +� +PV dV +���� +⋄ +. +(32) +To do this, we use the fact that PV = ∆ΦV †. Moreover, we know that J(Φ1l) = |1l⟩⟩⟨⟨1l| +and J(Φ⋆) = 1l/d, where Φ⋆ is the depolarizing channel defined in Eq. (2). Therefore, +calculating directly both lower and upper bounds for the diamond norm from Eq. (3), +we obtain +(33) +����P1l − +� +PV dV +���� +⋄ += 2 − 2 +d. +Finally +(34) +pH +succ = 1 +2 + 1 +4 +� +2 − 2 +d +� += 1 − 1 +2d. +□ +4.2. Asymmetric discrimination. In this subsection we will focus on asymmetric dis- +crimination between a fixed von Neumann measurement PU and a Haar-random mea- +surement PV . We will be interested in the scenario when the false positive error cannot +occur. The optimized probability of the false negative error is formulated as a theorem. +Theorem 4. Let P0 = PU be a fixed von Neumann measurement and P1 be some other +von Neumann measurement given without classical description. Let the H0 hypothesis +correspond to the case when P? = P0 and H1 hypothesis correspond to the case when + +10 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +P? = P1. Consider the discrimination scheme described in Subsection 2.2. If no false +positive error can occur, then the optimal probability of false negative error yields +(35) +pII = 1 +d. +Proof. This proof goes similar as the proof of Theorem 1. We will choose a fixed input +state on only two registers. We will also fix the final measurement and calculate the +probabilities of making the false positive and false negative errors. Later, from inequality +between errors in symmetric and asymmetric schemes in Appendix B we will see that +the calculated pII is the optimal one. +As the input state we take ψ := 1 +d|1l⟩⟩⟨⟨1l|. We calculate the output states +ρ(ψ) +0 +:= (PU ⊗ 1l) (ψ) = 1 +d (PU ⊗ 1l) (|1l⟩⟩⟨⟨1l|) = 1 +d +� +i +|i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ +(36) +and +ρ(ψ) +1 +:= +� +(PV ⊗ 1l) (ψ)dV = 1 +d +� +(PV ⊗ 1l) (|1l⟩⟩⟨⟨1l|)dV += 1 +d +� � +i +|i⟩⟨i| ⊗ |vi⟩⟨vi|⊤dV = 1 +d +� +i +|i⟩⟨i| ⊗ +� +|vi⟩⟨vi|⊤dV = 1 +d2 1l ⊗ 1l. +(37) +Recall that the measurement effect Ω correspond to H0 hypothesis and 1l − Ω corre- +spond to H1 hypothesis. Hence we have probabilities of false positive and false negative +errors (for given input state) equal +(38) +p(ψ,Ω) +I += 1 − tr +� +Ωρ(ψ) +0 +� +, +p(ψ,Ω) +II += tr +� +Ωρ(ψ) +1 +� +. +Without loss of generality we can consider Ω in the block-diagonal form, ie. +(39) +Ω := +� +i +|i⟩⟨i| ⊗ Ω⊤ +i . +As the unitary matrix U is known, we can use it to construct the final measurement. +Let +(40) +Ωi := |ui⟩⟨ui| +for every i = 1, . . . , d. +Then +tr +� +Ωρ(ψ) +0 +� += tr +� +� +�� +i +|i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ +� � +�1 +d +� +j +|j⟩⟨j| ⊗ |uj⟩⟨uj|⊤ +� +� +� +� += 1 +d +� +i +tr (|ui⟩⟨ui|ui⟩⟨ui|) = 1 +d +� +i +|⟨ui|ui⟩|2 = 1 +(41) +and hence +(42) +p(ψ,Ω) +I += 1 − tr +� +Ωρ(ψ) +0 +� += 0. + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +11 +Eventually +p(ψ,Ω) +II += tr +� +Ωρ(ψ) +1 +� += tr +��� +i +|i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ +� � 1 +d2 1l ⊗ 1l +�� += 1 +d2 +� +i +tr (|ui⟩⟨ui|) = 1 +d. +(43) +It remains to explain why p(ψ,Ω) +II += pII. Note that the arithmetic mean of probabilities +of both types of errors equals +1 +2d which is equal to the probability of erroneous discrim- +ination in the symmetric scheme (see Theorem 3). From the inequality between errors +in the symmetric and asymmetric schemes in Appendix B we conclude that pII = 1 +d. +□ +5. Conclusion +We were studying the problem whether the given von Neumann measurement is the +same as the reference one. We were considering the situation when the reference measure- +ment is given without classical description and when its classical description is known. +Both situations were studied in the symmetric and asymmetric scenarios. We proved +that in both cases one can achieve the probability of false positive error equal zero +and we calculated optimal probabilities of false negative errors. We also calculated the +probabilities of successful discrimination in the symmetric discrimination scheme. +Acknowledgements +This work was supported by the project ,,Near-term quantum computers Challenges, +optimal implementations and applications” under Grant Number POIR.04.04.00-00- +17C1/18-00, which is carried out within the Team-Net programme of the Foundation +for Polish Science co-financed by the European Union under the European Regional +Development Fund. +References +[1] J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Parekh, U. Chabaud, and E. Kashefi, +“Quantum certification and benchmarking,” Nature Reviews Physics, pp. 1–9, 2020. +[2] A. Acin, “Statistical distinguishability between unitary operations,” Physical Review Letters, vol. 87, +no. 17, p. 177901, 2001. +[3] J. Bae, “Discrimination of two-qubit unitaries via local operations and classical communication,” +Scientific Reports, vol. 5, no. 1, pp. 1–8, 2015. +[4] A. Kawachi, K. Kawano, F. Le Gall, and S. 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Krawiec, and R. Kukulski, “Strategies for optimal single-shot discrimi- +nation of quantum measurements,” Physical Review A, vol. 98, no. 4, p. 042103, 2018. +[17] Z. Pucha�la, �L. Pawela, A. Krawiec, R. Kukulski, and M. Oszmaniec, “Multiple-shot and unambigu- +ous discrimination of von Neumann measurements,” Quantum, vol. 5, p. 425, 2021. +[18] P. Lewandowska, A. Krawiec, R. Kukulski, �L. Pawela, and Z. Pucha�la, “On the optimal certification +of von Neumann measurements,” Scientific Reports, vol. 11, no. 1, pp. 1–16, 2021. +[19] M. Mikov´a, M. Sedl´ak, I. Straka, M. Miˇcuda, M. Ziman, M. Jeˇzek, M. Duˇsek, and J. Fiur´aˇsek, “Op- +timal entanglement-assisted discrimination of quantum measurements,” Physical Review A, vol. 90, +no. 2, p. 022317, 2014. +[20] M. Ziman, T. Heinosaari, and M. Sedl´ak, “Unambiguous comparison of quantum measurements,” +Physical Review A, vol. 80, no. 5, p. 052102, 2009. +[21] M. Sedl´ak and M. Ziman, “Optimal single-shot strategies for discrimination of quantum measure- +ments,” Physical Review A, vol. 90, no. 5, p. 052312, 2014. +[22] I. Nechita, Z. Pucha�la, �L. Pawela, and K. ˙Zyczkowski, “Almost all quantum channels are equidis- +tant,” Journal of Mathematical Physics, vol. 59, no. 5, p. 052201, 2018. +[23] J. Watrous, The Theory of Quantum Information. Cambridge University Press, 2018. +[24] Z. Pucha�la and J. Miszczak, “Symbolic integration with respect to the Haar measure on the unitary +groups,” Bulletin of the Polish Academy of Sciences. Technical Sciences, vol. 65, no. 1, 2017. +[25] B. Collins and P. ´Sniady, “Integration with respect to the Haar measure on unitary, orthogonal and +symplectic group,” Communications in Mathematical Physics, vol. 264, no. 3, pp. 773–795, 2006. +Appendix A. Lemma 1 +Lemma 1. Let J be as defined in Eq. (16), T := ∆(S) and W := (2T − 1l) ⊗ S, where +S is the swap matrix of dimension d2. Then J2 = (WJ)2. +Proof. As +(44) +J2 = +� +1 +d2 − 1 +�2 � 1 +d2 1l ⊗ 1l − 1 +d1l ⊗ S + T ⊗ S − 1 +dT ⊗ 1l +�2 +, + +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +13 +we calculate +� 1 +d2 1l ⊗ 1l − 1 +d1l ⊗ S + T ⊗ S − 1 +dT ⊗ 1l +�2 += 1 +d4 1l ⊗ 1l − 1 +d3 1l ⊗ S + 1 +d2 T ⊗ S − 1 +d3 T ⊗ 1l +− 1 +d3 1l ⊗ S + 1 +d2 1l ⊗ 1l − 1 +dT ⊗ 1l + 1 +d2 T ⊗ S ++ 1 +d2 T ⊗ S − 1 +dT ⊗ 1l + T ⊗ 1l − 1 +dT ⊗ S +− 1 +d3 T ⊗ 1l + 1 +d2 T ⊗ S − 1 +dT ⊗ S + 1 +d2 T ⊗ 1l += d2 + 1 +d4 +1l ⊗ 1l − 2 +d3 1l ⊗ S + +� +1 + 1 +d2 − 2 +d − 2 +d2 +� +T ⊗ 1l + +� 4 +d2 − 2 +d +� +T ⊗ S += d2 + 1 +d4 +1l ⊗ 1l − 2 +d3 1l ⊗ S + (d2 + 1)(d − 2) +d3 +T ⊗ 1l + 4 − 2d +d2 +T ⊗ S, +(45) +and eventually +(46) +J2 = +� +1 +d2 − 1 +�2 �d2 + 1 +d4 +1l ⊗ 1l − 2 +d3 1l ⊗ S + (d2 + 1)(d − 2) +d3 +T ⊗ 1l + 4 − 2d +d2 +T ⊗ S +� +. +On the other hand +(47) +WJ = (2T ⊗ S − 1l ⊗ S) +1 +d2 − 1 +� 1 +d2 1l ⊗ 1l − 1 +d1l ⊗ S + T ⊗ S − 1 +dT ⊗ 1l +� +. +Hence we calculate +(2T ⊗ S − 1l ⊗ S) +� 1 +d2 1l ⊗ 1l − 1 +d1l ⊗ S + T ⊗ S − 1 +dT ⊗ 1l +� += 2 +d2 T ⊗ S − 2 +dT ⊗ 1l + 2T ⊗ 1l − 2 +dT ⊗ S +− 1 +d2 1l ⊗ S + 1 +d1l ⊗ 1l − T ⊗ 1l + 1 +dT ⊗ S += 1 +d1l ⊗ 1l − 1 +d2 1l ⊗ S + d − 2 +d +T ⊗ 1l − d − 2 +d2 T ⊗ S. +(48) + +14 +DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS +and thus +�1 +d1l ⊗ 1l − 1 +d2 1l ⊗ S + d − 2 +d +T ⊗ 1l − d − 2 +d2 T ⊗ S +�2 += 1 +d2 1l ⊗ 1l − 1 +d3 1l ⊗ S + d − 2 +d2 T ⊗ 1l − d − 2 +d3 T ⊗ S +− 1 +d3 1l ⊗ S + 1 +d4 1l ⊗ 1l − d − 2 +d3 T ⊗ S + d − 2 +d4 T ⊗ 1l ++ d − 2 +d2 T ⊗ 1l − d − 2 +d3 T ⊗ S + (d − 2)2 +d2 +T ⊗ 1l − (d − 2)2 +d3 +T ⊗ S +− d − 2 +d3 T ⊗ S + d − 2 +d4 T ⊗ 1l − (d − 2)2 +d3 +T ⊗ S + (d − 2)2 +d4 +T ⊗ 1l += d2 + 1 +d4 +1l ⊗ 1l − 2 +d3 1l ⊗ S + (d2 + 1)(d − 2) +d3 +T ⊗ 1l + 4 − 2d +d2 +T ⊗ S. +(49) +Eventually +(50) +(WJ)2 = +� +1 +d2 − 1 +�2 �d2 + 1 +d4 +1l ⊗ 1l − 2 +d3 1l ⊗ S + (d2 + 1)(d − 2) +d3 +T ⊗ 1l + 4 − 2d +d2 +T ⊗ S +� +and hence (WJ)2 = J2. +□ +Appendix B. Inequality between errors +We will show that +(51) +pH +e ≤ 1 +2(p1 + p2), +where pH +e = 1 − pH +succ is the probability of error from the Holevo-Helstrom Theorem. +Let us recall that from Holevo-Helstrom Theorem we have +(52) +1 +2 Tr(Ω0ρ0) + 1 +2 Tr(Ω1ρ1) ≤ 1 − pH +e , +hence +(53) +pH +e ≤ 1 − 1 +2 (Tr(Ω0ρ0) + Tr(Ω1ρ1)) . +On the other hand we know that +Tr(Ω0ρ0) + Tr(Ω1ρ0) = 1 +Tr(Ω0ρ1) + Tr(Ω1ρ1) = 1 +(54) +and hence +(55) +Tr(Ω0ρ0) + Tr(Ω1ρ1) = 2 − (p1 + p2). +Therefore +pH +e ≤ 1 − 1 +2 (Tr(Ω0ρ0) + Tr(Ω1ρ1)) = 1 − 1 +2 (2 − (p1 + p2)) += 1 +2(p1 + p2). +(56) + diff --git a/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/load_file.txt b/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff8c3ca4310c6b3eead0abc9de74a111880f39fd --- /dev/null +++ b/VdE4T4oBgHgl3EQfMgyS/content/tmp_files/load_file.txt @@ -0,0 +1,560 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf,len=559 +page_content='DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS ALEKSANDRA KRAWIEC1,∗, �LUKASZ PAWELA1, AND ZBIGNIEW PUCHA�LA1 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We study the discrimination of von Neumann measurement in the scenario when we are given a reference measurement and some other measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The aim of the discrimination is to determine whether the other measurement is the same as the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We consider the cases when the reference measurement is given without the classical description and when its classical description is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Both cases are studied in the symmetric and asymmetric discrimination setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Moreover, we provide optimal certification schemes enabling us to certify a known quantum measurement against the unknown one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Introduction The need for appropriate certification tools is one of the barriers to the development of large-scale quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' [1] In this work, we propose tests that verify if a given device corresponds to its classical description or the reference device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' But why should we care about the discrimination of devices which description we do not know?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' A lot is known about discrimination of quantum states, channels and mea- surements, which description we do know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In the standard discrimination problem, there are two quantum objects, and one of them is secretly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The goal of discrimination is to decide which of the objects was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' These objects can be quantum states but also quantum channels and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' However, what if we were given a reference quantum measurement or channel instead of its classical description?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Then we may want to discriminate them regardless of their classical descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Therefore, we arrive at the new problem of discrimination of unknown objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Discrimination of known quantum channels was mainly studied for certain classes of channels like unitary channels [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Advantage of using entangled states for minimum- error discrimination of quantum channels was studied in [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' General conditions when quantum channels can be discriminated in the minimum error, unambiguous and asymmetric scenarios were derived in [7], [8] and [9] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Another formalism used for for studying discrimination of quantum channels is based on process POVM (PPOVM) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' It was applied to discrimination of unitary channels in [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Discrimination of unknown unitary channels was first studied in the work [13] in both minimum-error and unambiguous setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The authors calculated that the probability of successful minimum-error discrimination between two random qubit unitary channels 1 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Ba�ltycka 5, 44-100 Gliwice, Poland E-mail address: akrawiec@iitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='04948v1 [quant-ph] 12 Jan 2023 2 DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS equals 7/8 and they made use of the input state |ψ−⟩ = 1 √ 2 (|01⟩ − |10⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The authors of [14] proved that the probability 7/8 is optimal in the sense that it cannot be improved by the use of any (even adaptive) discrimination strategy for the qubit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Recent results concerning discrimination of unknown unitary channels can be found in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Minimum error discrimination of quantum measurements was studied in single-shot [16] and multiple-shot [17] regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Asymmetric discrimination of von Neumann measure- ments was studied in [18] The advantage of using entangled stated for single-shot dis- crimination between qubit measurements was experimentally shown in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Application of process POVMs for discrimination of quantum measurements can be found in [20,21] In this work we study discrimination of unknown von Neumann measurements in symmetric and asymmetric scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We begin with preliminaries in Section 2 and detailed setups for symmetric and asymmetric discrimination of quantum measurements will be presented therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Next, we will study the problem when one of the measurements is given without classical description and we want to verify if the other measurement is a copy of the same measurements or it is some other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' This problem will be studied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Later, we will assume that one copy of a measurement is given with its classical description and we want to know whether the other measurement is a copy of the same measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' This problem will be studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Preliminaries Let X, Y and Z be Hilbert spaces where dim(X) = dim(Y) = d, dim(Z) = d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let L(X) be a set of linear operators acting from X to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let U(X) denote the set of unitary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let D(X) denote the set of quantum states, C(X) denote the set of quantum channels and T (X) denote the set of quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' For U ∈ U(X), a unitary channel will be denoted ΦU(·) := U · U †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will also utilize two special quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The first one is the depolarizing channel, which transforms every quantum state into the maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Formally, it is defined for X ∈ L(X) as Φ∗(X) := Tr(X) 1l dim(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The second one is the dephasing channel defined as ∆(X) := � i |i⟩⟨i|X|i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' A quantum measurement is defined as a collection of positive semidefinite operator P = {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' , Em} which satisfy �m i=1 = 1l, where 1l is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Operators Ei are called effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' When a quantum state ρ is measured by the measurement P, then we obtain a label i with probability p(i) = tr (Eiρ) and the state ρ ceases to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will be particularly interested in von Neumann measurements, which effects are of the form PU = {|u1⟩⟨u1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' , |ud⟩⟨ud|}, where |ui⟩ = U|i⟩ is the i-th column of the unitary matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Every quantum measurement can be associated with a quantum channel (1) P(ρ) = � i |i⟩⟨i| tr(Eiρ), DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 3 which outputs a diagonal matrix where i-th entry on the diagonal corresponds to the probability of obtaining i-th label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The Choi-Jamio�lkowski representation of quantum operation Ψ ∈ T (X) is defined as J (Ψ) := (Ψ ⊗ 1lX ) (|1l⟩⟩⟨⟨1l|), where 1lX is the identity channel on the space L(X) and |X⟩⟩ denotes the (lexicographical) vectorization of the operator X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The diamond norm of a quantum operation Ψ ∈ T (X) is defined as (2) ∥Ψ∥⋄ := max X:∥X∥1=1 ∥(Ψ ⊗ 1lX ) (X)∥1 , where 1lX is, as previously, the identity channel on the space L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will often use the bounds on the diamond norm [22,23] (3) 1 d∥J(Ψ)∥1 ≤ ∥Ψ∥⋄ ≤ ∥ Tr1 |J(Ψ)|∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In this work we will focus on two approaches to discrimination of quantum measure- ments, which are symmetric and asymmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Symmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The goal of symmetric discrimination is to maximize the probability of correct discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' It is also known as minimum-error discrimi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The schematic representation of symmetric discrimination of quantum measure- ments is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' X P0 Y P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Z Ω decision � � � � � � � � � |ψ⟩ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Entanglement-assisted discrimination of von Neumann mea- surements There are two black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In the first black box there is a measurement P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In the second box there is a measurement P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=', which can either the same measurement P0, or some other measurement, P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In other words P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' ∈ {P0, P1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As the input state to the discrimination procedure we take a state |ψ⟩ ∈ X ⊗ Y ⊗ Z and we will write ψ := |ψ⟩⟨ψ| for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The measurement in the first black box acts on the register X and the second black box acts on the register Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Basing on the outcomes of both measurements in the black boxes, we prepare a final measurement on the register Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Having the output of the final register, we make a decision whether P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0 or P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' To calculate the probability of the successful discrimination between quantum mea- surements, we will make use of the Holevo-Helstrom theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' It states that the op- timal probability of successful discrimination between any quantum channels Ψ0 and Ψ1 ∈ C(X) is upper-bounded by (4) psucc ≤ 1 2 + 1 4 ∥Ψ0 − Ψ1∥⋄ and this bound can be saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' This optimal probability of successful discrimination will be denoted pH succ := 1 2 + 1 4 ∥Ψ0 − Ψ1∥⋄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 4 DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Asymmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Asymmetric discrimination is based on hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The null hypothesis H0 corresponds to the situation when P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The converse situation, P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P1 corresponds to alternative hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The scheme of asymmetric discrimination is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We begin with preparing an input state |ψ⟩ ∈ L(X ⊗ Y ⊗ Z) and apply P0 and P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' on registers X and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Therefore, in the case when P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0, we obtain as the output (P0 ⊗ P0 ⊗ 1l) (ψ) and if P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P1, then the output state yields (P0 ⊗ P1 ⊗ 1l) (ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Having the output states, we prepare a binary measurement {Ω, 1l − Ω}, where the effect Ω accepts the null hypothesis and the effect 1l − Ω accepts the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The type I error (false positive) happens when we reject the correct null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' When the input state ψ and measurement Ω are fixed, then the probability of making the type I error is given by the expression (5) p(ψ,Ω) I := Tr ((1l − Ω) (P0 ⊗ P0 ⊗ 1l) (ψ)) = 1 − Tr (Ω (P0 ⊗ P0 ⊗ 1l) (ψ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimized probability of the type I error yields (6) pI := min ψ,Ω p(ψ,Ω) I The probability of making the type II error (also known as false negative) for fixed input state and measurement equals (7) p(ψ,Ω) II = Tr (Ω (P0 ⊗ P1 ⊗ 1l) (ψ)) and corresponds to the situation when we accept the null hypothesis when the alternative one was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimized probability of making the type II error yields (8) pII := min ψ,Ω p(ψ,Ω) II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' For both symmetric and asymmetric schemes we will study two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' First we will assume that both measurements are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Later, we will assume that we know the description of the reference measurement and the other measurement is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will be also interested whether the additional register is necessary for optimal discrimi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The summary of results is presented in the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' pH succ pH err pI pII additional register both unknown 1 2 + 1 2d 1 2 − 1 2d 0 1 − 1 d no one fixed 1 − 1 2d 1 2d 0 1 d yes Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Summary of for symmetric and asymmetric discrimination of unknown von Neumann measurements 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Discrimination of both unknown von Neumann measurements In this section we will study a situation when we are given a von Neumann measure- ment P0 but no classical description of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' This measurement will be our reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We also have another von Neumann measurement P1, which can be the same as the reference one, but it does not have to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In this section we will study the problem how to verify DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 5 whether the second measurement is the same as the first one or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Similar problem of discrimination of both unknown unitary channels was recently studied in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Symmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will be calculating the success probability for the discrimination of von Neumann measurements in the scenario depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' There- fore we will be actually discriminating between P0⊗P0 and P0⊗P1 in the entanglement- assisted scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Thus, in order to use Holevo-Helstrom theorem we will need to cal- culate the value of the diamond norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As we do not have classical description of either P0 or P1, we will assume that both measurement are Haar-random, that is we will be discriminating between � PU ⊗PUdU and � PU ⊗PV dUdV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The probability of successful discrimination is formulated as the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P0 be a reference von Neumann measurement of dimension d given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P1 be another von Neumann measurement of the same dimension, also given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimal probability of correct verification if P1 is the same as the reference channel in the scheme described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1 equals (9) pH succ = 1 2 + 1 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The above theorem is a direct application of Holevo-Helstrom Theorem (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (4)) for discrimination between channels � PU ⊗ PUdU and � PU ⊗ PV dUdV , that is (10) pH succ = 1 2 + 1 4 ���� � PU ⊗ PUdU − � PU ⊗ PV dUdV ���� ⋄ = 1 2 + 1 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let U ∈ U(X), V ∈ U(Y) be unitary operators and dim(X) = dim(Y) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The probability of successful discrimination is given by the Holevo-Helstrom theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' To calculate this probability (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (4)), we need to calculate the diamond norm distance between the averaged channels (11) ���� � PU ⊗ PUdU − � PU ⊗ PV dUdV ���� ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As the von Neumann measurement PU can be seen as ∆ΦU†, where ∆ is a dephasing channel defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (2), we will actually be discriminating between (12) � (∆ ⊗ ∆)(ΦU† ⊗ ΦU†)dU and � (∆ ⊗ ∆)(ΦU† ⊗ ΦV †)dUdV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Using [24,25] we calculate the Choi-Jamio�lkowski representations of averaged unitary channels J �� ΦU ⊗ ΦUdU � = 1 d2 − 1 (1l ⊗ 1l + S ⊗ S) − 1 d(d2 − 1) (S ⊗ 1l + 1l ⊗ S) , J �� ΦU ⊗ ΦV dUdV � = 1 d2 1l ⊗ 1l, (13) where, unless said otherwise, S is the Swap matrix of dimension d2 and identity matrices 1l-s are also of dimension d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 6 DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS Using the above, we can calculate the Choi-Jamio�lkowski representations of the aver- aged measurements, that is (14) J �� PU ⊗ PUdU � = 1 d2 − 1 � 1l ⊗ � 1l − 1 dS � + T ⊗ � S − 1 d1l �� where T := ∆(S), and (15) J �� PU ⊗ PV dUdV � = 1 d2 1l ⊗ 1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' For later convenience, we introduce J as a difference of Choi matrices of both ran- domized measurements, that is J := J �� PU ⊗ PUdU � − J �� PU ⊗ PV dUdV � = 1 d2 − 1 � 1l ⊗ � 1 d2 1l − 1 dS � + T ⊗ � S − 1 d1l �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (16) The remaining part of the proof goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will first calculate the upper bound on the diamond norm ∥ � PU ⊗ PUdU − � PU ⊗ PV dUdV ∥⋄ ≤ ∥TrX,Y |J|∥ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Later, we will show that this inequality is saturated by Proposition 3 in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Now we will focus on the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' To calculate the upper bound we first need to find |J| = √ J†J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' From Lemma 1 in Appendix A, taking W := (2T − 1l) ⊗ S it holds that (WJ)2 = J2, and this gives a polar decomposition of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' To calculate the upper bound for the diamond norm from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (3) we need to calculate ∥TrX,Y |J|∥ = ∥TrX,Y WJ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Hence we calculate TrX,Y(WJ) = 1 d2 − 1 TrX,Y �1 d1l ⊗ 1l − 1 d2 1l ⊗ S + d − 2 d T ⊗ 1l − d − 2 d2 T ⊗ S � = 1 d2 − 1 �d2 d 1l − d2 d2 S + d(d − 2) d 1l − d(d − 2) d2 S � = 1 d2 − 1 � (2d − 2)1l − 2d − 2 d S � = 2 d + 1 � 1l − 1 dS � (17) and eventually we have (18) ∥TrX,Y |J|∥ = ���� 2 d + 1 � 1l − 1 dS ����� = 2 d + 1 ����1l − 1 dS ���� = 2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Now we proceed to proving that the upper bound is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' By Proposition 3 in [22] we need to check whether there exist a vector |a⟩ and a unitary matrix W such that (i) ⟨a| TrX,Y √ J†J|a⟩ = ���TrX,Y √ J†J ��� (ii) (1l ⊗ |a⟩⟨a|) W = W (1l ⊗ |a⟩⟨a|) (iii) W is the angular part of some polar decomposition of J (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' J = WP for some positive semidefinite P) As the matrix W we take W := (2T − 1l) ⊗ S and as the vector |a⟩ we take some vector 1 √ 2 (|ij⟩ − |ji⟩) ∈ Z, where i > j and dim(Z) = d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The condition (ii) translates to DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 7 (1l ⊗ |a⟩⟨a|) S ⊗S = S ⊗S (1l ⊗ |a⟩⟨a|) hence it suffices to note that |a⟩⟨a|S = S|a⟩⟨a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The condition (iii) follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Therefore (19) ���� � PU ⊗ PUdU − � PU ⊗ PV dUdV ���� ⋄ = 2 d and eventually (20) pH succ = 1 2 + 1 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Asymmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In the asymmetric discrimination we will consider two types of errors separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We would like to verify whether measurements in both black boxes are the same (which corresponds to H0 hypothesis) or they are different (which corresponds to H1 hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Formally, when the measurement in the first black box, P0, is unknown, we say that P0 = � PUdU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The measurement in the second black box can be either the same as in the first black box (P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0) or it can be some other measurement, that is P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = � PV dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' When performing asymmetric discrimina- tion, we prepare an input state |ψ⟩ ∈ X ⊗ Y ⊗ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If in both black boxes there were the same measurements, then the output state yields ρ(ψ) 0 = � (PU ⊗ PU ⊗ 1lZ) (ψ)dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If the measurements in the black boxes were different, when the output state is ρ(ψ) 1 = � (PU ⊗ PV ⊗ 1lZ) (ψ)dUdV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Next, we measure the output state by a binary measure- ment {Ω, 1l − Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will focus on the case when he type I error cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimal probability of the type II error is formulated as the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P0 be a reference von Neumann measurement of dimension d given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P1 be another von Neumann measurement of the same dimension, also given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Consider the hypotheses testing problem described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let H0 hypothesis state that P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0 and let the alternative H1 hypothesis state that P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If no false positive error can occur, then the optimal probability of false negative error yields (21) pII = 1 − 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Moreover, no additional register is needed to obtain this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As the input state to the discrimination procedure we take some state |ψ⟩ ∈ X ⊗Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Note that we assumed that this state is only on two registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In this proof we will calculate the probability of the type II error assuming that the register Z is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Later, we will prove that this gives the optimal probability and the additional register is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If both measurements are the same, then the output state will be (22) ρ(ψ) 0 = � (PU ⊗ PU) (ψ)dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If the measurement in the black boxes are different, then the output state will be (23) ρ(ψ) 1 = � (PU ⊗ PV ) (ψ)dUdV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 8 DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS We begin with calculating � (PU ⊗ PU) (ψ)dU by the use of formula for recovering the action of a quantum channel given its Choi matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Using the formula for the Choi matrix from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (14) and using the notation T := ∆(S) we calculate ρ(ψ) 0 = TrZ � J �� PU ⊗ PUdU � � 1l ⊗ ψ⊤�� = 1 d(d2 − 1) �� d − tr � Sψ⊤�� 1l + � d tr � Sψ⊤� − 1 � T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (24) Let us take the input state to be antisymmetric, that is it satisfies tr � Sψ⊤� = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We calculate ρ(ψ) 0 = 1 d(d2 − 1) ((d + 1) 1l − (d + 1) T) = 1 d(d − 1) (1l − T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (25) By similar calculation, using the antisymmetric input state we have ρ(ψ) 1 = TrZ � J �� PU ⊗ PV dU � � 1l ⊗ ψ⊤�� = TrZ �� 1 d2 1l ⊗ 1l � � 1l ⊗ ψ⊤�� = 1 d2 TrZ � 1l ⊗ ψ⊤� = 1 d2 1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (26) As the measurement effect we take Ω := 1l − T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Hence p(ψ,Ω) I = 1 − tr � Ωρ(ψ) 0 � = 1 − 1 d(d − 1) tr ((1l − T) (1l − T)) = 0, (27) and p(ψ,Ω) II = tr � Ωρ(ψ) 1 � = 1 d2 tr (1l − T) = d(d − 1) d2 = 1 − 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (28) From Appendix B we know that the probability of erroneous discrimination is the symmetric scheme (which equals 1 − pH succ) is never bigger than the arithmetic mean of probabilities of the type I and type II errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As (29) 1 2 � p(ψ,Ω) I + p(ψ,Ω) II � = 1 2 − 1 2d, then we conclude that our value of p(ψ,Ω) II = 1 − 1 d is optimal and hence pII = p(ψ,Ω) II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Finally, note the optimal value pII can be achieved for the input state |ψ⟩ ∈ X ⊗ Y, that is when the register Z is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Hence, the additional register is not needed for asymmetric discrimination in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Discrimination between a fixed and unknown von Neumann measurements In this section we assume that instead of the unknown reference measurement from the previous section, we are given P0 as a fixed von Neumann measurement PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will begin with studying symmetric discrimination and later proceed to studying the asymmetric discrimination scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Symmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Now we focus on the situation when we want to dis- tinguish between a fixed von Neumann measurement PU and a Haar-random measure- ment � PV dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The probability of successful discrimination is formulated as a theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P0 = PU be a reference von Neumann measurement of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P1 be another von Neumann measurement of the same dimension, but given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimal probability of correct verification whether P1 = P0 or P1 ̸= P0 in the scheme described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1 equals (30) pH succ = 1 − 1 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Without loss of generality we can take U = 1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' To calculate the bound from Holevo-Helstrom theorem (4), we want to calculate the diamond norm distance between quantum measurements (31) ����P1l ⊗ P1l − P1l ⊗ � PV dV ���� ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Using properties of the diamond norm [23] we calculate ����P1l ⊗ P1l − P1l ⊗ � PV dV ���� ⋄ = ����P1l ⊗ � P1l − � PV dV ����� ⋄ = ∥P1l∥⋄ ����P1l − � PV dV ���� ⋄ = ����P1l − � PV dV ���� ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (32) To do this, we use the fact that PV = ∆ΦV †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Moreover, we know that J(Φ1l) = |1l⟩⟩⟨⟨1l| and J(Φ⋆) = 1l/d, where Φ⋆ is the depolarizing channel defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Therefore, calculating directly both lower and upper bounds for the diamond norm from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (3), we obtain (33) ����P1l − � PV dV ���� ⋄ = 2 − 2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Finally (34) pH succ = 1 2 + 1 4 � 2 − 2 d � = 1 − 1 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Asymmetric discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' In this subsection we will focus on asymmetric dis- crimination between a fixed von Neumann measurement PU and a Haar-random mea- surement PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will be interested in the scenario when the false positive error cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' The optimized probability of the false negative error is formulated as a theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let P0 = PU be a fixed von Neumann measurement and P1 be some other von Neumann measurement given without classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let the H0 hypothesis correspond to the case when P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P0 and H1 hypothesis correspond to the case when 10 DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS P?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Consider the discrimination scheme described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' If no false positive error can occur, then the optimal probability of false negative error yields (35) pII = 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' This proof goes similar as the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will choose a fixed input state on only two registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We will also fix the final measurement and calculate the probabilities of making the false positive and false negative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Later, from inequality between errors in symmetric and asymmetric schemes in Appendix B we will see that the calculated pII is the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As the input state we take ψ := 1 d|1l⟩⟩⟨⟨1l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We calculate the output states ρ(ψ) 0 := (PU ⊗ 1l) (ψ) = 1 d (PU ⊗ 1l) (|1l⟩⟩⟨⟨1l|) = 1 d � i |i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ (36) and ρ(ψ) 1 := � (PV ⊗ 1l) (ψ)dV = 1 d � (PV ⊗ 1l) (|1l⟩⟩⟨⟨1l|)dV = 1 d � � i |i⟩⟨i| ⊗ |vi⟩⟨vi|⊤dV = 1 d � i |i⟩⟨i| ⊗ � |vi⟩⟨vi|⊤dV = 1 d2 1l ⊗ 1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (37) Recall that the measurement effect Ω correspond to H0 hypothesis and 1l − Ω corre- spond to H1 hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Hence we have probabilities of false positive and false negative errors (for given input state) equal (38) p(ψ,Ω) I = 1 − tr � Ωρ(ψ) 0 � , p(ψ,Ω) II = tr � Ωρ(ψ) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Without loss of generality we can consider Ω in the block-diagonal form, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (39) Ω := � i |i⟩⟨i| ⊗ Ω⊤ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As the unitary matrix U is known, we can use it to construct the final measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let (40) Ωi := |ui⟩⟨ui| for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Then tr � Ωρ(ψ) 0 � = tr � � �� i |i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ � � �1 d � j |j⟩⟨j| ⊗ |uj⟩⟨uj|⊤ � � � � = 1 d � i tr (|ui⟩⟨ui|ui⟩⟨ui|) = 1 d � i |⟨ui|ui⟩|2 = 1 (41) and hence (42) p(ψ,Ω) I = 1 − tr � Ωρ(ψ) 0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS 11 Eventually p(ψ,Ω) II = tr � Ωρ(ψ) 1 � = tr ��� i |i⟩⟨i| ⊗ |ui⟩⟨ui|⊤ � � 1 d2 1l ⊗ 1l �� = 1 d2 � i tr (|ui⟩⟨ui|) = 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (43) It remains to explain why p(ψ,Ω) II = pII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Note that the arithmetic mean of probabilities of both types of errors equals 1 2d which is equal to the probability of erroneous discrim- ination in the symmetric scheme (see Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' From the inequality between errors in the symmetric and asymmetric schemes in Appendix B we conclude that pII = 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Conclusion We were studying the problem whether the given von Neumann measurement is the same as the reference one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We were considering the situation when the reference measure- ment is given without classical description and when its classical description is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Both situations were studied in the symmetric and asymmetric scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We proved that in both cases one can achieve the probability of false positive error equal zero and we calculated optimal probabilities of false negative errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' We also calculated the probabilities of successful discrimination in the symmetric discrimination scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Acknowledgements This work was supported by the project ,,Near-term quantum computers Challenges, optimal implementations and applications” under Grant Number POIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='00-00- 17C1/18-00, which is carried out within the Team-Net programme of 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' ´Sniady, “Integration with respect to the Haar measure on unitary, orthogonal and symplectic group,” Communications in Mathematical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 264, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' 773–795, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Lemma 1 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let J be as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (16), T := ∆(S) and W := (2T − 1l) ⊗ S, where S is the swap matrix of dimension d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Then J2 = (WJ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' As (44) J2 = � 1 d2 − 1 �2 � 1 d2 1l ⊗ 1l − 1 d1l ⊗ S + T ⊗ S − 1 dT ⊗ 1l �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='we calculate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 1l ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d1l ⊗ S + T ⊗ S − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='dT ⊗ 1l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 1l ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ S − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ 1l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 1l ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='dT ⊗ 1l + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ S − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='dT ⊗ 1l + T ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='dT ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ 1l + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ S − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='dT ⊗ S + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ 1l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='= d2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1l ⊗ 1l − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='� 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='= d2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1l ⊗ 1l − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + (d2 + 1)(d − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l + 4 − 2d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (45) and eventually (46) J2 = � 1 d2 − 1 �2 �d2 + 1 d4 1l ⊗ 1l − 2 d3 1l ⊗ S + (d2 + 1)(d − 2) d3 T ⊗ 1l + 4 − 2d d2 T ⊗ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' On the other hand (47) WJ = (2T ⊗ S − 1l ⊗ S) 1 d2 − 1 � 1 d2 1l ⊗ 1l − 1 d1l ⊗ S + T ⊗ S − 1 dT ⊗ 1l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Hence we calculate (2T ⊗ S − 1l ⊗ S) � 1 d2 1l ⊗ 1l − 1 d1l ⊗ S + T ⊗ S − 1 dT ⊗ 1l � = 2 d2 T ⊗ S − 2 dT ⊗ 1l + 2T ⊗ 1l − 2 dT ⊗ S − 1 d2 1l ⊗ S + 1 d1l ⊗ 1l − T ⊗ 1l + 1 dT ⊗ S = 1 d1l ⊗ 1l − 1 d2 1l ⊗ S + d − 2 d T ⊗ 1l − d − 2 d2 T ⊗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='(48) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='DISCRIMINATION AND CERTIFICATION OF UNKNOWN QUANTUM MEASUREMENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='and thus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d1l ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 1l ⊗ S + d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l − d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 1l ⊗ 1l − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ 1l − d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 1l ⊗ 1l − d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ S + d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 T ⊗ 1l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='+ d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 T ⊗ 1l − d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ S + (d − 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l − (d − 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='− d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 T ⊗ S + d − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 T ⊗ 1l − (d − 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ S + (d − 2)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='= d2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='1l ⊗ 1l − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 1l ⊗ S + (d2 + 1)(d − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ 1l + 4 − 2d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content='T ⊗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (49) Eventually (50) (WJ)2 = � 1 d2 − 1 �2 �d2 + 1 d4 1l ⊗ 1l − 2 d3 1l ⊗ S + (d2 + 1)(d − 2) d3 T ⊗ 1l + 4 − 2d d2 T ⊗ S � and hence (WJ)2 = J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Inequality between errors We will show that (51) pH e ≤ 1 2(p1 + p2), where pH e = 1 − pH succ is the probability of error from the Holevo-Helstrom Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Let us recall that from Holevo-Helstrom Theorem we have (52) 1 2 Tr(Ω0ρ0) + 1 2 Tr(Ω1ρ1) ≤ 1 − pH e , hence (53) pH e ≤ 1 − 1 2 (Tr(Ω0ρ0) + Tr(Ω1ρ1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' On the other hand we know that Tr(Ω0ρ0) + Tr(Ω1ρ0) = 1 Tr(Ω0ρ1) + Tr(Ω1ρ1) = 1 (54) and hence (55) Tr(Ω0ρ0) + Tr(Ω1ρ1) = 2 − (p1 + p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' Therefore pH e ≤ 1 − 1 2 (Tr(Ω0ρ0) + Tr(Ω1ρ1)) = 1 − 1 2 (2 − (p1 + p2)) = 1 2(p1 + p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} +page_content=' (56)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE4T4oBgHgl3EQfMgyS/content/2301.04948v1.pdf'} diff --git a/XtE1T4oBgHgl3EQfcAQQ/content/2301.03178v1.pdf 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b/YNFJT4oBgHgl3EQf6S0s/content/tmp_files/2301.11673v1.pdf.txt @@ -0,0 +1,1709 @@ +Bayesian Self-Supervised Contrastive Learning +Bin Liu 1 Bang Wang* 1 +Abstract +Recent years have witnessed many successful ap- +plications of contrastive learning in diverse do- +mains, yet its self-supervised version still remains +many exciting challenges. As the negative sam- +ples are drawn from unlabeled datasets, a ran- +domly selected sample may be actually a false +negative to an anchor, leading to incorrect en- +coder training. This paper proposes a new self- +supervised contrastive loss called the BCL loss +that still uses random samples from the unlabeled +data while correcting the resulting bias with im- +portance weights. The key idea is to design the +desired sampling distribution for sampling hard +true negative samples under the Bayesian frame- +work. The prominent advantage lies in that the +desired sampling distribution is a parametric struc- +ture, with a location parameter for debiasing false +negative and concentration parameter for mining +hard negative, respectively. Experiments validate +the effectiveness and superiority of the BCL loss 1. +1. Introduction and Contribution +Unsupervised learning has been extensively researched for +its advantages of learning representations without human +labelers for manually labeling data. How to learn good +representation without supervision, however, has been a +long-standing problem in machine learning. Recently, con- +trastive learning that leverages a contrastive loss (Chopra +et al., 2005; Hadsell et al., 2006) to train a representation +encoder has been promoted as a promising solution to this +problem (Oord et al., 2018; Tian et al., 2020; Liu et al., 2021; +Chen et al., 2020b). Remarkable successes of contrastive +learning have been observed for many applications in dif- +ferent domains (Alec et al., 2018; 2019; Misra & Maaten, +2020; He et al., 2020). Nonetheless, its potentials can be +further released for a better contrastive loss being designed. +We study the following self-supervised contrastive learning +problem (Oord et al., 2018; Chuang et al., 2020; Robin- +1liubin0606@hust.edu.cn; wangbang@hust.edu.cn; School of +Electronic Information and Communications, Huazhong Univer- +sity of Science and Technology (HUST), Wuhan, China. +Corresponding author: Bang Wang +son et al., 2021; Chen et al., 2020a; Chuang et al., 2020; +Liu et al., 2021): Consider an unlabeled dataset X and +a class label set C, let h : X → C be the classification +function assigning a data point x ∈ X with a class label +c ∈ C. Assume that observing a class label ρ(c) = τ + +is uniform and τ − = 1 − τ + is the probability of ob- +serving any different class. For a given data point x, let +p+(x+) = p(x+|h(x) = h(x+)) denote the probability of +another point x+ with the same label as x, and in such a +case, x+ is called a positive sample specific to x. Likewise, +let p−(x−) = p(x−|h(x) ̸= h(x−)) denote the probability +of another point x− with a different label to x, and in such +a case, x− is called a negative sample specific to x. Let +f : X → Rd denote the representation learning function +(i.e., encoder) to map a point x to an embedding f(x) on a +d-dimensional hypersphere. +Self-supervised contrastive learning is to contrast similar +pairs (x, x+) and dissimilar pairs (x, x−) to learn the en- +coder f (Wang & Isola, 2020; Wang & Liu, 2021; Chuang +et al., 2020); While the objective is to encourage representa- +tions of (x, x+) to be closer than that of (x, x−). In training +the encoder, we randomly draw a point from the underlying +data distribution pd on X, i.e., x ∼ pd, and its positive sam- +ple x+ can be easily obtained from some semantic-invariant +operation on x like image masking, written as x+ ∼ p+. In +practice, a negative sample x− is drawn from the unlabeled +dataset and x− ∼ pd. However, the sample x− could be +potentially with the same label as x, i.e., it is a false negative +to x. In such a case, the construction of a dissimilar pair +(x, x−) would degrade the learned representations (Wang +& Liu, 2021; Chuang et al., 2020). As no prior knowledge +about the label of x−, we propose to include an importance +weight ω to measure the credibility of a constructed pair +(x, x−) for contrastive learning. +Contribution: In this paper, we propose the following +Bayesian self-supervised Contrastive Learning objective +function, viz., the BCL loss: +LBCL = E x∼pd +x+∼p+ +x−∼pd +−log[ +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ωi · ef(x)T f(x− +i ) ], +Our main contributions include (i) the derivation of para- +metric structural sampling distribution conditioned on hard +principle and true principle, (ii) the posterior estimation +arXiv:2301.11673v1 [cs.LG] 27 Jan 2023 + +Bayesian Self-Supervised Contrastive Learning +of a unlabeled sample being true negative sample, and (iii) +the stochastic process depiction to simulate predictions of +neural networks. +Compared with the InfoNCE (Oord et al., 2018), we include +the importance weight ω in the contrastive loss, which is +designed to down-weight a constructed pair (x, x−) for +x− being a false negative sample or to up-weight (x, x−) +for x− being a true negative sample. The key idea is to +design a desired sampling distribution for hard true negative +samples under the Bayesian framework. The prominent +advantage lies in that we derive the parametric structure for +the desired sampling distribution with a location parameter +for debiasing false negative and concentration parameter for +mining hard negative, respectively. More detailed analysis +on LBCL and derivation of ω computation are given in the +subsequent sections. +We summarize the computation of ω in each training epoch +as follows. In a training epoch, we randomly draw N sam- +ples {x− +i }N +i=1 from training set, which are assumed as neg- +ative samples specific to x. Let ˆx− +i = exp(f(x)T f(x− +i )) +denote the power exponent of similarity between x and +x− +i . We compute the importance weight ωi for x− +i by the +following three steps: +Step-1: Compute ΦN(ˆx− +i ), called the empirical distribution +function of ˆx− +i , +ΦN(ˆx− +i ) = 1 +N +N +� +j=1 +I(ˆx− +j ≤ ˆx− +i ), +(1) +where I(·) is the indicator function. +Step-2: Compute p(TN|ˆx− +i ), the posterior probability of +x− +i being a true negative (TN) to x, +p(TN|ˆx− +i ) = +ατ − + (1 − 2α)ΦN(ˆx− +i )τ − +ατ − + (1 − α)τ + + (1 − 2α)ΦN(ˆx− +i )(τ − − τ +), +(2) +where α is a hyperparameter to be explained latter. +Step-3: Compute ωi(ˆx− +i ), the importance weight of x− +i for +correcting the bias between actual sampling distribution and +desired sampling distribution +ωi(ˆx− +i ) = +p(TN|ˆx− +i ) · ˆxβ +i +1 +N +�N +i=1 p(TN|ˆx− +i ) · ˆxβ +i +, +(3) +where β is a hyperparameter to be explained latter. +2. Contrastive Loss and Analysis +2.1. Contrastive Loss +In the context of supervised contrastive learning, dissimilar +pairs (x, x−) can be easily constructed by randomly draw- +ing a true negative sample x− specific to x, i.e., x− ∼ p−, +based on the sample label. The contrastive predictive coding +(CPC) (Oord et al., 2018) introduces the following InfoNCE +loss (Gutmann & Hyv¨arinen, 2010; 2012): +LSUP = E +x∼pd +x+∼p+ +x−∼p− +[− log +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ef(x)T f(x− +i ) ] +(4) +to learn an encoder f : X → Rd/t that maps a data point +x to the hypersphere Rd of radius 1/t, where t is the tem- +perature scaling. As in the CPC, we also set t = 1 in our +theoretical analysis. +In the context of self-supervised contrastive learning, how- +ever, as samples’ labels are not available, i.e., p−(x′) = +p(x′|h(x) ̸= h(x′)) is not accessible, the standard approach +is to draw N samples from the data distribution pd, which +are supposed to be negative samples to x, to optimize the +following InfoNCE self-supervised contrastive loss: +LBIASED = E x∼pd +x+∼p+ +x−∼pd +[− log +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ef(x)T f(x− +i ) ]. +(5) +Following the DCL (Chuang et al., 2020), it is also called +as biased contrastive loss since those supposedly negative +samples x− drawn from pd might come from the same class +as the data point x with probability τ +. +2.2. Sampling Bias Analysis +Let x− ∈ TN denote x− being a true negative (TN) sample +specific to x. Let x− ∈ FN denote x− being a false negative +(FN) sample specific to x, i.e. x− and x are with the same +ground truth class label. Note that whether x− is a TN or +FN is specific to a particular anchor point x, and in what +follows, we omit the specific to x for brevity. It has been +proven that for {x− +i +∈ TN}N +i=1, optimizing the InfoNCE +loss LSUP will result in the learning model estimating and +optimizing the density ratio p+ +p− (Oord et al., 2018; Poole +et al., 2019). Denote ˆx+ = ef(x)T f(x+). The CPC (Oord +et al., 2018) proves that minimizing LSUP leads to +ˆx+ ∝ p+/p−. +(6) +As discussed by (Oord et al., 2018), p+/p− preserves the +mutual information (MI) of future information and present +signals, where MI maximization is a fundamental problem +in science and engineering (Poole et al., 2019; Belghazi +et al., 2018) . +Now consider the InfoNCE loss LBIASED, which can be re- +garded as the categorical cross-entropy of classifying one +positive sample x+ from unlabeled samples. For analysis +purpose, we rewrite x+ as x0. Given N + 1 unlabeled data +points, the posterior probability of one data point x0 being + +Bayesian Self-Supervised Contrastive Learning +Figure 1. Illustration of LBIASED and mutual information optimiza- +tion by Eq. (10). +a positive sample can be derived by +P(x0 ∈ pos|{xi}N +i=0) += +p+(x0) �N +i=1 pd(xi) +�N +j=0 p+(xj) � +i̸=j pd(xi) += +p+(x0)/pd(x0) +p+(x0)/pd(x0) + �N +j=1 p+(xj)/pd(xj) +(7) +To minimize LBIASED, the optimal value for this poste- +rior probability is 1, which is achieved in the limit of +p+(x0)/pd(x0) → +∞ or p+(xj)/pd(xj) → 0. Mini- +mizing LBIASED leads to +ˆx+ ∝ p+/pd. +(8) +Note that this is different from Eq. (6), since x− +i may not be +TN for lack of ground truth label. +Denote ˆx+ = m · p+/pd, m ≥ 0. We investigate the +gap between optimizing ˆx+ and the optimization objective +p+/p−. Inserting pd = p−τ − + p+τ + back to Eq. (8), we +obtain +ˆx+ = m · +p+ +p−τ − + p+τ + . +(9) +Rearranging the above equation yields +p+/p− = +ˆx+ · τ − +m − ˆx+ · τ + . +(10) +Fig. 1 illustrates the approximate shape of Eq. (10) as a frac- +tional function, which reveals the inconsistency between +InfoNCE LBIASED loss optimization and MI optimization. +That is, when optimizing InfoNCE loss, the increase of ˆx+ +does not lead to the monotonic increase of p+/p−. Indeed, +the existence of jump discontinuity indicates that the opti- +mization of LBIASED does not necessarily lead to the tractable +MI optimization. The reason for the intractable MI optimiza- +tion is from the fact that not all {x− +i }N +i=1 are TN samples, as +they are randomly drawn from the data distribution pd. This +leads to the inclusion of p+ in the denominator of Eq. (9) +when decomposing the data distribution pd. Fig. 6 in Ap- +pendix provides an intuitive explanation. The four sampled +data points actually contain one FN sample. Such a FN sam- +ple should be pulled closer to the anchor x. However, as it is +mistakenly treated as a negative sample, during model train- +ing it will be pushed further apart from the anchor, which +breaks the semantic structure of embeddings (Wang & Liu, +2021). +3. The Proposed Method +In this paper, we consider to randomly draw negative sam- +ples {x− +i }N +i=1 from the unlabeled dataset, i.e., x− +i ∼ pd. As +the class label is not accessible, x− +i could be either a TN +sample or a FN sample. We propose to include and compute +an importance weight ωi into the InfoNCE contrastive loss +for correcting the resulting bias of drawing negative samples +from pd. The ideal situation is that we can set ω = 0 to +each FN sample, so that only the hard true negative samples +contribute to the calculation of contrastive loss, which relies +on the design of desired sampling distribution. +We consider the following two design principles of the sam- +pling distribution for drawing {x− +i }N +i=1. The true princi- +ple (Wang & Liu, 2021; Robinson et al., 2021) states that +the FN samples should not be pushed apart from the anchor +x in the embedding space. The hard principle (Yannis et al., +2020; Robinson et al., 2021; Florian et al., 2015; Hyun et al., +2016) states that the hard TN samples should be pushed +further apart in the embedding space. +3.1. False Negative Debiasing +We first consider the true principle for the design of sam- +pling distribution. We denote the power exponent of similar- +ity between an anchor x and another unlabeled sample x′ as +ˆx = ef(x)Tf(x′). Assume that ˆx is independently and identi- +cally distributed with a probability density function φ and +cumulative distribution function Φ(ˆx) = +� ˆx +−∞ φ(t)dt. As +x′ can be either a TN sample or a FN sample, so φ contains +two populations, denoted as φTN and φFN. The problem of +computing the LBCL loss Eq. (1) is reduced to estimating +the sum over ˆx ∼ φTN, i.e., �N +i=1 ef(x)T f(x− +i ), while using +samples ˆx ∼ φ. +Existing approaches for solving above problem is the density +estimation to fit φ (Xia et al., 2022), where φ is parameter- +ized as a two-component mixture of φTN and φFN, such as +the Gaussian Mixture Model (Lindsay, 1995), Beta Mixture +Model (Xia et al., 2022). To make the analysis possible, φTN +and φFN are postulated to follow a simple density function +with fixed parameters, which is a too strong assumption. In +addition, the learning algorithm for estimating φ is expen- + +0Bayesian Self-Supervised Contrastive Learning +sive, since the mixture coefficients that indicate the probabil- +ity of ˆx ∈ TN or FN are hidden variables. The parameters of +φTN and φFN can only be obtained through the iterative nu- +merical computation of the EM algorithm (Dempster et al., +1977) that are sensitive to initial values. +In this paper, we propose an analytic method without explic- +itly estimating φ, also called the nonparametric method in +the statistical theory. Consider n random variables from φ ar- +ranged in the ascending order according to their realizations. +We write them as X(1) ≤ X(2) ≤ · · · ≤ X(n), and X(k) +is called the k-th (k = 1, · · · , n) order statistics (David & +Nagaraja, 2004). The probability density function (PDF) of +X(k) is given by: +φ(k)(ˆx) = +n! +(k − 1)!(n − k)!Φk−1(ˆx)φ(ˆx)[1 − Φ(ˆx)]n−k +By conditioning on n = 2 we obtain: +φ(1) += +2φ(ˆx)[1 − Φ(ˆx)] +(11) +φ(2) += +2φ(ˆx)Φ(ˆx) +(12) +Next, we investigate the position of positive and negative +samples on the hypersphere, so as to get a deep insight into +φTN. Consider a (x, x+, x−) triple, there exists a closed +ball B[f(x), d+] = {f(·)|d(f(x), f(·)) ≤ d+} with center +f(x) and radius d+, where d+ = ∥f(x) − f(x+)∥ is the +distance of anchor embedding f(x) and positive sample +embedding f(x+). Two possible cases arise: f(x−) ∈ +B[f(x), d+] or f(x−) /∈ B[f(x), d+], as illustrated by +Fig. 2. We can describe the two cases with the Euclidean +distance: Fig 2(a) corresponds to d+ < d−, and Fig 2(b) cor- +responds to d− ≤ d+, where d− = ∥f(x) − f(x−)∥. Note +that the Euclidean distance d± = +� +2/t2 − 2f(x)Tf(x±) +since all embeddings f(·) are on the surface of a hyper- +sphere of radius 1/t, so we have ˆx− < ˆx+ for case (a), and +ˆx+ ≤ ˆx− for case (b). Expressed in the notation of order +statistics, ˆx− (marked in blue in Fig 2) is a realization of +X(1) for case (a), or ˆx− is a realization of X(2) for case (b), +respectively. +Figure 2. Two possible cases for the relative positions of anchor, +positive, and negative triples. +The generation process of observation ˆx from φTN can be +described as follows: Select case (a) with probability α, and +Figure 3. φTN(ˆx) and φFN(ˆx) with different α settings. +then generate an observation ˆx from φ(1); Or select case (b) +with probability 1 − α, and then generate an observation ˆx +from φ(2). That is, φTN is the component of X(1) and X(2) +with a mixture coefficient α +φTN(ˆx) = αφ(1)(ˆx) + (1 − α)φ(2)(ˆx) +(13) +Similarly, φFN is the component of X(2) and X(1) with +mixture coefficient α: +φFN(ˆx) = αφ(2)(ˆx) + (1 − α)φ(1)(ˆx) +(14) +Note that the way of taking ˆx− as a realization of X(1) for +case (a) omits the situation of ˆx− = ˆx+. The probability +measure of ˆx− for such case is 0 as φ is continuous density +function dominated by Lebesgue measure. +Proposition 3.1 (Class Conditional Density). If φ(ˆx) is +continuous density function that satisfy φ(ˆx) ≥ 0 and +� +∞ +−∞ φ(ˆx)dˆx = 1, then φTN(ˆx) and φFN(ˆx) are probability +density functions that satisfy φTN(ˆx) ≥ 0, φFN(ˆx) ≥ 0, and +� +∞ +−∞ φTN(ˆx)dˆx = 1, +� +∞ +−∞ φFN(ˆx)dˆx = 1. +Proof. See Appendix C.1. +There may need a further understanding and clarification of +the mixture coefficient α by reviewing Fig. 2. Intuitively, α +is the probability that f(x−) falls out of B[f(x), d+]. For +a worst encoder f that randomly guesses, α = 0.5; While +for a perfect encoder α = 1. Therefor, the reasonable value +of α ∈ [0.5, 1]. In fact, α reflects the encoder’s capability +of scoring a positive sample higher than that of a negative +sample, which admits the empirical macro-AUC metric over +all anchors x in the training data set D: +α += +� +x∈X ++∞ +� +0 ++∞ +� +0 +I(ˆx+ ≥ ˆx−)p(ˆx+, ˆx−)p(x)dˆx+dˆx−dx +≃ +1 +|D| +1 +|D+||D−| +� +D+ +� +D− +I(ˆx+ ≥ ˆx−) += +1 +|D|AUC +(15) + +f(x) +f(x +f(x) o: +f(x)o +if(xt) +if(xt) +(a) [α + (1 − 2α) · cdf]/α do +ˆxj = generate observation ˆxj from φ +cdf = +� ˆxj +−∞ φ(t)dt +u = random.uniform(0, 1) +Result: ˆxj +Likewise, the acceptance probability for φFN +pFN = [1 − α + (2α − 1) · Φ(ˆx)]/α +can be calculated in the similar way. An observation ˆx from proposal distribution φ(ˆx) is accepted with probability pFN, +formulates the empirical observations ˆx ∼ φFN(ˆx) as described in Algorithm 3. +Algorithm 3 AccRejetSamplingFN(φFN). +Input: location parameter α, proposal distrition φ +Output: samples ˆx ∼ φFN. +ˆxj = generate observation ˆxj from φ +cdf = +� ˆxj +−∞ φ(t)dt +u = random.uniform(0, 1) +while u > [1 − α + (2α − 1) · cdf]/α do +ˆxj = generate observation ˆxj from φ +cdf = +� ˆxj +−∞ φ(t)dt +u = random.uniform(0, 1) +Result: ˆxj +Parameter Settings. For baseline estimator ˆθDCL in Eq (30), the number of positive samples K is fixed as 10. We start from +fixing the parameters as: α = 0.9, β = 0.9, γ = 0.1 τ + = 0.1, temperature scaling t = 0.5, number of anchors M = 1e3, +and number of negative samples N = 64 to investigate the performance of different estimators under different parameters +settings. +Mean values. Aside from MSE, we report the mean values of the estimators ˆθ. Our primary concern is whether the chosen +of proposal distribution φ or encoder f affects the consistency of estimators. Fig 8(a) shows the influence of different +variation of anchor-specific proposal distribution φ on mean values of estimators. Fig 8(b) shows the influence of encoder’s +performance on mean values of estimators. It can be seen that the mean value of ˆθ BCL, ˆθ DCL and θ sequences are very +close, which guaranteed by Chebyshev law of large numbers, namely lim +M→∞ P{| 1 +M +�M +m=1 ˆθm − 1 +M +�M +m=1 θm| < ϵ} = 1 as +E ˆθm = θm. This conclusion is un-affected by the chosen of proposal distribution φ or encoder f. Aside from consistency of +estimators, Fig 8(b) can be seen as a simulation of training process: a better trained encoder with higher macro-AUC embeds + +Bayesian Self-Supervised Contrastive Learning +(a) +(b) +Figure 8. Mean values of estimators ˆθ over all anchors. +true negative samples dissimilar with anchor, while more similar to anchor for false negative samples. So we observe the +decrease of ¯θ, which corresponds to the decrease of loss values in the training process, and increase of bias |¯θBIASED − ¯θ| +since the contained false negative samples in ¯θBIASED are scored higher as α increases. +B.2. Real data experiment +Experimental setup: We perform the experiments on vision tasks using using CIFAR10 (Krizhevsky & Hinton, 2009) +and STL10 (Adam et al., 2011) datasets. All the experimental settings are identically as DCL (Chuang et al., 2020) and +HCL (Robinson et al., 2021). Specifically, we implement SimCLR (Chen et al., 2020a) with ResNet-50 (He et al., 2016) as +the encoder architecture and use the Adam optimizer (Diederik & Jimmy, 2015) with learning rate 0.001. The temperature +scaling and the dimension of the latent vector were set as t = 0.5 and d=128. All the models are trained for 400 epochs, and +evaluated by training a linear classifier after fixing the learned embedding (Lajanugen & Honglak, 2018; Robinson et al., +2021). The source codes are available at https://github.com/liubin06/BNS +Location parameter α: recall that α corresponds to the encoder’s macro-AUC, we recommend the following two basic +strategies for setting α: +• Timely-adjust strategy: setting α as empirical macro-AUC, and updated every 20 training epochs. The gradual +increase of α means that the confidence level of sample information Φn(·) increases, and a sample with relative closer +embedding distance to the anchor are more likely to be a FN sample. +• Warm-start strategy: setting α = 0.5 at first 300 training epochs, that is, only uses prior information τ for debiasing +false negative samples to warm-up the encoder, then set α = 0.7 at the later 100 training epochs to introduce sample +information Φn(ˆx) to debiase false negative samples. Since the encoder has been warmed up, the sample information +is more reliable. +We adopt the warm-start strategy for setting α, since the sample information is not reliable at the initial training phase, the +performance gain of debiasing false negative samples is less than that of mining hard negative samples. +Concentration parameter β: it corresponds to concentration degree of embeddings. Note that contrastive loss optimizes +the negative samples for uniformity asymptotically (Wang & Isola, 2020) in the training phase, which indicating the +concentration degree of embeddings are gradually decreased. We therefore set β to be a decreasing function of training +epoch: β = 2 − 0.005 · epoch, which is also updated every 20 training epochs. beta = 2 at epoch 0 indicating that BCL +lays more emphasize on hard negative mining task at initial training phase, while beta = 0 at epoch 400 indicating that BCL +lays more emphasize on false negative debiasing task at later training phase. + +40 +- +OBCL +35 +épCL +30 +25 +15 +10 +5 +0. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +y120 +OBCL +ODCL +100 +80 +- +60 +40 +20 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +αBayesian Self-Supervised Contrastive Learning +C. Proofs +Proposition C.1 (Class Conditional Density). If φ(ˆx) is continuous density function that satisfy φ(ˆx) ≥ 0 and +� +∞ +−∞ φ(ˆx)dˆx = 1, then φTN(ˆx) and φFN(ˆx) are probability density functions that satisfy φTN(ˆx) ≥ 0, φFN(ˆx) ≥ 0, +and +� +∞ +−∞ φTN(ˆx)dˆx = 1, +� +∞ +−∞ φFN(ˆx)dˆx = 1. +Proof. Since φ(ˆx) ≥ 0 and +� +∞ +−∞ φ(ˆx)dˆx = 1, so +φTN(ˆx) += +αφ(1)(ˆx) + (1 − α)φ(2)(ˆx) += +2αφ(ˆx)[1 − Φ(ˆx)] + 2(1 − α)φ(ˆx)Φ(ˆx) +≥ +0, +(32) +where α ∈ [0.5, 1] and Φ(ˆx) ∈ [0, 1]. +� +∞ +−∞ +φTN(ˆx)dˆx += +� +∞ +−∞ +2αφ(ˆx)[1 − Φ(ˆx)] + 2(1 − α)φ(ˆx)Φ(ˆx)dˆx += +2α +� +∞ +−∞ +φ(ˆx)[1 − Φ(ˆx)]dˆx + 2(1 − α) +� +∞ +−∞ +φ(ˆx)Φ(ˆx)dˆx += +2α +� +∞ +−∞ +[1 − Φ(ˆx)]dΦ(ˆx) + 2(1 − α) +� +∞ +−∞ +Φ(ˆx)dΦ(ˆx) += +2α +� 1 +0 +(1 − µ)dµ + 2(1 − α) +� 1 +0 +µdµ +(33) += +[α(2µ − µ2) + (1 − α)µ2] +��1 +0 += +1, +(34) +where Eq 33 is integration by substitution of Φ(ˆx) and µ. Likewise, +φFN(ˆx) += +αφ(2)(ˆx) + (1 − α)φ(1)(ˆx) += +2αφ(ˆx)Φ(ˆx) + 2(1 − α)φ(ˆx)[1 − Φ(ˆx)] +≥ +0, +(35) +and +� +∞ +−∞ +φFN(ˆx)dˆx += +2α +� +∞ +−∞ +φ(ˆx)Φ(ˆx)dˆx + 2(1 − α) +� +∞ +−∞ +φ(ˆx)[1 − Φ(ˆx)]dˆx += +2α +� +∞ +−∞ +Φ(ˆx)dΦ(ˆx) + 2(1 − α) +� +∞ +−∞ +[1 − Φ(ˆx)]dΦ(ˆx) += +2α +� 1 +0 +µdµ + 2(1 − α) +� 1 +0 +(1 − µ)dµ += +[αµ2 + (1 − α)(2µ − µ2)] +��1 +0 += +1. +(36) +Lemma C.2 (Asymptotic Unbiased Estimation). For any encoder f and as N → ∞, we have L SUP → L BCL. + +Bayesian Self-Supervised Contrastive Learning +Proof. We draw the conclusion by the Lebesgue Dominant Convergence Theorem and the properties of important sampling: +lim +N→∞ L SUP += +lim +N→∞ E x∼pd +x+∼p+ +x−∼p− +[− log +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ef(x)T f(x− +j ) ] += +E x∼pd +x+∼p+ +x−∼p− +lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ef(x)T f(x− +j ) ] += +E x∼pd +x+∼p+ lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + NEx−∼p−ef(x)T f(x− +j ) ] += +E x∼pd +x+∼p+ lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + N · +� +∞ +0 +ˆxψ(ˆx)dˆx +] += +E x∼pd +x+∼p+ lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + N · +� +∞ +0 +ˆx ψ(ˆx) +φ(ˆx) φ(ˆx)dˆx +] += +E x∼pd +x+∼p+ lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + N · Eˆx∼φ(ˆx)ωˆx] += +E x∼pd +x+∼p+ lim +N→∞[− log +ef(x)T f(x+) +ef(x)T f(x+) + N · Eˆx∼φ(ˆx)ωˆx] += +E x∼pd +x+∼p+ +x−∼pd +[− log +ef(x)T f(x+) +ef(x)T f(x+) + �N +i=1 ωiˆxi +] += +L BCL +(37) + diff --git a/YNFJT4oBgHgl3EQf6S0s/content/tmp_files/load_file.txt b/YNFJT4oBgHgl3EQf6S0s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3e9a941f3f34716a2ce1576423f7b0dd43e9bb0 --- /dev/null +++ b/YNFJT4oBgHgl3EQf6S0s/content/tmp_files/load_file.txt @@ -0,0 +1,1143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf,len=1142 +page_content='Bayesian Self-Supervised Contrastive Learning Bin Liu 1 Bang Wang* 1 Abstract Recent years have witnessed many successful ap- plications of contrastive learning in diverse do- mains, yet its self-supervised version still remains many exciting challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' As the negative sam- ples are drawn from unlabeled datasets, a ran- domly selected sample may be actually a false negative to an anchor, leading to incorrect en- coder training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' This paper proposes a new self- supervised contrastive loss called the BCL loss that still uses random samples from the unlabeled data while correcting the resulting bias with im- portance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The key idea is to design the desired sampling distribution for sampling hard true negative samples under the Bayesian frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The prominent advantage lies in that the desired sampling distribution is a parametric struc- ture, with a location parameter for debiasing false negative and concentration parameter for mining hard negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Experiments validate the effectiveness and superiority of the BCL loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Introduction and Contribution Unsupervised learning has been extensively researched for its advantages of learning representations without human labelers for manually labeling data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' How to learn good representation without supervision, however, has been a long-standing problem in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Recently, con- trastive learning that leverages a contrastive loss (Chopra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Hadsell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2006) to train a representation encoder has been promoted as a promising solution to this problem (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Remarkable successes of contrastive learning have been observed for many applications in dif- ferent domains (Alec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Misra & Maaten, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Nonetheless, its potentials can be further released for a better contrastive loss being designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We study the following self-supervised contrastive learning problem (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Robin- 1liubin0606@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' wangbang@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' School of Electronic Information and Communications, Huazhong Univer- sity of Science and Technology (HUST), Wuhan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Corresponding author: Bang Wang son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2021): Consider an unlabeled dataset X and a class label set C, let h : X → C be the classification function assigning a data point x ∈ X with a class label c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Assume that observing a class label ρ(c) = τ + is uniform and τ − = 1 − τ + is the probability of ob- serving any different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' For a given data point x, let p+(x+) = p(x+|h(x) = h(x+)) denote the probability of another point x+ with the same label as x, and in such a case, x+ is called a positive sample specific to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Likewise, let p−(x−) = p(x−|h(x) ̸= h(x−)) denote the probability of another point x− with a different label to x, and in such a case, x− is called a negative sample specific to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Let f : X → Rd denote the representation learning function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', encoder) to map a point x to an embedding f(x) on a d-dimensional hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Self-supervised contrastive learning is to contrast similar pairs (x, x+) and dissimilar pairs (x, x−) to learn the en- coder f (Wang & Isola, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Wang & Liu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' While the objective is to encourage representa- tions of (x, x+) to be closer than that of (x, x−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In training the encoder, we randomly draw a point from the underlying data distribution pd on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', x ∼ pd, and its positive sam- ple x+ can be easily obtained from some semantic-invariant operation on x like image masking, written as x+ ∼ p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In practice, a negative sample x− is drawn from the unlabeled dataset and x− ∼ pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' However, the sample x− could be potentially with the same label as x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', it is a false negative to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In such a case, the construction of a dissimilar pair (x, x−) would degrade the learned representations (Wang & Liu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' As no prior knowledge about the label of x−, we propose to include an importance weight ω to measure the credibility of a constructed pair (x, x−) for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Contribution: In this paper, we propose the following Bayesian self-supervised Contrastive Learning objective function, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', the BCL loss: LBCL = E x∼pd x+∼p+ x−∼pd −log[ ef(x)T f(x+) ef(x)T f(x+) + �N i=1 ωi · ef(x)T f(x− i ) ], Our main contributions include (i) the derivation of para- metric structural sampling distribution conditioned on hard principle and true principle, (ii) the posterior estimation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='11673v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='LG] 27 Jan 2023 Bayesian Self-Supervised Contrastive Learning of a unlabeled sample being true negative sample, and (iii) the stochastic process depiction to simulate predictions of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Compared with the InfoNCE (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018), we include the importance weight ω in the contrastive loss, which is designed to down-weight a constructed pair (x, x−) for x− being a false negative sample or to up-weight (x, x−) for x− being a true negative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The key idea is to design a desired sampling distribution for hard true negative samples under the Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The prominent advantage lies in that we derive the parametric structure for the desired sampling distribution with a location parameter for debiasing false negative and concentration parameter for mining hard negative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' More detailed analysis on LBCL and derivation of ω computation are given in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We summarize the computation of ω in each training epoch as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In a training epoch, we randomly draw N sam- ples {x− i }N i=1 from training set, which are assumed as neg- ative samples specific to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Let ˆx− i = exp(f(x)T f(x− i )) denote the power exponent of similarity between x and x− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We compute the importance weight ωi for x− i by the following three steps: Step-1: Compute ΦN(ˆx− i ), called the empirical distribution function of ˆx− i , ΦN(ˆx− i ) = 1 N N � j=1 I(ˆx− j ≤ ˆx− i ), (1) where I(·) is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Step-2: Compute p(TN|ˆx− i ), the posterior probability of x− i being a true negative (TN) to x, p(TN|ˆx− i ) = ατ − + (1 − 2α)ΦN(ˆx− i )τ − ατ − + (1 − α)τ + + (1 − 2α)ΦN(ˆx− i )(τ − − τ +), (2) where α is a hyperparameter to be explained latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Step-3: Compute ωi(ˆx− i ), the importance weight of x− i for correcting the bias between actual sampling distribution and desired sampling distribution ωi(ˆx− i ) = p(TN|ˆx− i ) · ˆxβ i 1 N �N i=1 p(TN|ˆx− i ) · ˆxβ i , (3) where β is a hyperparameter to be explained latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Contrastive Loss and Analysis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Contrastive Loss In the context of supervised contrastive learning, dissimilar pairs (x, x−) can be easily constructed by randomly draw- ing a true negative sample x− specific to x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', x− ∼ p−, based on the sample label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The contrastive predictive coding (CPC) (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018) introduces the following InfoNCE loss (Gutmann & Hyv¨arinen, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2012): LSUP = E x∼pd x+∼p+ x−∼p− [− log ef(x)T f(x+) ef(x)T f(x+) + �N i=1 ef(x)T f(x− i ) ] (4) to learn an encoder f : X → Rd/t that maps a data point x to the hypersphere Rd of radius 1/t, where t is the tem- perature scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' As in the CPC, we also set t = 1 in our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In the context of self-supervised contrastive learning, how- ever, as samples’ labels are not available, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', p−(x′) = p(x′|h(x) ̸= h(x′)) is not accessible, the standard approach is to draw N samples from the data distribution pd, which are supposed to be negative samples to x, to optimize the following InfoNCE self-supervised contrastive loss: LBIASED = E x∼pd x+∼p+ x−∼pd [− log ef(x)T f(x+) ef(x)T f(x+) + �N i=1 ef(x)T f(x− i ) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (5) Following the DCL (Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020), it is also called as biased contrastive loss since those supposedly negative samples x− drawn from pd might come from the same class as the data point x with probability τ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Sampling Bias Analysis Let x− ∈ TN denote x− being a true negative (TN) sample specific to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Let x− ∈ FN denote x− being a false negative (FN) sample specific to x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' x− and x are with the same ground truth class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Note that whether x− is a TN or FN is specific to a particular anchor point x, and in what follows, we omit the specific to x for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' It has been proven that for {x− i ∈ TN}N i=1, optimizing the InfoNCE loss LSUP will result in the learning model estimating and optimizing the density ratio p+ p− (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Denote ˆx+ = ef(x)T f(x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The CPC (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018) proves that minimizing LSUP leads to ˆx+ ∝ p+/p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (6) As discussed by (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018), p+/p− preserves the mutual information (MI) of future information and present signals, where MI maximization is a fundamental problem in science and engineering (Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Belghazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2018) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Now consider the InfoNCE loss LBIASED, which can be re- garded as the categorical cross-entropy of classifying one positive sample x+ from unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' For analysis purpose, we rewrite x+ as x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Given N + 1 unlabeled data points, the posterior probability of one data point x0 being Bayesian Self-Supervised Contrastive Learning Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Illustration of LBIASED and mutual information optimiza- tion by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' a positive sample can be derived by P(x0 ∈ pos|{xi}N i=0) = p+(x0) �N i=1 pd(xi) �N j=0 p+(xj) � i̸=j pd(xi) = p+(x0)/pd(x0) p+(x0)/pd(x0) + �N j=1 p+(xj)/pd(xj) (7) To minimize LBIASED, the optimal value for this poste- rior probability is 1, which is achieved in the limit of p+(x0)/pd(x0) → +∞ or p+(xj)/pd(xj) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Mini- mizing LBIASED leads to ˆx+ ∝ p+/pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (8) Note that this is different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (6), since x− i may not be TN for lack of ground truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Denote ˆx+ = m · p+/pd, m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We investigate the gap between optimizing ˆx+ and the optimization objective p+/p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Inserting pd = p−τ − + p+τ + back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (8), we obtain ˆx+ = m · p+ p−τ − + p+τ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (9) Rearranging the above equation yields p+/p− = ˆx+ · τ − m − ˆx+ · τ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (10) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 1 illustrates the approximate shape of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (10) as a frac- tional function, which reveals the inconsistency between InfoNCE LBIASED loss optimization and MI optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' That is, when optimizing InfoNCE loss, the increase of ˆx+ does not lead to the monotonic increase of p+/p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Indeed, the existence of jump discontinuity indicates that the opti- mization of LBIASED does not necessarily lead to the tractable MI optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The reason for the intractable MI optimiza- tion is from the fact that not all {x− i }N i=1 are TN samples, as they are randomly drawn from the data distribution pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' This leads to the inclusion of p+ in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (9) when decomposing the data distribution pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 6 in Ap- pendix provides an intuitive explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The four sampled data points actually contain one FN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Such a FN sam- ple should be pulled closer to the anchor x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' However, as it is mistakenly treated as a negative sample, during model train- ing it will be pushed further apart from the anchor, which breaks the semantic structure of embeddings (Wang & Liu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The Proposed Method In this paper, we consider to randomly draw negative sam- ples {x− i }N i=1 from the unlabeled dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', x− i ∼ pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' As the class label is not accessible, x− i could be either a TN sample or a FN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We propose to include and compute an importance weight ωi into the InfoNCE contrastive loss for correcting the resulting bias of drawing negative samples from pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The ideal situation is that we can set ω = 0 to each FN sample, so that only the hard true negative samples contribute to the calculation of contrastive loss, which relies on the design of desired sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We consider the following two design principles of the sam- pling distribution for drawing {x− i }N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The true princi- ple (Wang & Liu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2021) states that the FN samples should not be pushed apart from the anchor x in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The hard principle (Yannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Florian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Hyun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2016) states that the hard TN samples should be pushed further apart in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' False Negative Debiasing We first consider the true principle for the design of sam- pling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We denote the power exponent of similar- ity between an anchor x and another unlabeled sample x′ as ˆx = ef(x)Tf(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Assume that ˆx is independently and identi- cally distributed with a probability density function φ and cumulative distribution function Φ(ˆx) = � ˆx −∞ φ(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' As x′ can be either a TN sample or a FN sample, so φ contains two populations, denoted as φTN and φFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The problem of computing the LBCL loss Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (1) is reduced to estimating the sum over ˆx ∼ φTN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', �N i=1 ef(x)T f(x− i ), while using samples ˆx ∼ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Existing approaches for solving above problem is the density estimation to fit φ (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2022), where φ is parameter- ized as a two-component mixture of φTN and φFN, such as the Gaussian Mixture Model (Lindsay, 1995), Beta Mixture Model (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' To make the analysis possible, φTN and φFN are postulated to follow a simple density function with fixed parameters, which is a too strong assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In addition, the learning algorithm for estimating φ is expen- 0Bayesian Self-Supervised Contrastive Learning sive, since the mixture coefficients that indicate the probabil- ity of ˆx ∈ TN or FN are hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The parameters of φTN and φFN can only be obtained through the iterative nu- merical computation of the EM algorithm (Dempster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=', 1977) that are sensitive to initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In this paper, we propose an analytic method without explic- itly estimating φ, also called the nonparametric method in the statistical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Consider n random variables from φ ar- ranged in the ascending order according to their realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We write them as X(1) ≤ X(2) ≤ · · · ≤ X(n), and X(k) is called the k-th (k = 1, · · · , n) order statistics (David & Nagaraja, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The probability density function (PDF) of X(k) is given by: φ(k)(ˆx) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='Φk−1(ˆx)φ(ˆx)[1 − Φ(ˆx)]n−k By conditioning on n = 2 we obtain: φ(1) = 2φ(ˆx)[1 − Φ(ˆx)] (11) φ(2) = 2φ(ˆx)Φ(ˆx) (12) Next, we investigate the position of positive and negative samples on the hypersphere, so as to get a deep insight into φTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Consider a (x, x+, x−) triple, there exists a closed ball B[f(x), d+] = {f(·)|d(f(x), f(·)) ≤ d+} with center f(x) and radius d+, where d+ = ∥f(x) − f(x+)∥ is the distance of anchor embedding f(x) and positive sample embedding f(x+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Two possible cases arise: f(x−) ∈ B[f(x), d+] or f(x−) /∈ B[f(x), d+], as illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' We can describe the two cases with the Euclidean distance: Fig 2(a) corresponds to d+ < d−, and Fig 2(b) cor- responds to d− ≤ d+, where d− = ∥f(x) − f(x−)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Note that the Euclidean distance d± = � 2/t2 − 2f(x)Tf(x±) since all embeddings f(·) are on the surface of a hyper- sphere of radius 1/t, so we have ˆx− < ˆx+ for case (a), and ˆx+ ≤ ˆx− for case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Expressed in the notation of order statistics, ˆx− (marked in blue in Fig 2) is a realization of X(1) for case (a), or ˆx− is a realization of X(2) for case (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Two possible cases for the relative positions of anchor, positive, and negative triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The generation process of observation ˆx from φTN can be described as follows: Select case (a) with probability α, and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' φTN(ˆx) and φFN(ˆx) with different α settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' then generate an observation ˆx from φ(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Or select case (b) with probability 1 − α, and then generate an observation ˆx from φ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' That is, φTN is the component of X(1) and X(2) with a mixture coefficient α φTN(ˆx) = αφ(1)(ˆx) + (1 − α)φ(2)(ˆx) (13) Similarly, φFN is the component of X(2) and X(1) with mixture coefficient α: φFN(ˆx) = αφ(2)(ˆx) + (1 − α)φ(1)(ˆx) (14) Note that the way of taking ˆx− as a realization of X(1) for case (a) omits the situation of ˆx− = ˆx+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' The probability measure of ˆx− for such case is 0 as φ is continuous density function dominated by Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='1 (Class Conditional Density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' If φ(ˆx) is continuous density function that satisfy φ(ˆx) ≥ 0 and � +∞ −∞ φ(ˆx)dˆx = 1, then φTN(ˆx) and φFN(ˆx) are probability density functions that satisfy φTN(ˆx) ≥ 0, φFN(ˆx) ≥ 0, and � +∞ −∞ φTN(ˆx)dˆx = 1, � +∞ −∞ φFN(ˆx)dˆx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' There may need a further understanding and clarification of the mixture coefficient α by reviewing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Intuitively, α is the probability that f(x−) falls out of B[f(x), d+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' For a worst encoder f that randomly guesses, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' While for a perfect encoder α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' Therefor, the reasonable value of α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content='5, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFJT4oBgHgl3EQf6S0s/content/2301.11673v1.pdf'} +page_content=' In fact, α reflects the encoder’s capability of scoring a positive sample higher than that of a negative sample, which admits the empirical macro-AUC metric over all anchors x in the training data set D: α = � x∈X +∞ � 0 +∞ � 0 I(ˆx+ ≥ ˆx−)p(ˆx+, ˆx−)p(x)dˆx+dˆx−dx ≃ 1 |D| 1 |D+||D−| � D+ � D− I(ˆx+ ≥ ˆx−) = 1 |D|AUC (15) f(x) f(x f(x) o: f(x)o if(xt) if(xt) (a) 푢1 >= 푢푖. +(5) +If it is believed that the site initiation rate starts from a high value and then follows a downward pattern, the following time-decay +rate function can be used: +휆(1) +푖푛,푖(푢) = Λ푘[푖] exp(−(휂 + 휖)(푢 − 푢푖)), Λ푘[푖] ∼ Γ(훼, 훽), 휂 > 0, 휖 = 10−5(offset), 푢 >= 푢푖. +(6) +The base site activation rate Λ푘[푖] is country-specific and drawn from a Gamma distribution. The posterior predictions of country- +specific site activation rates would provide meaningful information to the clinical operation team for their site planning and +optimization in different countries. If the exploratory analysis (e.g., times series plotting of site activation dates within countries) +shows that the site activation rate generally first increases to a peak and then decays, then the following quadratic rate function +would be a good choice, +휆(2) +푖푛,푖(푢) = Λ푘[푖] exp +( +−(푢 − 푢푖 − 푒)2 +2(휂 + 휖) +) +, Λ푘[푖] ∼ Γ(훼, 훽), 푒 > 0, 휂 >= 0, 휖 = 10−5(offset), 푢 >= 푢푖 +(7) +The quadratic rate function 휆(2) +푖푛,푖(푢) is more flexible than the time-decay rate function 휆(1) +푖푛,푖(푢) because 휆(2) +푖푛,푖(푢) can absorb the +decaying rate function as a special case when the parameter 푒 is close to 0. +For the 푖th study-country combination, the observed data regarding site activation in the time interval (푢푖, 푢′ +푖] include the +total number of sites activated, denoted by 푛푖푛,푖 +(푢′ +푖 +) and the corresponding site activation times for the 푖th combination, i.e., +⃖⃖⃖⃖⃖⃗ +푢푖푛,푖 = [푢푖푛,푖,푚|푚 ∈ {1, ..., 푛푖푛,푖 +(푢′ +푖 +)}]. It can be shown that the likelihood function of site activation data over all 푁 study-country +combinations with time-decay rate function has the following form, +퐿 (훼, 훽, 휂; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) = +푁 +∏ +푖=1 +퐿 (훼, 훽, 휂; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +)) ∝ +푁 +∏ +푖=1 +⎛ +⎜ +⎜⎝ +Γ (푛푖푛,푖 +(푢′ +푖 +) + 훼) +Γ (훼) 훽훼 +( +훽(휂 + 휖) +훽 (1 − exp (− (휂 + 휖) +(푢′ +푖 − 푢푖 +))) + (휂 + 휖) +)(푛푖푛,푖(푢′ +푖)+훼) ∏ +푚 +exp (− (휂 + 휖) +(푢푖푛,푖,푚 − 푢푖 +))⎞ +⎟ +⎟⎠ +(8) +Let 휋 (훼), 휋 (훽), and 휋 (휂) be the prior distributions for the parameters 훼, 훽, and 휂 respectively. Then the posterior distribution is, +휋 (훼, 훽, 휂|⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) ∝ 퐿 (훼, 훽, 휂; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) 휋 (훼) 휋 (훽) 휋 (휂) +(9) +By conditioning on 훼, 훽, and 휂 and the observed data, we can generate Λ푘(푘 ∈ {1, ..., 퐶}) for country 푘 from the following +gamma distribution, +푓 (Λ푘|훼, 훽, 휂; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) ∼ 퐺푎푚푚푎 +( +∑ +푖∶푘[푖]=푘 +푛푖푛,푖 +(푢′ +푖 +) + 훼, +훽(휂 + 휖) +훽 (1 − exp (− (휂 + 휖) ∑ +푖∶푘[푖]=푘 +(푢′ +푖 − 푢푖 +))) + (휂 + 휖) +) +(10) +The likelihood function based on the quadratic rate function and the corresponding posterior distribution are, +퐿 (훼, 훽, 휂, 푒; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) = +푁 +∏ +푖=1 +⎛ +⎜ +⎜ +⎜ +⎜⎝ +Γ (푛푖푛,푖 +(푢′ +푖 +) + 훼) +Γ (훼) 훽훼 +⎛ +⎜ +⎜ +⎜⎝ +훽 +훽 +(√ +2휋(휂 + 휖) +[ +Φ( +푢′ +푖−푢푖−푒 +√ +휂+휖 ) − Φ( +−푒 +√ +휂+휖) +]) ++ 1 +⎞ +⎟ +⎟ +⎟⎠ +(푛푖푛,푖(푢′ +푖)+훼) +∏ +푚 +exp +( +− +(푢푖푛,푖,푚 − 푢푖 − 푒)2 +2 (휂 + 휖) +)⎞ +⎟ +⎟ +⎟ +⎟⎠ +, +(11) +휋 (훼, 훽, 휂, 푒|⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) ∝ 퐿 (훼, 훽, 휂, 푒; ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) 휋 (훼) 휋 (훽) 휋 (휂) 휋 (푒) , +(12) + +Zhong ET AL +7 +where Φ (⋅) is the cdf of the standard normal distribution. By conditioning on 훼, 훽, 휂, 푒 and observed data, we can generate Λ푘 +for country 푘 from the following gamma distribution, +푓 (Λ푘|훼, 훽, 휂, 푒, ⃖⃖⃖⃖⃖⃗ +푢푖푛,푖, 푛푖푛,푖 +(푢′ +푖 +) , 푖 ∈ 푁) ∼ 퐺푎푚푚푎 +⎛ +⎜ +⎜ +⎜⎝ +∑ +푖∶푘[푖]=푘 +푛푖푛,푖 +(푢′ +푖 +) + 훼, +훽 +훽 +(√ +2휋(휂 + 휖) ∑ +푖∶푘[푖]=푘 +[ +Φ +( 푢′ +푖−푢푖−푒 +√ +휂+휖 +) +− Φ +( +−푒 +√ +휂+휖 +)]) ++ 1 +⎞ +⎟ +⎟ +⎟⎠ +. +(13) +In the modeling of the rate function for site initiation, we note that a study effect Ω푗[푖] may be considered. Unlike the country +effect, the study effect is not of direct interest for the simulation of new trials, because a new simulated trial is always a different +trial from the historical ones while a common set of countries are used in both historical and future studies. Due to much higher +computation complexity from adding an additional study effect and its marginal benefit, we choose to not include it in site +initiation modeling. +For the computation of posterior distribution of model parameters, we use the R function MCMCmetrop1R in the R package +MCMCpack19, which can produce samples from the user derived posterior distribution function (as in 9 and 12). +3.4 +Subject enrollment model +In terms of modeling subject enrollment process, our observed data only include site-level summary of subject enrollment +information from historical studies, where for each site in a historical study, we observe the total number of enrolled subjects +only instead of the actual date when each recruited subject was enrolled. This data limitation does not allow for the modeling of +the time dynamics of subject enrollment stochastic process within each site and restricts our approaches to those that model the +overall site-level enrollment rates. +Our historical study data set used for modeling enrollment consists of a number of 푁 study-country combinations from 푆 +different studies, where for each study-country combination 푖, the number of sites activated is 푛푖푛,푖. Let 푁푒푛,푖,푚 be the total number +of subjects enrolled at the site 푚 in the 푖th study-country combination. Then 푁푒푛,푖,푚 can be modeled by the following generalized +mixed-effects Poisson regression model14,15,20, +푁푒푛,푖,푚 ∼ independently as 푃 표푖푠푠표푛(휇푖푚) where log(휇푖푚) = 휇푒푛 + log(푑푖푚) + 훾푒푛,푘[푖]. +(14) +The parameter 휇푖푚 is the mean number of enrolled subjects for site 푚 in the 푖th study-country combination. 휇푒푛 is the grand inter- +cept for the model. The term log(푑푖푚) is the offset that accounts for the enrollment duration for the site 푚 in the 푖th study-country +combination, where 푑푖푚 is the corresponding total enrollment duration. Here the enrollment duration for a site is defined to be +the time duration between the site activation date and the last subject enrollment date that is standardized by the normalization +factor 푚. The random parameter 훾푒푛,푘[푖] ∼ 푁(0, 휎2 +푒푛,훾) denotes the country effect and 푘[푖] is the country index for the 푖th study- +country combination. If exploratory graphical analysis suggests that the average enrollment rate over sites in a country changes +greatly across different historical studies, then the random study effect 훽푒푛,푗[푖] can be added to the model. +It is interesting to note that when the parameter 휇푒푛 is absorbed in 훾푒푛,푘[푖] ∼ 푁(휇푒푛, 휎2 +푒푛,훾), then we would essentially assume a +lognormal distribution for the country effect 휆푒푛,푘[푖] = exp(훾푒푛,푘[푖]) where 푁푒푛,푖,푚 ∼ 푃 표푖푠푠표푛(푑푖푚휆푒푛,푘[푖]). This sheds light on the +connection of our proposed model to the popular approach of the Poisson-Gamma model in which, we would impose a Gamma +distribution on the parameter 휆푒푛,푘[푖]. +For the computation, the glmer function in the lme4 package18 is used to estimate the mixed-effect poisson regression model +with an offset included. +3.5 +Monte Carlo simulation to predict future recruitment process +The parameters of three enrollment related models (i.e., country start-up time model, site initiation model, and subject enrollment +model) need to be estimated before we can proceed to predict future recruitment timeline for the current study under planning. +We take the Monte Carlo simulation based approach to randomly generate a number of simulated trials, where the parameters of +three recruitment related models are simulated first for each trial and then the country start-up times, site initiation times within +each simulated country, and subject enrollment times within each simulated site from each simulated country are simulated +sequentially, up to a pre-specified upper time limit, for each simulated trial. We choose 1000 to be the total number of trials +to be simulated. The upper time limit can be chosen to be the maximum enrollment duration of all completed historical trials + +8 +Zhong ET AL +within an organization and it can be therapeutic area dependent. The computation complexity is highly dependent on this upper +time limit. Hence we suggest that it is chosen with the knowledge from expert users in the therapeutic area which the current +study belongs to, especially when a batch of enrollment scenarios need to be explored. In our case we set the upper time limit +to be 5 years. Once all recruitment related data are simulated for all 1000 simulated trials, the median, lower/upper percentiles, +and other statistics can be summarized for any quantities of interest. For instance, to provide the study enrollment completion +prediction, we can report the median last subject first dose date and its lower and upper percentiles as a prediction interval. +The following paragraphs describe in details how to simulate from each of the country start-up time, site initiation and subject +enrollment models. +To simulate country start-up times in a new trial, we need to simulate values for the fixed effect parameters 휇푐푠푢 and 훼푐푠푢 +and the random effects 훾푐푠푢,푘 and 훽푐푠푢,푗 as well as the random error 휖푖. For the fixed parameters 휇푐푠푢 and 훼푐푠푢, we draw from +the asymptotic normal distribution of their restricted maximum likelihood (REML) estimators, where the mean of the normal +distribution equals the REML estimates and the standard deviation equals the standard error of REML estimates. For the random +country effect 훾푐푠푢,푘 for a country 푘 ∈ {1, ..., 퐶}, we simulate from a normal distribution with mean and variance equal to the +conditional mode and conditional variance respectively of the random effect 훾푐푠푢,푘, because the countries we use in the new study +are the ones used in the historical studies. For the random study effect 훽푐푠푢,푗, we simulate from its unconditional distribution +푁(0, ̂휎2 +푐푠푢,훽) where ̂휎2 +푐푠푢,훽 is the REML estimate of 휎2 +푐푠푢,훽, because a study under planning is a new study different from the +historical ones. For the random error 휖푖, we simulate from the normal distribution 푁(0, ̂휎2 +휖) where ̂휎2 +휖 is the REML estimate of +휎2 +휖. Once all the parameters are simulated, they can be combined by, for instance, the equation 2, to produce the country start-up +time for country 푘 in a simulated trial 푗. +To conduct site initiation simulation, we need to first draw the values for model parameters (i.e., 훼, 훽, 휂 and 푒) from the +MCMC samples for the posterior distribution of either 9 or 12. The country-specific base site activation rate Λ푘 is drawn from +the Gamma distribution of either 10 or 13, conditional on the previously drawn model parameter values and observed historical +data. Then the total number of sites to be activated and the site activation times can be drawn following the non-homogeneous +Poisson process defined by 5. +To conduct subject enrollment simulation, we follow the similar approach as in country start-up time simulation to simulate +values for the fixed effect parameter 휇푒푛 and the random effects 훾푒푛,푘 and 훽푒푛,푗 for a country 푘 in a simulated trial 푗. The enrollment +duration offset is set according to the pre-specified upper time limit for simulation, 5 years. Then the total number of subjects +to be enrolled and the subject enrollment times can be drawn following the homogeneous Poisson process. +4 +PERFORMANCE EVALUATION +4.1 +General consideration +To validate the performance of our proposed modeling framework, we select a set of recently completed studies within our +organization, run our models to generate the predictions on enrollment duration, and compare the predictions to the ground-truth +values. The enrollment duration of a validation study is defined to be the time duration between the protocol approval date and +the last subject first dose date. To get the enrollment prediction for a validation study, the inputs provided to our model include +the total number of subjects, the total number of sites, the therapeutic area, disease indication and the patient population of the +validation study. In addition, the enrollment related data from historical studies with the same disease indication and patient +population as the validation study are used for estimation of model components. Here historical studies are those studies that +have their enrollment completed prior to the protocol approval date of the validation study. We compare the performance of our +proposed modeling framework to a previous internally-developed enrollment forecast system based on the traditional statistical +methods that fit simple probability distributions, described in the next subsection. Furthermore, we show that our modeling +and simulation framework calibrates the data variability correctly by comparing the nominal level vs. the coverage rate for +prediction intervals of various nominal levels. Finally, we demonstrate how to generate the predicted enrollment curves through +time, overlaid with confidence bands, which are deemed very informative to trial operation planning by our users. +4.2 +A previous enrollment forecast system +In a previous enrollment forecast system developed internally, the whole enrollment procedure is also divided into three segments +with different definitions. The first segment is called site start-up period-1, which is the period from the final protocol approval + +Zhong ET AL +9 +TABLE 1 Selection criteria for a candidate set of studies for model performance validation +Criteria +Value +Enrollment completion date +Within past 3.5 years +Interventional vs. observational studies +Interventional studies only +Study phase +2 and 3 +Therapeutic area +Oncology, neuroscience, immunology, and general medicine +Total number of sites +10-500 sites +Total number of subjects +50-1600 subjects +TABLE 2 Number of selected studies in different therapeutic areas +Immunology +General Medicine +Neuroscience +Oncology +Number of studies +14 +4 +3 +4 +date to the open date of each site. The second segment is site start-up period-2, which is defined as the time duration between the +site open date and the first subject enrollment date for the site. The last segment is the enrollment period which is from the first +subject enrollment date for the site to the last subject enrollment date of the whole study. The models underlying the previous +forecast system are primitive compared to our proposed modeling framework. For site start-up period-1, no formal statistical +modeling is employed and the simulation is simply based on bootstrapping the observed periods from the historical data in +the same therapeutic area and country with replacement. For the site start-up period-2, the modeling utilizes an log-normal +distribution to fit historical data for each country and then random samples are drawn from it for various sites in each country. +For the modeling of subject enrollment, a country-level modeling approach is taken. A separate Gamma distribution is fitted +to the historically observed enrollment rates of all sites in each country and then during the simulation stage, the enrollment +rate for each site is randomly drawn from the Gamma distribution and used as the parameter of Poisson distribution to simulate +subject enrollment in each day. +4.3 +Selection of evaluation studies +To ensure the objectivity of the study selection process, an independent steering committee other than the modeling team (which +the authors belong to) forms to lead the selection of a set of enrollment completed studies for model evaluation. First, a candidate +set of studies are pulled from our internal database, based on a list of search criteria given in Table 1. Then for each candidate +study, the actual enrollment curve is plotted for the purpose of checking unusual shapes, such as pauses and drastic enrollment +speed changes. These unusual shapes are often due to rare unforeseeable events. Whenever such events happen, our internal +users would manually assess the impact on a case-by-case basis and we shall not expect any models can still provide accurate +predictions. Based on the manual inspection of the enrollment curves, a list of studies with unusual shapes in enrollment curves +is proposed so as to be removed from model performance evaluation, and the steering committee makes the final decision on +whether a study should be included in the list of evaluation studies. Ultimately, a total of 25 studies are selected from four different +therapeutic areas to test the performance of our proposed modeling approach in comparison to the previous internally-developed +forecast system. Table 2 shows the break-up of 25 studies into four therapeutic areas. +4.4 +Prediction accuracy +To evaluate the model performance, we apply both our proposed modeling framework and the previous enrollment forecast +system to the 25 studies selected by the independent steering committee. In terms of the performance evaluation metrics, we +report the rates of coverage of the ground-truth enrollment duration by prediction intervals with fixed radii set to be +/- 1, 2, +and 3 months (so the widths of prediction intervals are 2, 4 and 6 months respectively) as well as the coverage rate by the + +10 +Zhong ET AL +TABLE 3 Predictions of enrollment duration for 25 studies by our new modeling framework vs. a previous internally-developed +enrollment forecast system +Coverage of +Coverage of +Coverage of +Coverage of +Width of +Median Absolute +Mean Absolute +Model +2-Month PI +4-Month PI +6-Month PI +95% PI +95% PI +Prediction Error +Prediction Error +New +7 (28%) +8 (32%) +13 (52%) +23 (92%) +20 months +2.8 months +4.8 months +Previous +5 (20%) +6 (24%) +9 (36%) +5 (20%) +2 months +5.6 months +5.8 months +TABLE 4 Nominal level vs. actual coverage rate of prediction intervals from our new modeling framework +Nominal Level of PI +40% +50% +70% +95% +Number of studies correctly predicted +13 +15 +17 +23 +Coverage rate +52% +60% +68% +92% +SE for coverage rate +10% +10% +9% +5% +95% CI for coverage rate +32%-72% +40%-80% +50%-86% +82%-100% +Prediction interval width +5.3 Months +7 Months +10 Months +20 months +prediction interval of non-fixed width with nominal confidence level 95%. In addition, we also look at the mean and median +absolute prediction error of the median predicted enrollment duration compared to the ground-truth. +From Table 3, our proposed approach achieves higher coverage rates for all prediction intervals of various fixed widths than +the previous enrollment forecast system. In terms of median absolute prediction error, our proposed approach achieves 50% +reduction from 5.6 months to 2.8 months, while for the mean absolute prediction error that is often influenced by large errors, +our method still provides 17% reduction from 5.8 months to 4.8 months. This suggests that the traditional distribution-based +approach is not adequate to capture the the sophisticated patterns behind recruitment related processes, compared to our more +complex modeling approach based on mixed-effects models and non-homogeneous Poisson processes models. Table 5 and 6 +provide the detailed outputs from our new modeling approach and the previous forecast system respectively. +4.5 +Coverage of prediction intervals +In term of the prediction interval of nominal level 95%, our method has achieved the actual coverage rate of 92% (see Table +3), while the previous forecast system is ill-calibrated and gives a coverage rate of 20% together with an extremely narrow +width. This implies that the variability in historical data is not accounted correctly in either modeling or simulation in the +previous enrollment forecast system. While the prediction intervals of other nominal levels are not available in the previous +forecast system, we further investigate the discrepancy between the normal level vs. actual coverage rate of our proposed new +modeling approach. Table 4 shows that the nominal levels are very consistent with the actual coverage rates at different nominal +levels. This is because our methodology properly accounts for the data variability at different levels (such as study and country +level) with appropriate random effects. In the simulation stage, the random effects are simulated from the correct conditional or +unconditional asymptotic sampling distributions correspondingly (see subsection 3.5). Although 95% is a widely used number +for the level of significance in statistical hypothesis testing, the width of 95% prediction intervals is in general too large to be +used in practice in our scenario. This is often due to the large amount of variability naturally in enrollment related processes +and the limited number of historical studies available for model estimation. The 40% or 50% prediction intervals tend to offer a +good balance between the coverage rate and the interval width in our application. +4.6 +Predicted enrollment curves +In addition to the predicted enrollment duration, our users also find it very informative to their planning job to have a predicted +enrollment curve throughout the whole time course from protocol approval to enrollment completion. This can be achieved by + +Zhong ET AL +11 +TABLE 5 Predictions of enrollment duration for 25 studies by our new modeling framework (unit:month) +Study +Actual +Median Predicted +Within +Within +Within +NO. +Duration +Duration (95% PI) +Residual ++/- 1 Month ++/- 2 Months ++/- 3 Months +1 +21.23 +23.94 ( 14.66 , 44.90 ) +2.70 +No +No +Yes +2 +20.93 +20.26 ( 14.02 , 34.03 ) +-0.67 +Yes +Yes +Yes +3 +21.87 +15.17 ( 10.38 , 22.68 ) +-6.69 +No +No +No +4 +15.63 +16.16 ( 9.88 , 30.08 ) +0.53 +Yes +Yes +Yes +5 +22.47 +29.22 ( 11.45 , Inf ) +6.75 +No +No +No +6 +26.37 +20.46 ( 13.71 , 31.27 ) +-5.90 +No +No +No +7 +23.20 +23.63 ( 17.58 , 34.35 ) +0.43 +Yes +Yes +Yes +8 +26.83 +24.31 ( 16.07 , 40.46 ) +-2.53 +No +No +Yes +9 +30.27 +20.91 ( 10.95 , 51.80 ) +-9.35 +No +No +No +10 +37.33 +25.31 ( 15.37 , 55.11 ) +-12.02 +No +No +No +11 +28.77 +26.02 ( 17.11 , 40.29 ) +-2.74 +No +No +Yes +12 +19.93 +19.18 ( 12.29 , 34.41 ) +-0.75 +Yes +Yes +Yes +13 +17.53 +20.18 ( 13.54 , 35.40 ) +2.65 +No +No +Yes +14 +11.27 +10.13 ( 6.38 , 17.34 ) +-1.14 +No +Yes +Yes +15 +10.03 +10.07 ( 6.89 , 17.29 ) +0.03 +Yes +Yes +Yes +16 +21.13 +22.11 ( 15.72 , 36.70 ) +0.98 +Yes +Yes +Yes +17 +19.93 +12.77 ( 7.96 , 20.31 ) +-7.16 +No +No +No +18 +16.77 +19.55 ( 9.96 , 60.24 ) +2.79 +No +No +Yes +19 +18.37 +14.86 ( 9.24 , 25.40 ) +-3.50 +No +No +No +20 +30.80 +19.14 ( 12.48 , 34.99 ) +-11.66 +No +No +No +21 +24.90 +18.61 ( 9.39 , 46.12 ) +-6.29 +No +No +No +22 +26.37 +31.81 ( 21.82 , 51.40 ) +5.44 +No +No +No +23 +20.63 +11.58 ( 6.90 , 17.76 ) +-9.05 +No +No +No +24 +16.10 +35.03 ( 16.21 , Inf ) +18.93 +No +No +No +25 +10.83 +11.58 ( 8.11 , 17.70 ) +0.75 +Yes +Yes +Yes +first selecting a grid of equally spaced time points (e.g., in months) and then summarizing the median, low percentile, upper +percentile total enrollment number for each time point in the grid. In Figure 2 and 3, we show the predicted enrollment curves +with 95% and 50% confidence bands for a well-predicted study. Note that the upper confidence band is capped by the target +number of subjects enrolled for the study, as preferred by our users. +5 +DISCUSSION +For the problem of forecasting enrollment at the planning stage, we have developed a novel statistical modeling framework, +which is based on GLMM and the use of non-homogeneous Poisson processes through Bayesian framework to systematically +model the country initiation, site activation and subject enrollment in sequential steps. Our new modeling framework shows a +substantial improvement in prediction accuracy in comparison with the traditional statistical approach that fits simple probability +distributions, based on a collection of 25 pre-selected studies from four therapeutic areas. Furthermore, we have showed that our +modeling and simulation framework calibrates the data variability appropriately and gives correct coverage rates for prediction +intervals of various nominal levels. With the Monte Carlo simulation, we demonstrated how to generate the median predicted +enrollment curve with confidence bands. It is no harder to generate predictions on other recruitment related quantities such as +the median number of sites for each country and the median country start date. +In terms of the modeling of country start-up time, we utilize a generalized linear mixed-effects model to account for the +country-level and study-level variability in historical studies as well as the effects of fixed covariates. Machine learning models + +12 +Zhong ET AL +TABLE 6 Predictions of enrollment duration for 25 studies by a previous internally-developed enrollment forecast system +(unit:month) +Study +Actual +Median Predicted +Within +Within +Within +NO. +Duration +Duration (95% PI) +Residual ++/-1 Month ++/-2 Months ++/-3 Months +1 +21.23 +24.07 ( 22.20 , 26.10 ) +2.83 +No +No +Yes +2 +20.93 +21.37 ( 19.90 , 22.83 ) +0.43 +Yes +Yes +Yes +3 +21.87 +13.53 ( 12.87 , 14.17 ) +-8.33 +No +No +No +4 +15.63 +14.23 ( 13.47 , 14.93 ) +-1.40 +No +Yes +Yes +5 +22.47 +19.00 ( 18.03 , 20.13 ) +-3.47 +No +No +No +6 +26.37 +17.67 ( 16.87 , 18.37 ) +-8.70 +No +No +No +7 +23.20 +23.70 ( 22.50 , 24.70 ) +0.50 +Yes +Yes +Yes +8 +26.83 +14.33 ( 13.47 , 15.20 ) +-12.50 +No +No +No +9 +30.27 +26.50 ( 24.37 , 28.63 ) +-3.77 +No +No +No +10 +37.33 +25.80 ( 24.47 , 27.60 ) +-11.53 +No +No +No +11 +28.77 +16.30 ( 15.73 , 16.83 ) +-12.47 +No +No +No +12 +19.93 +20.40 ( 19.00 , 22.17 ) +0.47 +Yes +Yes +Yes +13 +17.53 +13.50 ( 12.67 , 14.23 ) +-4.03 +No +No +No +14 +11.27 +9.07 ( 8.47 , 9.90 ) +-2.20 +No +No +Yes +15 +10.03 +9.60 ( 8.83 , 10.40 ) +-0.43 +Yes +Yes +Yes +16 +21.13 +15.33 ( 14.47 , 16.10 ) +-5.80 +No +No +No +17 +19.93 +13.43 ( 12.63 , 14.13 ) +-6.50 +No +No +No +18 +16.77 +14.40 ( 13.37 , 15.43 ) +-2.37 +No +No +Yes +19 +18.37 +12.77 ( 11.73 , 13.63 ) +-5.60 +No +No +No +20 +30.80 +19.90 ( 18.83 , 21.23 ) +-10.90 +No +No +No +21 +24.90 +32.07 ( 29.07 , 36.37 ) +7.17 +No +No +No +22 +26.37 +34.73 ( 30.27 , 39.97 ) +8.37 +No +No +No +23 +20.63 +11.40 ( 10.63 , 12.20 ) +-9.23 +No +No +No +24 +16.10 +31.07 ( 27.37 , 35.00 ) +14.97 +No +No +No +25 +10.83 +11.73 ( 10.80 , 12.70 ) +0.90 +Yes +Yes +Yes +(e.g., gradient boosting) can be exploited if more covariates are available for modeling. But appropriately accounting for the +variability of predictions from machine learning models is not a easy task. Another potential caveat is that more covariate +information is needed at the time of enrollment prediction. For a study at the portfolio planning stage, often limited information +is known about the study. +With regard to the modeling of site activation process within a country, if a specific country initiation date is available, we may +need to consider the fact that the time duration between the country initiation date and its first site activation date could be much +longer than the times between successive activation dates of subsequently activated sites. In this case, we would recommend to +model the time between country initiation and its first site activation separately from the rest of the activated sites. An alternative +approach is to incorporate the time between country initiation and its first site activation into the definition of country start-up +time and handle it implicitly in the modeling of the country start-up times. +For the modeling of subject enrollment, our observed data only include site-level aggregate subject enrollment information +from historical studies. If instead the actual date when each subject was enrolled is observed, a non-homogeneous Poisson process +can be used to estimate the time-varying rate function for subject enrollment process in a site with each country. Likewise, we +may need to take into consideration that it often takes longer time to enroll the first subject due to the overhead site preparation +work and hence modeling the time to the first subject enrollment separately may be more appropriate. +When using an enrollment forecast system that depends on historical studies for model estimation, it is important for the end +user (e.g., clinical operation team) to conduct data-checking for the historical studies being used. If the summary statistics on, +for instance, enrollment speed, in historical studies are systematically slower or faster than what the user expects, caution needs + +Zhong ET AL +13 +FIGURE 2 Predicted enrollment curve with 95% confidence bands +to be taken when interpreting the predicted enrollment duration. Certain ad-hoc measures may be considered, such as excluding +some studies with extreme statistics or using a multiplier to adjust the speed of enrollment. +Disclosure +This manuscript was sponsored by AbbVie. AbbVie contributed to the design, research, and interpretation of data, writing, +reviewing, and approved the content. All authors are employees of AbbVie Inc. and may own AbbVie stock. +Data availability statement +The authors elect to not share data. + +300 +100 +0- +- +200 +400 +800 +DaysFromProtocol ApprovalDate14 +Zhong ET AL +FIGURE 3 Predicted enrollment curve with 40% confidence bands +References +1. Heitjan DF, Ge Z, Ying sG. Real-time prediction of clinical trial enrollment and event counts: A review. Contemporary +Clinical Trials 2015; 45: 26 - 33. 10th Anniversary Special Issuedoi: https://doi.org/10.1016/j.cct.2015.07.010 +2. Anisimov VV. Discussion on the paper “real-time prediction of clinical trial enrollment and event counts: a review”, by DF +Heitjan, Z Ge, and GS Ying. Contemporary clinical trials 2016; 46: 7–10. +3. Bagiella E, Heitjan DF. Predicting analysis times in randomized clinical trials. 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Statistics in medicine 2018; 37(4): 572–589. + +16 +Zhong ET AL + diff --git a/atAzT4oBgHgl3EQfZfwz/content/tmp_files/load_file.txt b/atAzT4oBgHgl3EQfZfwz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c68cfd264372b3acb84dff24602ae6604acea25 --- /dev/null +++ b/atAzT4oBgHgl3EQfZfwz/content/tmp_files/load_file.txt @@ -0,0 +1,727 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf,len=726 +page_content='Received: Added at production Revised: Added at production Accepted: Added at production DOI: xxx/xxxx RESEARCH ARTICLE Enrollment Forecast for Clinical Trials at the Portfolio Planning Phase Based on Site-Level Historical Data Sheng Zhong | Yunzhao Xing | Mengjia Yu | Li Wang Statistical Innovation Group, Data and Statistical Sciences, AbbVie Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', Illinois, United States Correspondence Li Wang, Email: wangleelee@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='com Sheng Zhong, Email: zhongever@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='com Present Address 1 N Waukegan Rd, North Chicago, IL, 60064 Summary Accurate forecast of a clinical trial enrollment timeline at the planning phase is of great importance to both corporate strategic planning and trial operational excel- lence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' While predictions of key milestones such as last subject first dose date can inform strategic decision-making, detailed predictive insights (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', median number of enrolled subjects by month for a country) can facilitate the planning of clinical trial operation activities and promote execution excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The naïve approach often calculates an average enrollment rate from historical data and generates an inaccu- rate prediction based on a linear trend with the average rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The traditional statistical approach utilizes the simple Poisson-Gamma model that assumes time-invariant site activation rates and it can fail to capture the underlying nonlinear patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', up- and-down site activation pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We present a novel statistical approach based on generalized linear mixed-effects models and the use of non-homogeneous Poisson processes through Bayesian framework to model the country initiation, site activa- tion and subject enrollment sequentially in a systematic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We validate the performance of our proposed enrollment modeling framework based on a set of pre- selected 25 studies from four therapeutic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Our modeling framework shows a substantial improvement in prediction accuracy in comparison to the traditional sta- tistical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Furthermore, we show that our modeling and simulation approach calibrates the data variability appropriately and gives correct coverage rates for pre- diction intervals of various nominal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Finally, we demonstrate the use of our approach to generate the predicted enrollment curves through time with confidence bands overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' KEYWORDS: Non-homogeneous Poisson processes, Bayesian hierarchical models, Generalized linear mixed-effects models, Enrollment, Portfolio planning 1 INTRODUCTION In the portfolio planning stage of clinical development, projected trial enrollment duration and associated costs are crucial feasibility factors that senior management has to consider before deciding whether to fund the trial or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Therefore, a predictive modeling algorithm capable of providing accurate enough enrollment forecast across all portfolios to facilitate management’s decision is highly desirable in any pharmaceutical company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The naive and simplest approach currently utilized in practice rests arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='01351v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='AP] 3 Jan 2023 2 Zhong ET AL on the concept of average enrollment rate defined as the mean number of subjects enrolled per site per month for a study, often abbreviated as 푝푠푚, 푝푠푚 = number of enrolled subjects number of sites × enrollment time and the assumption that every site in a trial will have such a constant enrollment rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The projected enrollment duration for a planned trial is simply planned sample size number of sites proposed × 푝푠푚, where the 푝푠푚 rate in the denominator can be directly taken from the average rate in historical studies, specified based on expert knowledge or obtained in a mixed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It is a fast and easy way to project enrollment duration based on sample size and the total number of sites proposed for the new trial and no deep statistical thinking or theory is involved behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Methodology-wise, this approach is purely empirical, does not provide quantification of uncertainty such as confidence intervals, and replies on strong assumption on the constant enrollment rate for all sites overtime, which is often not met in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This approach is usually the first tool implemented within a pharmaceutical company, mostly by a team with less formal statistical training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' With unsatisfactory predictive performance from this naive approach, a more statistics modeling oriented team is often consulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Thus the need and the gap of developing a more systematic and accurate statistical predictive modeling approach pose both opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In the literature, various statistical approaches to modeling and predicting patient accrual have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The Heitjan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1,2 review paper provides an excellent systematic literature review summarizing existing patient accrual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Bagiella and Heitjan3 used a homogenous Poisson process to model the recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Anisimov and Fedorov4 improved the recruitment model by using a Poisson-Gamma mixture model to handle the variation in recruitment rate across multiple centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Such type of methods is basically utilizing random effects models5,6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It aims to capture the heterogeneity in enrollment rates across different centers whereas the enrollment pattern within each center is described by a homogeneous Poisson process with gamma distributed rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 implemented a time-varying rate function that allows modeling the time decay trend in recruitment while taking site initiation into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='9 investigated a Bayesian approach to accrual modeling using a non- homogeneous Poisson process where region-specific accrual is accounted in their framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Zhang and Long10,11 employed a non-homogeneous Poisson process to model patient accrual where the underlying accrual rates are allowed to change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='12 defined time to endpoint maturation framework and linked the concept to key milestone dates in clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' They proposed a simulation based non-homogeneous Poisson process with a normal kernel enrollment rate which can capture the up-and-down enrollment trend in reality and provided improved prediction performance in both simulated and real study enrollment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The last few works mentioned above have motivated us to apply a non-homogeneous Poisson process through Bayesian framework to model site activation process with some technical adaptation to account for multiple historical studies as input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' However, the modeling frameworks above are mainly proposed to be used in trial monitoring at the execution stage where actual patients have already been enrolled into the trial and the goal is to use the accumulated in-trial data up to a certain time point to predict or re-forecast future enrollment beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' There are not many published literature for enrollment forecast in the portfolio planning phase yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To fill the gap, we propose a novel statistical framework based on generalized linear mixed-effects models (GLMM) and the use of non-homogeneous Poisson processes through Bayesian hierarchical framework to model and predict the country initiation, site activation and subject enrollment sequentially in a systematic fashion, utilizing historical site-level enrollment related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We randomly selected 25 completed studies across four therapeutic areas to validate the performance of our proposed modeling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In particular, we compare the prediction accuracy of our proposed modeling framework vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' a previous forecast system developed in our company based on the traditional statistical methods that fit simple probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Furthermore, we show that our modeling and simulation approach calibrates the data variability appropriately and gives correct coverage rates for prediction intervals of various nominal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Finally, we demonstrate the use of our approach to generate the predicted enrollment curves through time with overlaid confidence bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 2 INPUT DATA AND FORECAST WORKFLOW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1 Enrollment framework The whole enrollment procedure in clinical trials is not just a simple step about subject enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Instead, it is a complicated process with multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To facilitate the subsequent statistical model development, a comprehensive enrollment framework that clearly defines each step of the whole enrollment procedure needs to be specified first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In the literature, people mainly focused on the subject enrollment process5,6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' However in real-world practice, the time of several important operational steps prior to subject enrollment like country start-up and site start-up would have a crucial impact on the trial duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In this paper, we dissected the trial duration and established a comprehensive enrollment framework taking into account of all the Zhong ET AL 3 important operational steps of a clinical trial enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Advanced statistical models are developed (depending on historical data availability) for each segment of the whole enrollment framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The enrollment framework is defined from the final protocol approval date to the last subject first dose (LSFD) date of a study with three sequential segments as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' FIGURE 1 Three Sequential Segments of Enrollment Framework The first segment of the enrollment framework is the country start-up process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The country start-up time is defined as the time from the study start up date to the country start up date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In our consideration, all the selected countries in a clinical trial will share the same study start date which is the final protocol approval date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' On the other hand, each country will have its own country start date which is defined in our case as the country’s 1st site activation date due to the lack of a specific country start date in our historical data for modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The country start-up is a country-level procedure that involves country approval and preparation activities, such as country/region regulatory approval and IRB approval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It can be affected significantly by a couple of factors, including but not limited to sponsors, countries/regions, therapeutic areas, and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For modeling purposes, data availability is also an important factor to be considered, since the final protocol approval date and site activation date are typically not available in the most common public databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Therefore, dedicated integrated data from multiple data sources are required to estimate the country’s start-up time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Immediately following the country start-up, the second segment of the enrollment framework is the site activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It can be viewed as a stochastic process of site openings starting from the start-up date of the country where the sites locate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The site activation procedure describes a couple of site-level preparation activities for a clinical trial, such as documentation transferring, site training, and drug delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It can reflect the relative agility of a site and usually vary among different therapeutic areas, indications, countries, and regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The site activation date in the definition is the date when the site is ready and opened for patient recruitment in a clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The last part of the enrollment framework is the subject enrollment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It is a site-specific procedure that starts from the site activation date to the last subject enrolled date of the entire clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' As it’s mentioned above, the site starts to recruit patients after the site activation date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In our consideration, the patient enrollment process in each activated site will last until the time of the last patient recruitment of the entire trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This means that the LSFD date of the entire clinical trial is used as the end of patient enrollment of all sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Although the last subject enrolled date of each site is also available in our current date set, it is not recommended to be used as the end of the patient enrollment procedure, because once a site is activated, it is capable of recruiting patients until the end of the entire trial enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' During the time interval between the last patient enrolled in one particular site and the end of patient enrollment of the entire trial, that particular site is still under the patient enrollment procedure (although not able to recruit any more patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If that interval is not taken into account, it will introduce extra bias to the length of patient enrollment duration in each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='2 Historical data Two major historical data sources are utilized to model the duration of different segments in the enrollment framework: the site level data from the internal clinical trial management system (CTMS) and the site level data from proprietary Data Query System (DQS) from IQVIA Inc, which contains multiple sponsors’ enrollment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Although there are different limitations in each data source, the combination of these two data sources provides the most comprehensive site level enrollment information in each segment enabling the modeling of the proposed enrollment framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In CTMS data, detailed site level timeline from each clinical trial are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It provides the required anchor dates and process information in the enrollment framework, in particular, the final protocol approval date of each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' However, the data Enroment Framework Country (1st Site) (Study) Last Subject Final Protocol Site Activation First Dose Date Start-Up Date Approval Dates Country Start-Up Process Site Activation Process Subject Enrollment Process4 Zhong ET AL is limited to the historical clinical trials conducted by AbbVie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It would be challenging to use it alone to plan scenarios where there is no prior AbbVie experience in the indication or country/site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' On the other hand, in the DQS system, 17 major pharmaceutical companies including AbbVie have signed up to share the operational data from their clinical trials with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The data from various sponsors are transferred and integrated into one database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Although the specific names and definitions of fields may vary in the sponsors’ internal systems, they are well-aligned and mapped into one standard set of field names and definitions before being transferred into DQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Moreover, DQS utilized a well-designed algorithm to identify a clinical site across clinical trials and sponsors and assign a unique identifier to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It allows overlaying the historical information of the same site from different clinical trials and sponsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In the proposed enrollment framework, AbbVie CTMS data is used to model country start-up time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The DQS data that includes both AbbVie internal and external studies are used to model both the site activation process and patient enrollment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 3 METHODOLOGY DETAILS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1 General consideration Our primary objective is to forecast the enrollment timeline of a study at the portfolio planning phase where very limited information on the planned study is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This means any detailed recruitment planning information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', number of planned sites in each country) is not available and actual in-trial enrollment data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', actual number of activated sites up to a cutoff date) are not observed yet for the study, where in-trial enrollment data can include those on country initiation, site activation and subject enrollment etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the enrollment forecast purpose, very basic information on the planned study is provided including the total number of subjects to be enrolled, the total number of planned sites, the patient population, the disease indication and therapeutic area of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Since the in-trial enrollment data is not available at the planning stage, to achieve our primary objective, it is necessary to utilize the enrollment related information from historical studies, both internal or external, to model the underlying enrollment related processes defined in our enrollment framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' With an estimated model for each segment, the Monte Carlo simulation based approach is applied to simulate and forecast the future enrollment activities for the planned study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' As a secondary objective, the prediction of the list of specific countries that enrolled subjects originate from as well as the number of sites activated in each country are often of great interest to the clinical operation team, in addition to the overall subject enrollment timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Such country-level detailed insights can facilitate the planning of clinical trial operation activities and promote execution excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The implication on statistical modeling from the need of the country-level predictive insights is that it is sufficient to model the country-specific effects, although our historical study enrollment data distinguishes sites by their unique site identifiers across different studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Thus observed sites within a country for a study would be treated indistinguishably as drawn randomly from the pool of sites available in the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This country-level modeling consideration avoids the unnecessary technical complication of modeling the site-specific effects, which is not essential to addressing either our primary or secondary objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='2 Country start-up time model Suppose we observe the enrollment related data from 푆 historical studies, where these historical studies are usually selected based on having the same therapeutic area, disease indication and patient sub-population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', pediatric patients) as the current planned study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Other study-level features such as study phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', Phase 2b) or study start year can be used to select recent studies with potentially similar recruitment characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For each historical study 푗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 푆}, 푁푗 countries were observed to be initiated and their country start-up dates are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The initiated countries from different historical studies form a candidate pool of 퐶 unique countries, denoted by {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 퐶}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The observed historical study data come from a list of 푁 = ∑ 푗∈푆 푁푗 study-country combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the 푖th study-country combination, the study index mapping 푗[푖] ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 푆} gives the study index and the country index mapping 푘[푖] ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 퐶} gives the country index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We note that this notation is employed because different sets of countries (with overlapping) were initiated for subject enrollment for different historical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For a historical clinical trial 푗, let 푡0,푗 denote the protocol approval date, which is treated as the starting point for the clinical trial recruitment related activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We define the country start-up time for a country in a study to be the time duration between the study protocol approval date and the country start-up date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In our situation, the country start-up date is chosen to be the country’s first site activation date, because our historical Zhong ET AL 5 data do not include a specific date for country initiation (before its first site activation date).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We provide some discussion on other modeling choices in the final section for the scenario that such a specific date for country initiation is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Let 푡푖 denote the observed country start-up date for the 푖th study-country combination and 푢푖 = (푡푖 −푡0,푗[푖])∕푚 denote the corresponding observed country start-up time, where 푚 is a normalization parameter that provides a desired time unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Note that the observation 푢푖 is the country start-up time for the country 푘[푖] in the historical study 푗[푖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The country start-up time depends on the effects of several factors: country index (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', United States), therapeutic area of the study and maybe other covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We take a linear mixed-model approach13,14,15 to modeling the logarithm of the country start-up time, log(푢푖) = 휇푐푠푢 + 훾푐푠푢,푘[푖] + 훼푐푠푢푋푖 + 휖푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (1) 휇푐푠푢 is the grand intercept for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 훾푐푠푢,푘[푖] ∼ 푁(0, 휎2 푐푠푢,훾) denotes the country effect on country start-up times collected from different countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 훾푐푠푢,푘[푖] is treated as a random effect, because it is common to enroll subjects from tens of countries for a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 푋푖 denotes the therapeutic area for the 푖th study-country combination and since only four therapeutic areas (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', immunology, oncology, neurology and general medicine) are under consideration in our scenario, 푋푖 is treated as a fixed factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' As 푋푖 has 4 levels, the model matrix for 푋푖 has 3 columns (since the intercept is included) and 훼푐푠푢 is a 3 dimensional vector of fixed effect parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Finally, 휖푖 ∼ 푁(0, 휎2 휖) denotes the independently and identically distributed normal random error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Other covariates may be added to the model depending on their availability in both the historical data set and at the time of portfolio planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If it is believed that country start-up times vary greatly across historical studies, then a random study effect 훽푐푠푢,푗[푖] ∼ 푁(0, 휎2 푐푠푢,훽) (independent of 훾푐푠푢,푘[푖]’s) can be included, as it is common to observe tens of historical studies available from the internal and external databases for modeling purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' So the country start-up time model can be, log(푢푖) = 휇푐푠푢 + 훽푐푠푢,푗[푖] + 훾푐푠푢,푘[푖] + 훼푐푠푢푋푖 + 휖푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (2) Furthermore, if exploratory analysis by graphic plots suggests country-dependent therapeutic area effects, then random country slopes can be included with the following model, log(푢푖) = 휇푐푠푢 + 훽푐푠푢,푗[푖] + 훾푐푠푢,푘[푖] + (훼푐푠푢 + 훿푘[푖])푋푖 + 휖푖, (3) where 훿푘[푖] ∼ 푁(0, 휎2 훿) denotes a random effect in the slope of 푋푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' While the linear mixed-effects model of the logarithm of the country start-up time benefits from its fast computational time, this does not limit us to consider the generalized linear mixed-effects model14,16,17 to directly model the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For instance, we can model the country start-up time with a Gamma distribution based on the generalized linear mixed-effects model with the log link function, 푢푖 ∼ Gamma(휇푖, 휙) with the Gamma distribution mean 휇푖 = exp(휇푐푠푢 + 훽푐푠푢,푗[푖] + 훾푐푠푢,푘[푖] + 훼푐푠푢푋푖), (4) where 휙 is the dispersion parameter for the Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For mixed-model estimation, the R package lme418 can be used for estimation of both linear mixed-effects models and gen- eralized linear mixed-effects models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In our case, the study and country effects are modeled as crossed random effects and lme4 package provides a simple syntax and efficient implementation for fitting models with crossed random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='3 Site initiation model In our problem, we observe the activation dates for all activated sites within each country in a historical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' So the Bayesian hierarchical non-homogeneous Poisson process approach can be utilized to model the time-varying site activation pattern within a particular country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For instance, in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='12, this approach has been used to model the subject enrollment process within a specific site based on the observed in-trial data up to a certain time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In their application, the input data for model estimation is based on the observed data in a single study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We extend this approach to account for multiple historical studies from which recruitment related data are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the 푖th study-country combination defined in the last section, 푗[푖] and 푘[푖] are the corresponding study index and the country index respectively for the 푖th combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 푢푖, the country start-up time for the 푖th combination, is the starting point of site activation for the 푖th combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Let 푡′ 푖 be the end date of site activation for the 푖th combination, where 푡′ 푖 can be chosen to be the activation date of the last site activated at either the study-country or study level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' While the use of the study-country level site activation end date is based on the assumption that the site activation period is country-specific, the use of the study-level site activation end date replies on the assumption that the site activation of all countries would not stop until the target number of sites to be activated has reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To accommodate the study starting 6 Zhong ET AL point and the desired time scale, we standardize 푡′ 푖 and get 푢′ 푖 = (푡′ 푖 − 푡0,푗[푖])∕푚 just as we did for 푢푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Let 푁푖푛,푖(푢) be the number of sites activated in the time interval (푢푖, 푢] for 푖th study-country combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Note that the first activated site is not included in 푁푖푛,푖(푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then the number of sites activated in (푢1, 푢2] for the 푖th study-country combination can be modeled by a Poisson process as follows, 푁푖푛,푖 (푢2 ) − 푁푖푛,푖 (푢1 ) ∼ 푃 표푖푠푠표푛 ⎛ ⎜ ⎜⎝ 푢2 ∫ 푢1 휆푖푛,푖(푣)푑푣 ⎞ ⎟ ⎟⎠ , 푢2 > 푢1 >= 푢푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (5) If it is believed that the site initiation rate starts from a high value and then follows a downward pattern, the following time-decay rate function can be used: 휆(1) 푖푛,푖(푢) = Λ푘[푖] exp(−(휂 + 휖)(푢 − 푢푖)), Λ푘[푖] ∼ Γ(훼, 훽), 휂 > 0, 휖 = 10−5(offset), 푢 >= 푢푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (6) The base site activation rate Λ푘[푖] is country-specific and drawn from a Gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The posterior predictions of country- specific site activation rates would provide meaningful information to the clinical operation team for their site planning and optimization in different countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If the exploratory analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' times series plotting of site activation dates within countries) shows that the site activation rate generally first increases to a peak and then decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' then the following quadratic rate function would be a good choice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 휆(2) 푖푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='푖(푢) = Λ푘[푖] exp ( −(푢 − 푢푖 − 푒)2 2(휂 + 휖) ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Λ푘[푖] ∼ Γ(훼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 훽),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 푒 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 휂 >= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 휖 = 10−5(offset),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 푢 >= 푢푖 (7) The quadratic rate function 휆(2) 푖푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='푖(푢) is more flexible than the time-decay rate function 휆(1) 푖푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='푖(푢) because 휆(2) 푖푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='푖(푢) can absorb the decaying rate function as a special case when the parameter 푒 is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the 푖th study-country combination, the observed data regarding site activation in the time interval (푢푖, 푢′ 푖] include the total number of sites activated, denoted by 푛푖푛,푖 (푢′ 푖 ) and the corresponding site activation times for the 푖th combination, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖 = [푢푖푛,푖,푚|푚 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 푛푖푛,푖 (푢′ 푖 )}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It can be shown that the likelihood function of site activation data over all 푁 study-country combinations with time-decay rate function has the following form, 퐿 (훼, 훽, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) = 푁 ∏ 푖=1 퐿 (훼, 훽, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 )) ∝ 푁 ∏ 푖=1 ⎛ ⎜ ⎜⎝ Γ (푛푖푛,푖 (푢′ 푖 ) + 훼) Γ (훼) 훽훼 ( 훽(휂 + 휖) 훽 (1 − exp (− (휂 + 휖) (푢′ 푖 − 푢푖 ))) + (휂 + 휖) )(푛푖푛,푖(푢′ 푖)+훼) ∏ 푚 exp (− (휂 + 휖) (푢푖푛,푖,푚 − 푢푖 ))⎞ ⎟ ⎟⎠ (8) Let 휋 (훼), 휋 (훽), and 휋 (휂) be the prior distributions for the parameters 훼, 훽, and 휂 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then the posterior distribution is, 휋 (훼, 훽, 휂|⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) ∝ 퐿 (훼, 훽, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) 휋 (훼) 휋 (훽) 휋 (휂) (9) By conditioning on 훼, 훽, and 휂 and the observed data, we can generate Λ푘(푘 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 퐶}) for country 푘 from the following gamma distribution, 푓 (Λ푘|훼, 훽, 휂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) ∼ 퐺푎푚푚푎 ( ∑ 푖∶푘[푖]=푘 푛푖푛,푖 (푢′ 푖 ) + 훼, 훽(휂 + 휖) 훽 (1 − exp (− (휂 + 휖) ∑ 푖∶푘[푖]=푘 (푢′ 푖 − 푢푖 ))) + (휂 + 휖) ) (10) The likelihood function based on the quadratic rate function and the corresponding posterior distribution are, 퐿 (훼, 훽, 휂, 푒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) = 푁 ∏ 푖=1 ⎛ ⎜ ⎜ ⎜ ⎜⎝ Γ (푛푖푛,푖 (푢′ 푖 ) + 훼) Γ (훼) 훽훼 ⎛ ⎜ ⎜ ⎜⎝ 훽 훽 (√ 2휋(휂 + 휖) [ Φ( 푢′ 푖−푢푖−푒 √ 휂+휖 ) − Φ( −푒 √ 휂+휖) ]) + 1 ⎞ ⎟ ⎟ ⎟⎠ (푛푖푛,푖(푢′ 푖)+훼) ∏ 푚 exp ( − (푢푖푛,푖,푚 − 푢푖 − 푒)2 2 (휂 + 휖) )⎞ ⎟ ⎟ ⎟ ⎟⎠ , (11) 휋 (훼, 훽, 휂, 푒|⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) ∝ 퐿 (훼, 훽, 휂, 푒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) 휋 (훼) 휋 (훽) 휋 (휂) 휋 (푒) , (12) Zhong ET AL 7 where Φ (⋅) is the cdf of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' By conditioning on 훼, 훽, 휂, 푒 and observed data, we can generate Λ푘 for country 푘 from the following gamma distribution, 푓 (Λ푘|훼, 훽, 휂, 푒, ⃖⃖⃖⃖⃖⃗ 푢푖푛,푖, 푛푖푛,푖 (푢′ 푖 ) , 푖 ∈ 푁) ∼ 퐺푎푚푚푎 ⎛ ⎜ ⎜ ⎜⎝ ∑ 푖∶푘[푖]=푘 푛푖푛,푖 (푢′ 푖 ) + 훼, 훽 훽 (√ 2휋(휂 + 휖) ∑ 푖∶푘[푖]=푘 [ Φ ( 푢′ 푖−푢푖−푒 √ 휂+휖 ) − Φ ( −푒 √ 휂+휖 )]) + 1 ⎞ ⎟ ⎟ ⎟⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (13) In the modeling of the rate function for site initiation, we note that a study effect Ω푗[푖] may be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Unlike the country effect, the study effect is not of direct interest for the simulation of new trials, because a new simulated trial is always a different trial from the historical ones while a common set of countries are used in both historical and future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Due to much higher computation complexity from adding an additional study effect and its marginal benefit, we choose to not include it in site initiation modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the computation of posterior distribution of model parameters, we use the R function MCMCmetrop1R in the R package MCMCpack19, which can produce samples from the user derived posterior distribution function (as in 9 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='4 Subject enrollment model In terms of modeling subject enrollment process, our observed data only include site-level summary of subject enrollment information from historical studies, where for each site in a historical study, we observe the total number of enrolled subjects only instead of the actual date when each recruited subject was enrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This data limitation does not allow for the modeling of the time dynamics of subject enrollment stochastic process within each site and restricts our approaches to those that model the overall site-level enrollment rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Our historical study data set used for modeling enrollment consists of a number of 푁 study-country combinations from 푆 different studies, where for each study-country combination 푖, the number of sites activated is 푛푖푛,푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Let 푁푒푛,푖,푚 be the total number of subjects enrolled at the site 푚 in the 푖th study-country combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then 푁푒푛,푖,푚 can be modeled by the following generalized mixed-effects Poisson regression model14,15,20, 푁푒푛,푖,푚 ∼ independently as 푃 표푖푠푠표푛(휇푖푚) where log(휇푖푚) = 휇푒푛 + log(푑푖푚) + 훾푒푛,푘[푖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' (14) The parameter 휇푖푚 is the mean number of enrolled subjects for site 푚 in the 푖th study-country combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 휇푒푛 is the grand inter- cept for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The term log(푑푖푚) is the offset that accounts for the enrollment duration for the site 푚 in the 푖th study-country combination, where 푑푖푚 is the corresponding total enrollment duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Here the enrollment duration for a site is defined to be the time duration between the site activation date and the last subject enrollment date that is standardized by the normalization factor 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The random parameter 훾푒푛,푘[푖] ∼ 푁(0, 휎2 푒푛,훾) denotes the country effect and 푘[푖] is the country index for the 푖th study- country combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If exploratory graphical analysis suggests that the average enrollment rate over sites in a country changes greatly across different historical studies, then the random study effect 훽푒푛,푗[푖] can be added to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It is interesting to note that when the parameter 휇푒푛 is absorbed in 훾푒푛,푘[푖] ∼ 푁(휇푒푛, 휎2 푒푛,훾), then we would essentially assume a lognormal distribution for the country effect 휆푒푛,푘[푖] = exp(훾푒푛,푘[푖]) where 푁푒푛,푖,푚 ∼ 푃 표푖푠푠표푛(푑푖푚휆푒푛,푘[푖]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This sheds light on the connection of our proposed model to the popular approach of the Poisson-Gamma model in which, we would impose a Gamma distribution on the parameter 휆푒푛,푘[푖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the computation, the glmer function in the lme4 package18 is used to estimate the mixed-effect poisson regression model with an offset included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='5 Monte Carlo simulation to predict future recruitment process The parameters of three enrollment related models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', country start-up time model, site initiation model, and subject enrollment model) need to be estimated before we can proceed to predict future recruitment timeline for the current study under planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We take the Monte Carlo simulation based approach to randomly generate a number of simulated trials, where the parameters of three recruitment related models are simulated first for each trial and then the country start-up times, site initiation times within each simulated country, and subject enrollment times within each simulated site from each simulated country are simulated sequentially, up to a pre-specified upper time limit, for each simulated trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We choose 1000 to be the total number of trials to be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The upper time limit can be chosen to be the maximum enrollment duration of all completed historical trials 8 Zhong ET AL within an organization and it can be therapeutic area dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The computation complexity is highly dependent on this upper time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Hence we suggest that it is chosen with the knowledge from expert users in the therapeutic area which the current study belongs to, especially when a batch of enrollment scenarios need to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In our case we set the upper time limit to be 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Once all recruitment related data are simulated for all 1000 simulated trials, the median, lower/upper percentiles, and other statistics can be summarized for any quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For instance, to provide the study enrollment completion prediction, we can report the median last subject first dose date and its lower and upper percentiles as a prediction interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The following paragraphs describe in details how to simulate from each of the country start-up time, site initiation and subject enrollment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To simulate country start-up times in a new trial, we need to simulate values for the fixed effect parameters 휇푐푠푢 and 훼푐푠푢 and the random effects 훾푐푠푢,푘 and 훽푐푠푢,푗 as well as the random error 휖푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the fixed parameters 휇푐푠푢 and 훼푐푠푢, we draw from the asymptotic normal distribution of their restricted maximum likelihood (REML) estimators, where the mean of the normal distribution equals the REML estimates and the standard deviation equals the standard error of REML estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the random country effect 훾푐푠푢,푘 for a country 푘 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 퐶}, we simulate from a normal distribution with mean and variance equal to the conditional mode and conditional variance respectively of the random effect 훾푐푠푢,푘, because the countries we use in the new study are the ones used in the historical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the random study effect 훽푐푠푢,푗, we simulate from its unconditional distribution 푁(0, ̂휎2 푐푠푢,훽) where ̂휎2 푐푠푢,훽 is the REML estimate of 휎2 푐푠푢,훽, because a study under planning is a new study different from the historical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the random error 휖푖, we simulate from the normal distribution 푁(0, ̂휎2 휖) where ̂휎2 휖 is the REML estimate of 휎2 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Once all the parameters are simulated, they can be combined by, for instance, the equation 2, to produce the country start-up time for country 푘 in a simulated trial 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To conduct site initiation simulation, we need to first draw the values for model parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', 훼, 훽, 휂 and 푒) from the MCMC samples for the posterior distribution of either 9 or 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The country-specific base site activation rate Λ푘 is drawn from the Gamma distribution of either 10 or 13, conditional on the previously drawn model parameter values and observed historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then the total number of sites to be activated and the site activation times can be drawn following the non-homogeneous Poisson process defined by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To conduct subject enrollment simulation, we follow the similar approach as in country start-up time simulation to simulate values for the fixed effect parameter 휇푒푛 and the random effects 훾푒푛,푘 and 훽푒푛,푗 for a country 푘 in a simulated trial 푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The enrollment duration offset is set according to the pre-specified upper time limit for simulation, 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then the total number of subjects to be enrolled and the subject enrollment times can be drawn following the homogeneous Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4 PERFORMANCE EVALUATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1 General consideration To validate the performance of our proposed modeling framework, we select a set of recently completed studies within our organization, run our models to generate the predictions on enrollment duration, and compare the predictions to the ground-truth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The enrollment duration of a validation study is defined to be the time duration between the protocol approval date and the last subject first dose date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' To get the enrollment prediction for a validation study, the inputs provided to our model include the total number of subjects, the total number of sites, the therapeutic area, disease indication and the patient population of the validation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In addition, the enrollment related data from historical studies with the same disease indication and patient population as the validation study are used for estimation of model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Here historical studies are those studies that have their enrollment completed prior to the protocol approval date of the validation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' We compare the performance of our proposed modeling framework to a previous internally-developed enrollment forecast system based on the traditional statistical methods that fit simple probability distributions, described in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Furthermore, we show that our modeling and simulation framework calibrates the data variability correctly by comparing the nominal level vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' the coverage rate for prediction intervals of various nominal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Finally, we demonstrate how to generate the predicted enrollment curves through time, overlaid with confidence bands, which are deemed very informative to trial operation planning by our users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='2 A previous enrollment forecast system In a previous enrollment forecast system developed internally, the whole enrollment procedure is also divided into three segments with different definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The first segment is called site start-up period-1, which is the period from the final protocol approval Zhong ET AL 9 TABLE 1 Selection criteria for a candidate set of studies for model performance validation Criteria Value Enrollment completion date Within past 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='5 years Interventional vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' observational studies Interventional studies only Study phase 2 and 3 Therapeutic area Oncology, neuroscience, immunology, and general medicine Total number of sites 10-500 sites Total number of subjects 50-1600 subjects TABLE 2 Number of selected studies in different therapeutic areas Immunology General Medicine Neuroscience Oncology Number of studies 14 4 3 4 date to the open date of each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The second segment is site start-up period-2, which is defined as the time duration between the site open date and the first subject enrollment date for the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The last segment is the enrollment period which is from the first subject enrollment date for the site to the last subject enrollment date of the whole study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The models underlying the previous forecast system are primitive compared to our proposed modeling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For site start-up period-1, no formal statistical modeling is employed and the simulation is simply based on bootstrapping the observed periods from the historical data in the same therapeutic area and country with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the site start-up period-2, the modeling utilizes an log-normal distribution to fit historical data for each country and then random samples are drawn from it for various sites in each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the modeling of subject enrollment, a country-level modeling approach is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' A separate Gamma distribution is fitted to the historically observed enrollment rates of all sites in each country and then during the simulation stage, the enrollment rate for each site is randomly drawn from the Gamma distribution and used as the parameter of Poisson distribution to simulate subject enrollment in each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='3 Selection of evaluation studies To ensure the objectivity of the study selection process, an independent steering committee other than the modeling team (which the authors belong to) forms to lead the selection of a set of enrollment completed studies for model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' First, a candidate set of studies are pulled from our internal database, based on a list of search criteria given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Then for each candidate study, the actual enrollment curve is plotted for the purpose of checking unusual shapes, such as pauses and drastic enrollment speed changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' These unusual shapes are often due to rare unforeseeable events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Whenever such events happen, our internal users would manually assess the impact on a case-by-case basis and we shall not expect any models can still provide accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Based on the manual inspection of the enrollment curves, a list of studies with unusual shapes in enrollment curves is proposed so as to be removed from model performance evaluation, and the steering committee makes the final decision on whether a study should be included in the list of evaluation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Ultimately, a total of 25 studies are selected from four different therapeutic areas to test the performance of our proposed modeling approach in comparison to the previous internally-developed forecast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Table 2 shows the break-up of 25 studies into four therapeutic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='4 Prediction accuracy To evaluate the model performance, we apply both our proposed modeling framework and the previous enrollment forecast system to the 25 studies selected by the independent steering committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In terms of the performance evaluation metrics, we report the rates of coverage of the ground-truth enrollment duration by prediction intervals with fixed radii set to be +/- 1, 2, and 3 months (so the widths of prediction intervals are 2, 4 and 6 months respectively) as well as the coverage rate by the 10 Zhong ET AL TABLE 3 Predictions of enrollment duration for 25 studies by our new modeling framework vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' a previous internally-developed enrollment forecast system Coverage of Coverage of Coverage of Coverage of Width of Median Absolute Mean Absolute Model 2-Month PI 4-Month PI 6-Month PI 95% PI 95% PI Prediction Error Prediction Error New 7 (28%) 8 (32%) 13 (52%) 23 (92%) 20 months 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months Previous 5 (20%) 6 (24%) 9 (36%) 5 (20%) 2 months 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='6 months 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months TABLE 4 Nominal level vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' actual coverage rate of prediction intervals from our new modeling framework Nominal Level of PI 40% 50% 70% 95% Number of studies correctly predicted 13 15 17 23 Coverage rate 52% 60% 68% 92% SE for coverage rate 10% 10% 9% 5% 95% CI for coverage rate 32%-72% 40%-80% 50%-86% 82%-100% Prediction interval width 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='3 Months 7 Months 10 Months 20 months prediction interval of non-fixed width with nominal confidence level 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In addition, we also look at the mean and median absolute prediction error of the median predicted enrollment duration compared to the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' From Table 3, our proposed approach achieves higher coverage rates for all prediction intervals of various fixed widths than the previous enrollment forecast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In terms of median absolute prediction error, our proposed approach achieves 50% reduction from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='6 months to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months, while for the mean absolute prediction error that is often influenced by large errors, our method still provides 17% reduction from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='8 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This suggests that the traditional distribution-based approach is not adequate to capture the the sophisticated patterns behind recruitment related processes, compared to our more complex modeling approach based on mixed-effects models and non-homogeneous Poisson processes models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Table 5 and 6 provide the detailed outputs from our new modeling approach and the previous forecast system respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='5 Coverage of prediction intervals In term of the prediction interval of nominal level 95%, our method has achieved the actual coverage rate of 92% (see Table 3), while the previous forecast system is ill-calibrated and gives a coverage rate of 20% together with an extremely narrow width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This implies that the variability in historical data is not accounted correctly in either modeling or simulation in the previous enrollment forecast system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' While the prediction intervals of other nominal levels are not available in the previous forecast system, we further investigate the discrepancy between the normal level vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' actual coverage rate of our proposed new modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Table 4 shows that the nominal levels are very consistent with the actual coverage rates at different nominal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This is because our methodology properly accounts for the data variability at different levels (such as study and country level) with appropriate random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In the simulation stage, the random effects are simulated from the correct conditional or unconditional asymptotic sampling distributions correspondingly (see subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Although 95% is a widely used number for the level of significance in statistical hypothesis testing, the width of 95% prediction intervals is in general too large to be used in practice in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This is often due to the large amount of variability naturally in enrollment related processes and the limited number of historical studies available for model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' The 40% or 50% prediction intervals tend to offer a good balance between the coverage rate and the interval width in our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='6 Predicted enrollment curves In addition to the predicted enrollment duration, our users also find it very informative to their planning job to have a predicted enrollment curve throughout the whole time course from protocol approval to enrollment completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' This can be achieved by Zhong ET AL 11 TABLE 5 Predictions of enrollment duration for 25 studies by our new modeling framework (unit:month) Study Actual Median Predicted Within Within Within NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Duration Duration (95% PI) Residual +/- 1 Month +/- 2 Months +/- 3 Months 1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='23 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='94 ( 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='66 , 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='90 ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 No No Yes 2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='93 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='26 ( 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='02 , 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='03 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='67 Yes Yes Yes 3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='87 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='17 ( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='38 , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='68 ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='69 No No No 4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='63 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='16 ( 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='88 , 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='08 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='53 Yes Yes Yes 5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='22 ( 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='45 , Inf ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='75 No No No 6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='31 ( 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 , 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='11 ) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='02 No No No 11 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='77 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='02 ( 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='11 , 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Yes Yes Yes 13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='53 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='18 ( 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='54 , 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='40 ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='65 No No Yes 14 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='61 ( 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='39 , 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='12 ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='29 No No No 22 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='81 ( 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='82 , 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='40 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='44 No No No 23 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='63 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='58 ( 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='90 , 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='76 ) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='05 No No No 24 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='10 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='03 ( 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='21 , Inf ) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='93 No No No 25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='83 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='58 ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='11 , 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='75 Yes Yes Yes first selecting a grid of equally spaced time points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', in months) and then summarizing the median, low percentile, upper percentile total enrollment number for each time point in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In Figure 2 and 3, we show the predicted enrollment curves with 95% and 50% confidence bands for a well-predicted study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Note that the upper confidence band is capped by the target number of subjects enrolled for the study, as preferred by our users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 5 DISCUSSION For the problem of forecasting enrollment at the planning stage, we have developed a novel statistical modeling framework, which is based on GLMM and the use of non-homogeneous Poisson processes through Bayesian framework to systematically model the country initiation, site activation and subject enrollment in sequential steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Our new modeling framework shows a substantial improvement in prediction accuracy in comparison with the traditional statistical approach that fits simple probability distributions, based on a collection of 25 pre-selected studies from four therapeutic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Furthermore, we have showed that our modeling and simulation framework calibrates the data variability appropriately and gives correct coverage rates for prediction intervals of various nominal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' With the Monte Carlo simulation, we demonstrated how to generate the median predicted enrollment curve with confidence bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' It is no harder to generate predictions on other recruitment related quantities such as the median number of sites for each country and the median country start date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In terms of the modeling of country start-up time, we utilize a generalized linear mixed-effects model to account for the country-level and study-level variability in historical studies as well as the effects of fixed covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Machine learning models 12 Zhong ET AL TABLE 6 Predictions of enrollment duration for 25 studies by a previous internally-developed enrollment forecast system (unit:month) Study Actual Median Predicted Within Within Within NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Duration Duration (95% PI) Residual +/-1 Month +/-2 Months +/-3 Months 1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='23 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='07 ( 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='20 , 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='10 ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='83 No No Yes 2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='93 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 ( 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='90 , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='83 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='43 Yes Yes Yes 3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='87 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='53 ( 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='87 , 14.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='00 ( 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='03 , 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='13 ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 No No No 6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='67 ( 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='87 , 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 ) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 No No No 7 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='20 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 ( 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='50 , 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='50 Yes Yes Yes 8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='83 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='33 ( 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 , 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='20 ) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='50 No No No 9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 No No No 12 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='93 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='40 ( 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='00 , 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='17 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 Yes Yes Yes 13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='53 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='50 ( 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='67 , 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='23 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='03 No No No 14 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='07 ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 , 9.' 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Yes 16 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='33 ( 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='47 , 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='10 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='80 No No No 17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='93 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='07 ( 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='37 , 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='00 ) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='97 No No No 25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='83 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='73 ( 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='80 , 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='70 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='90 Yes Yes Yes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', gradient boosting) can be exploited if more covariates are available for modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' But appropriately accounting for the variability of predictions from machine learning models is not a easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Another potential caveat is that more covariate information is needed at the time of enrollment prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For a study at the portfolio planning stage, often limited information is known about the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' With regard to the modeling of site activation process within a country, if a specific country initiation date is available, we may need to consider the fact that the time duration between the country initiation date and its first site activation date could be much longer than the times between successive activation dates of subsequently activated sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In this case, we would recommend to model the time between country initiation and its first site activation separately from the rest of the activated sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' An alternative approach is to incorporate the time between country initiation and its first site activation into the definition of country start-up time and handle it implicitly in the modeling of the country start-up times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' For the modeling of subject enrollment, our observed data only include site-level aggregate subject enrollment information from historical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If instead the actual date when each subject was enrolled is observed, a non-homogeneous Poisson process can be used to estimate the time-varying rate function for subject enrollment process in a site with each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Likewise, we may need to take into consideration that it often takes longer time to enroll the first subject due to the overhead site preparation work and hence modeling the time to the first subject enrollment separately may be more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' When using an enrollment forecast system that depends on historical studies for model estimation, it is important for the end user (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=', clinical operation team) to conduct data-checking for the historical studies being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' If the summary statistics on, for instance, enrollment speed, in historical studies are systematically slower or faster than what the user expects, caution needs Zhong ET AL 13 FIGURE 2 Predicted enrollment curve with 95% confidence bands to be taken when interpreting the predicted enrollment duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Certain ad-hoc measures may be considered, such as excluding some studies with extreme statistics or using a multiplier to adjust the speed of enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Disclosure This manuscript was sponsored by AbbVie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' AbbVie contributed to the design, research, and interpretation of data, writing, reviewing, and approved the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' All authors are employees of AbbVie Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' and may own AbbVie stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Data availability statement The authors elect to not share data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 300 100 0- 200 400 800 DaysFromProtocol ApprovalDate14 Zhong ET AL FIGURE 3 Predicted enrollment curve with 40% confidence bands References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Heitjan DF, Ge Z, Ying sG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Real-time prediction of clinical trial enrollment and event counts: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Contemporary Clinical Trials 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 45: 26 - 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 10th Anniversary Special Issuedoi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='cct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='07.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Bagiella E, Heitjan DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Predicting analysis times in randomized clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Statistics in Medicine 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 20(14): 2055- 2063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='org/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='1002/sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content='2956 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Anisimov VV, Downing D, Fedorov VV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Recruitment in multicentre trials: prediction and adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' In: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 2007 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 1–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Anisimov VV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Statistical modeling of clinical trials (recruitment and randomization).' metadata={'source': 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Austin PC, Stryhn H, Leckie G, Merlo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Measures of clustering and heterogeneity in multilevel P oisson regression analyses of rates/count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' Statistics in medicine 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 37(4): 572–589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} +page_content=' 16 Zhong ET AL' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAzT4oBgHgl3EQfZfwz/content/2301.01351v1.pdf'} diff --git a/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/2301.01831v1.pdf.txt b/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/2301.01831v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5ce5f9a85555b0a8852a2688f6f5e9a5b6998ed --- /dev/null +++ b/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/2301.01831v1.pdf.txt @@ -0,0 +1,2973 @@ +Ab initio determination of thermal conductivity in crystals +Krzysztof.Parlinski1, 2 +1)Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, +Poland +2)Computing for Materials, Kraków, Poland +(*Electronic mail: Krzysztof.Parlinski@ifj.edu.pl) +(Dated: 04 January 2023) +The calculations of thermal conductivity requires to know anharmonic properties of the crystal. For this purpose a +non-perturbative anharmonic theory is applied, which do not make use of the potential energy expansion over atomic +displacements, but instead, runs ab initio calculations of Hellmann-Feynman forces for atomic patterns of atoms with +specific displacements to rebuild the anharmonic phonon frequencies, and group velocities. see [K.Parlinski, Phys.Rev. +B 98, 054305 (2018),] The Green-Kubo equation for the thermal conductivity needs to know the above quantities +and the phonon relaxation times, which are related to the 4th-order phonon correlation function expressed in terms of +phonon anihilation and creation Bose operators. In currect formulation of anharmonic theory the relaxation times can +be derived as analitical expression. The Green-Kubo formulae was succesfully applied to find thermal conductivity of +Si and conductivities, related to the phonon and elastic waves, respectivily, were computed. +I. +INTRODUCTION +The understanding of thermal conductivity in solids is +needed for applications of technically relevant materials to +nanofabrication technology, to manufacture electronic devices +for nanoscale demands, to understand the mechanisms, pre- +dict the properties of solid thermal condunctivities and to be +able to run related computations. Similarly, handle of ther- +mal conductivity describes partly the behaviour of thermo- +electrics, electron-mediate superconductors and thermal con- +ductivity materials, which govern the heat transfer processes +in the Earth’s interior. +The heat transport properties of solids are usually divided +into two mechanisms: First kind is called Lattice thermal +conductivity (LTC). It is calculated applying phonon anhar- +monicity. The method seems to be rather well known, and +in this case the Green-Kubo linear-response theory1 is mainly +used. There are some variants in formulating this method. +In one of them the harmonic phonon frequencies, the group +velocities for phonon modes, and some relaxation times are +used. Another way is to find the input data from the anhar- +monic perturbation method, usually with help of the triple +and quatric order terms2–10, where relaxation time comes from +solving the Boltzmann equation11,12, generally using third, or +third and fourth order anharmonic terms only. Next method +is to run molecular dynamics (MD)13–18 provided that the po- +tential of the studied system is known. The last mentioned ap- +proach solves the classical equations of motions for the sys- +tem, tracing particle’s evolution and then collecting the nec- +essary quantities, which are required by Green-Kubo formu- +lae. Typically, the Green-Kubo equations describe properly +the thermal conductivity of solids for LTC, in the interval +from around room to melting temperature. The anharmonic +effects alone can be also studied applying the stochastic self- +consistent harmonic approximation method19,20, which ac- +cording to the Gibbs – Bogoliubov variational principle re- +quires that the true free energy of the system reaches the min- +imum of the functional F[ ˜ρ] with respect to all possible trial +density matrices [ ˜ρ]. +The second kind of heat transport will be in this case called +High thermal conductivity (HTC), for which the complete +theory is still under construction. The HTC typically occurs +in simple crystal structures. At low temperature (below 200 - +300K) HTC materials exhibit usually two, even three order of +magnitude higher thermal conductivity values than the same +material at high temperature range only (above 300K). To this +group of crystals belongs: C (diamond), Si, Ge, AlN, AlP, +BAs, BN, BP, BeS, GaN, MgO21. Thermal properties of sev- +eral HTC crystals have been measured by Slack et.al.21–25. +These HTC materials attracted special attention and called for +relevant theory. In 1964 Glassbrenner and Slack25 proposed a +mechanism of HTC for silicon Si and Germanium Ge, based +on phenomenological approach24,25. Later, a similar consid- +eration on ab initio level was published by A.Ward et.al.12. +Recently, Esfarjani et.al.26, studying Si, have discussed HTC +mechanism as arising from large mean free path of phonons, +determined by size of sample. It was shown that HTC of Si +arises for more than order of magnitude, if mean free path +spans from about nanometers to 100 microns. +It is generally accepted that the LTC is totally described by +the acoustic and optic phonon modes, and therefore the LTC +heat transfer is described by Green-Kubo formulae, where +usually the relaxation time is found by Boltzmann equation +or MD simulation. In articles27 and28 it was shown how the +Green function of the anharmonic perturbation theory may +lead to the typical Lorentzian term, with shift and width of +anharmonic peak28. +In the article29 it was attempted to decouple the fourth-order +correlation function, responsible for the relaxation times and +needed for thermal conductivity, using the pairing Wick’s the- +orem, but finally MD runs validated the results for silicon. +In the present article we reformulate the Green-Kubo ap- +proach to use so called displacement patterns (DPs) of atomic +configurations to derive the LTC directly from phonon disper- +sion curves created from DP and simultaneously determine +phonon relaxation times. Moreover, derivation of the relax- +ation time from solution of the Boltzmann equation or MD +calculations is not needed. +arXiv:2301.01831v1 [cond-mat.mtrl-sci] 4 Jan 2023 + +2 +In next sections the method has been extended to handle +also the HTC phenomena. In this case the low frequency and +very long elastic waves are used to govern the HTC process. +To calculate such long wavelength states we compute ab ini- +tio the elastic constant tensors with the equilibrium atoms in +the supercell and for series of similar supercells with atoms +displaced from equilibrium positions due to presence of tem- +perature, like in DP. From these elastic supercells one calcu- +lates the frequencies and group velocities of the elastic waves, +and apply them to the Green-Kubo expression to find HTC. In +this case it is obvious that the accounted wavelength of elas- +tic waves could be considerably longer than wavelength of +ordinary phonons, therefore one must introduce limit to the +longest active wavevector which could be accommodated in +the sample size. The LTC and HTC calculation of Green- +Kubo relations, as derived in this paper, have the same formal +forms. +A. +Anharmonicity +The thermal conductivity in solids is determined by anhar- +monicity of the system, therefore, one should start from dis- +cussion how to handle anharmonicity. In present article a pro- +cedure, which takes also advantage of the ab initio calcula- +tions considers anharmonic properties of crystals within a new +non-perturbative approach. (see Ref.30). It would be much +much easier to understand the current article first looking to +Ref.30 and glance at the examples presented there. There, the +procedure begins from selecting the supercell of the studied +crystal and calculating the harmonic phonons, using PHONON +software31,32. +At equilibrium every atom of the crystal resides near the +potential energy minimum. Displacing an atom from its equi- +librium position by a vector u one creates so called Hellmann- +Feynman (HF) forces computed using VASP33, and acting +on the surrounding atoms, in particular atoms of the super- +cell. PHONON computes in this way the harmonic phonon +frequencies ω(0)(k, j) and eigenvectors e(0)(k, j). +The HF +forces could be calculated with the ab initio program. The +same HF forces are used to build all force constants and dy- +namical matrix elements, which are the essential quantities in +lattice dynamics theory since more as a century3. One should +only keep in mind that the used atomic displacements am- +plitudes u should probe only small interval of the harmonic +potential around the atoms. From these data the mentioned +software calculates harmonic phonon dispersion curves in the +whole Brillouin zone. In the harmonic calculations the used +atomic displacements u are small, of order of 0.03 − 0.04 Å, +which is close to zero-temperature phonon vibration. +In this harmonic theory30 the method uses first the exact +wavevectors k, with wavelengths, being commensurate with +the supercell size. At such exact wavevectors30–32 the peri- +odic structure of the crystals ensures that the harmonic fre- +quencies ω(0)(k, j) and eigenvectors e(0)(k, j) are calculated +exactly, independent on the size of the supercell. +Unfor- +tunately, the list of exact wavevectors diminishes with de- +creasing size of the supercell. +Of course, certain balance +between computational time and accuracy of the result will +determine the selected supercell size. The phonon frequen- +cies and eigenvectors beyond exact wavevectors are interpola- +tions between exact wavevectors. The interpolations are sup- +ported by a traditional analytical derivation of dynamical ma- +trix elements, which must be solved, what in practice leads +to the valid results in the whole Brillouin zone. The interpo- +lated procedure uses the singular value decomposition (SVD) +method32,34, which simply assures that the finale phonon dis- +persion curves are the best fit in the mean square sens to the +exact phonons frequencies of the exact points within the con- +strains of classical phonon dispersion curves. As a matter of +fact this approach32 to phonon theory was already equipped +in 1996 with the procedure similar to the machine learning +method. +The PHONON software30 is also able to calculate the +phonon dispersion curves from supercell with many atoms, +which are displaced simultaneously out from their equilibrium +positions. Moreover, if the atomic displacements stay small, it +means they do not enter the non-parabolic part of the potential, +then the resulting phonon dispersion curves look like in the +harmonic case. Indeed, the force constants are determined by +proportionality coefficient between atomic displacement and +HF force and in harmonic regime do not depend on the ampli- +tude of displacements. +However, if in the above procedure the displacements are +larger, some deviation of the phonon frequencies might be ob- +served because in reality atoms during vibrations visit the non- +parabolic parts of the potential. These changes of frequencies +and eigenvectors manifest the anharmonicity. +Hence, the +deviation of the particular phonon frequency (ω(anh)(k, j) − +ω(0)(k, j)), for the same (k, j), could be considered as a mea- +sure of the anharmonicity +Of course, it is well known that atoms vibrate in the crystal +sites due to finite temperature T. For a given T, one should +displace the atoms from their equilibrium positions and create +the displacements pattern (DP) next used to find the phonon +vibrations. +At a given T, the DP could be represented as a snap- +shot of supercell with many atoms displaced. +One would +like to create sets of Ni atomic displacement patterns DP(i), +i = 1,2,...Ni, which might arise in the crystal at a given T, +and in different moments and locations. The proposition given +in30 is as follows: Each supercell DP should be filled with +the phonon waves, determined by the well known expression +of atomic displacements u(m,µ,γ) and supplemented by the +phase factor φ(k, j) of traveled phonon waves, where meaning +of indices is later given before Eq.(8). +u(m,µ,γ) = Q(k, j) +�Mµ +eγ(k, j | µ)exp[2πi(k·R(m,µ)−φ(k, j)] +(1) +The phase φ(k, j) of the phonon wave could be taken at ran- +dom from the interval [0.0 − 1.0) to mimic different atomic +displacement pattern labelled by the same (k, j). The mean +square displacement amplitude < Q2(k, j) > of the phonon +wave was determined in35,36 by + +3 +FIG. 1. A schematic set of DP(i) in single anharmonic phonon peak. +Si +Si +(a) +(b) +k=X +k=K +T=1000K +T=1000K +PEAK INTENSITY +0 +.05 +.40 +.30 +.20 +.10 +0 +2 +4 +6 +-2 +8 +10 +12 +14 +16 +18 +.15 +.25 +.35 +.45 +.50 +PEAK INTENSITY +0 +.05 +.20 +.10 +.15 +.25 +0 +2 +4 +6 +-2 +8 +10 +12 +14 +16 +18 +FREQUENCY (THz) +FREQUENCY (THz) +FIG. 2. Silicon Si. Anharmonic phonon peaks calculated for crystal +at T = 1000K and wavevectors (a) X = (0.5,0.5,0.0) and (b) K = +(0.375,0.375,0.725). The plots arrived from DP(i) i = 1,2,...500. +< Q2(k, j) >= +¯h +2ω(k, j)coth +� ¯hω(k, j) +2kBT +� +(2) +In the harmonic approximation the above relation is exact. +Si +T= 200K +T=1000K +(a) +(b) +WAVE VECTOR +WAVE VECTOR +FREQUENCY (THz) +FREQUENCY (THz) +X +K +L +Γ +Γ +X +K +L +Γ +Γ +18 +0 +-2 +2 +16 +14 +12 +10 +8 +6 +4 +0 +2 +16 +14 +12 +10 +8 +6 +4 +FIG. 3. +Silicon Si. +The maps of anharmonic phonon dispersion +curves along the line of wavevector Γ−X −K−Γ−L for temperature +(a) T = 200K and (b) T = 1000K calculated from DP(i) 1,2,...500 +each. Blue-green-red colours indicates intensity. +https://www.overleaf.com/project/63668958d1320f3c7c9a7540 +The Si and MgO the 2 × 2 × 2 supercell contains 64 atoms, +32 exact wavevectors each with 6 degree of freedom. More- +over, the phonon waves may still be supplemented by random +number of phase φ(k, j) from the interval [0.0 − 1.0). For +Si and MgO, the atomic displacement changes with T, from +Eq.(2), it follows Q = 0.05 − 0.16 Å in temperature range +T = 40 − 1500K. This is only 0.02 − 0.07%, respectively, of +the nearest neighbor interatomic distance. +Using the above method it is rather easy to obtain the an- +harmonic peaks for any wavevector k and phonon branch j. +These can be any wavevectors, although those which do not +belong to list of exact wavevectors. One needs to create dis- +placement patterns DP (i), i = 1,2,...Ni in the range from +Ni = 20 to 500, depending on the requested precision. For +conventional anharmonic peaks it could be limited to about +Ni = 50 DP, but to study a peculiar form of the anharmonic +peak, such as asymmetric shape, particularly high background +under the peak of non - Lorentzian shape, or even splitting of +the single anharmonic peak, the value of Ni should be larger +Ni = 200−500. The amplitudes of the vibrating atoms caus- +ing anharmonic effects and estimated above occur in real crys- +tals and create many HF forces. These multiplicity of forces +create multiplicity of force constants, which in turn, are used +to solve the equations pf classical lattice dynamic. Schemat- +ically the construction of anharmonic phonon mode can be + +0 +10110 +Reference +9 +12 +Frequency +13 +8 +14 +15 +16 +6 +17 +5 +18 +19 +20 +FREQUENCY (i)(k,j) (THz).25 +.20 +.15 +PEAK +10 +.05 +2 +0 +2 +6 +8 +10 +12 +14 +16 +18 +FREQUENCY (THz).50 +45 +.40 +INTENSITY +.35 +.30 +.25 +PBAK +.20 +15 +.10 +.05 +0 +-2 +0 +2 +6 +8 +10 +12 +14 +16 +18 +FREQUENCY (THz)r +x +K +T +16 +14 +(zHL) +12 +FREQUENCY +10 +8 +6 +4 +2 +0 +WAVE VECTORr +x +K +r +18 +16 +14 +FREQUENCY (THz) +12 +10 +8 +6 +4 +2 +0 +-2 +WAVE VECTOR4 +performed as shown on Fig.1. Examples of calculated an- +harmonic phonon peaks are shown on Fig.2. It is a set of +δ(ω(i)) functions of DP(i)1,2,i = ···20, Every segment i rep- +resents single snapshot of atomic displacements for the same +anharmonic phonon mode. The phonon waves have differ- +ent phases counting against the fixed sites of the atoms, hence +the frequencies and intensities may vary a little. The δ(ω(i)) +frequencies together with intensities (amplitude) are solutions +of the lattice dynamic equations for the selected wavevector +k and accompanied displacements corresponding to tempera- +ture T. In the above scheme a set of 20 δs mimic envelope of +single anharmonic phonon mode. In further one calculation +of anharmonic phonon mode with wavevector k being located +in between the already plotted one can be added to increase +statistic and precision of phonon peak. The envelope of the +delta set should give the form of the anharmonic peak. Refer- +ence frequency on the plot corresponds to harmonic frequency +used letter in the conductivity theory. +There appear more profits, following this method. Namely, +in this theory the symmetry of each obtained anharmonic peak +is uniquely labeled by the irreducible representation of the +crystal space group. Normally, it is done only for the har- +monic phonon δ-kind peaks. Here, however, the calculated +area under the anharmonic phonon peaks is characterized by +the same irreducible representation. +From the same DP(i), i = 1,2,...Ni, with value Ni as dis- +cussed above, one may construct histograms for the phonon +dispersion curves along any path of the reciprocal space, +which next can be plotted as a map of the phonon dispersion +curves. Such maps for Si at T = 200K and 1000K are shown +on Fig.3. +B. +Harmonic and anharmonic hamiltonians +The vibrational hamiltonian for a crystal in harmonic +approximation3 can be written as +H(0) = ∑ +m,µ,γ +P2,(0)(m,µ,γ) +2Mµ ++ 1 +2 ∑ +m,µ,γ ∑ +n,ν,δ +Φ(0)(m,µ,γ;n,ν,δ) +× (U(0)(m,µ,γ)(U(0)n,nu,δ) +(3) +where the harmonic force constants Φ(0) have been calculated +from the Hellman-Feynman forces of the perfect crystal with +atoms preserving the crystal symmetry. The H(0) hamilto- +nian describes the harmonic phonons. Solving the eigenvalue +equation for H(0) one arrives to harmonic phonon frequencies +ω(0)(k, j) and polarization vectors e(0) +µ (k, j). These collection +of harmonic phonons are used as a reference set of data when +analysing the thermal conductivity. +The current method requires also to find phonon frequen- +cies from the hamiltonians H(i),where "anharmonic" force +constants Φ(i), i > 0, lead to larger/smaller displacement +amplitudes, then in harmonic case. +Now, one creates the +Hellmann-Feynman forces for all displaced atoms collected +in DP (i), Eqs (1,2). Solution of these eigenvector equations +leads to little different phonon frequencies and one may write +H(i) = ∑ +m,µ,γ +P2,(i)(m,µ,γ) +2Mm,µ ++ 1 +2 ∑ +m,µ,γ ∑ +n,ν,δ +Φ(i)(m,µ,γ;n,ν,δ) +× (U(i)(m,µ,γ)(U(i)n,nu,δ) +(4) +If the anharmonic system converts to the harmonic one, +then the force constants converge Φ(i) → Φ(0), and the forces +are reduced to harmonic one. From the relations given above +we conclude that in similar conditions as proclaimed above +occurs H(i) → H(0), and therefore the anharmonic hamiltoni- +ans disappears HA = 0. Anharmonic hamiltonian vanishes if +the phonons of crystal become harmonic. Then, the thermal +conductivity becomes infinity.. +Above, +the +two +body +anharmonic +force +constants, +Φ(i)(m,µ,γ;n,ν,δ), are labelled also by index (i) of DP(i), +which indicates that the anharmonic force constant acting +on the atom (m,µ,γ) arises not only due to displacing a +single atom (n,ν,δ) (as was in the harmonic case), but it +really senses also forces coming from all other displaced +atoms of supercell according to the configuration imposed by +DP(i). This suggests that all atoms affects the anharmonic +force constant Φ(i)(m,µ,γ;n,ν,δ) as well. This means that +Φ(i)(m,µ,γ;n,ν,δ) is in some sens a many body force con- +stant, which feels simultaneous displacements of all other +atoms in the crystal. +In other words all anharmonic force +constants are computed not in the perfect crystal, but in the +crystal being represented by a series of i = ···, supercells , +having atoms shifted out from equilibrium positions, due to +finite temperature, and from that configuration one computes +the contributions to anharmonicity. +The hamiltonian H(0) provides harmonic phonon frequen- +cies only. The harmonic potential for perfect insulator should +lead to infinity thermal conductivity of the crystal. This state- +ment has been expressed in the textbook of Ashcroft and +Mermin37, in Callaway’s11 and Maradudin38 papers. Ashcroft +and Mermin says that ” in perfect harmonic insulator crystal +the phonon scattering does not occur, so such a crystal should +have infinite thermal conductivity. Scattering of phonons from +lattice imperfections would produce a finite thermal conduc- +tivity, but with a wrong temperature dependence. The only +way to explain the realistic thermal conductivity data is to ad- +mit that phonons can be scattered by other phonons”. Thus, +the relevant thermal conductivity should exhibit the following +properties: (i) demonstrate infinite thermal conductivity for +strictly harmonic crystals. (ii) describe the finite thermal con- +ductivity for crystal with anharmonicity. Consequently, one +may propose to treat the thermal conductivity using the fol- +lowing approach. The anharmonic effects are described by +the excess of effects arising from H(i) hamiltonians, superim- +posed on the harmonic modes coming from H(0). Thus, the +anharmonicity effects of a crystal can be determined by the + +5 +following hamiltonian +HA = 1 +Ni +Ni +∑ +i=1 +� +H(i) −H(0)� +(5) +From the relations given above we may conclude that for +vanishing anharmonicity, when H(i) → H(0), the anharmonic +hamiltonians disappear HA = 0 and the crystal exhibits infinite +thermal conductivity. +Because the hamiltonians H(0) and H(i), Eqs(3,4) are sums +of two positively definite quadratic forms, one in the com- +ponents of the momenta and the other in the components of +the atomic displacements, it follows from a theorem of ma- +trix algebra39 that it is possible to find principal axes, or nor- +mal coordinate transformations which simultaneously diago- +nalized the kinetic and potential energies in these hamiltoni- +ans. Such a principal axis transformations are generated by +the conventional expansion of displacements and momenta in +terms of plane waves and next Bose annihilation b(k, j) and +creation b+(k, j) operators. +In therms of these operators, the hamiltonians Eqs(3,4) take +the simple forms +H(0) = ∑k, j ¯hω(0)(k, j)[b+(k, j)b(k, j)+ 1 +2] +H(i) = ∑k, j ¯hω(i)(k, j)[b+(k, j)b(k, j)+ 1 +2] +(6) +From Eqs (5, 6) the anharmonic hamiltonian HA, with sub- +tructed harmonic phonon contribution H0 reads +HA = ∑ +k, j +Ni +∑ +i=0 +� +¯hω(i)(k, j)− ¯hω(0)(k, j) +� +× b+(k, j)b(k, j) +(7) +where it has been assumed that the Bose operators b+(k, j) +and b(k, j) for the same mode (k, j) with close frequen- +cies should be, respectively, very similar and further we +assume that they remain the same. +Indeed, in this ap- +proach the anharmonicity is determined by the differences of +� +¯hω(i)(k, j)− ¯hω(0)(k, j) +� +. +These frequencies could be systematized and collected to +histograms, labeled by a wavevector and phonon branch (k, j) +and finally to present as a Lorenzian-kind anharmonic peaks. +Such peaks could be measured by inelastic neutron scattering, +Raman spectra, or infrared absorption. Below we shall use +this method to model the thermal conductivity as well. It is +essential to remind that the path from the DP (i) to phonon +frequencies is performed by the solution of lattice dynamics +equation of motion only. +Here, a single DP (i) for fixed i can be treated as an anhar- +monic perturbation cluster, arising from simultaneously dis- +placements of many atoms. In traditional perturbation the- +ory, DP(i) is typically limited to triple or quatric interactions. +Here, a crystal with supercell of 64 atoms provides single +DP (i) data for all wavevectors (k, j) of the Brillouin zone, so +some cross interaction therms are included. +II. +FORMULAE FOR THERMAL CONDUCTIVITY +A. +Phonons +The +Green-Kubo +approach +is +based +on +statistical +thermodynamics40–43. +A derivation of basic formulae +can be found in references8,26,44–46. +The heat flux J(t), +for simplicity, is usually determined without contribution +from diffusion and convection, (see Ref.8). Here also, we +adapt the formalism of the anharmonic theory described in +previous section, to apply the set of anharmonic hamiltonians +H(i) Eq.(5). +The mentioned method expects the crystal +to be presented as a set of Ni supercell’s subsystems with +atoms randomly displaced patterns DP (i), i = 1,2,...Ni, +corresponding to studied temperature T, +J(i) +α (t) = 1 +2 ∑ +m,µ,γ ∑ +n,ν,δ +(Rα(m,µ,γ)−Rα(n,ν,δ)) +× +� +U(i)(m,µ,γ | t)·Φ(i)(m,µ,γ;n,ν,δ) +· 1 +Mν +P(i)(n,ν,δ | t) +� +(8) +Here, we use indexing of atoms: first atom: (m,µ,γ), sec- +ond atom:(n,ν,δ), where m,n are coordinates of primi- +tive unit cells, µ,ν are atomic indices within primitive unit +cells, and γ,δ stay for coordinate x,y,z. The force constants +Φ(i)(m,µ,γ;n,ν,δ) may have contributions from harmonic +and/or anharmonic regions of the interatomic potentials. In +this sens the force constants may contain contributions from +any higher order anharmonic therms. +Moreover, the force +constants might also have contributions from other displaced +atoms of used DP (i), and not shown explicitly in the now dis- +cussed form of Φ(i). The same force constant may also repre- +sent harmonic force constants. +As argued in Sec.I B the thermal conductivity should be cal- +culated according to Eq.(9), over thermal fluctuations repre- +sented by the harmonic and anharmonic hamiltonians Eq.(5), +determined by the components DP (i) (i = 1,···Ni), all gen- +erated for the same T. The Green Kubo expression is then +written as +κα,β = +1 +VkBT 2 +1 +Ni +Ni +∑ +i=1 +� ∞ +0 +< J(i) +α (t)J(i) +β (0) > dt +(9) +Averaging the above correlation function over DP (i) one may +use it to study also anharmonic phonon peaks. +Using the +expansions of atom displacements and momenta over plane + +6 +waves Q(i)(k, j) and +• +Q +(i) +(k, j), respectively,3, one has +U(i)(m,µ,γ | t) = +� +¯h +NMµ ∑ +k,j +e(i) +γ (k, j | µ) +× exp[2πi(k·R(m,µ)]Q(i)(k, j | t) +P(i)(n,ν,δ | t) = 1 +i +� +¯hMν +N ∑ +k,j +e(i) +δ (k, j | ν) +× exp[2πi(k·R(n,ν)] +• +Q +(i) +(k, j | t), +(10) +where (bold i = √−1), N is the number of wavevectors k used +in the summation of Eqs (10), and j is the index of phonon +branches. Now, recalculating Eq.(8) one can rewrite it in the +form +J(i) +α (t) = ¯h +iN ∑ +k, j +ω(i)(k, j)v(i)α +gr (k, j) +×Q(i)(k, j | t) +• +Q +(i) +(k, j | t) +(11) +Here, imaginary unit i appears since it was added to the expo- +nent of the dynamical matrix D(i)(k), when used to define the +group velocity, Eq(13) +In next steps one finds the phonon frequencies and eigen- +vectors for perfect crystal (i = 0) and for crystal modified +with DP(i), (i > 0). Both are lattice dynamic solutions of +the eigenvalue phonon equation +ω(i)2(k, j) = e(i)T(k, j)D(i)(k)·e(i)(k, j) +(12) +Of course they need different values of the elements of dy- +namical matrix D(i)(k). +Further, the group velocity vectors can be found from rele- +vant dynamical matrices using +v(i) +gr (k, j) = +1 +2ω(i)(k, j) +� +e(i)T(k, j) +� ∂ +∂kD(i)(k) +� +·e(i)k, j) +� +(13) +Notice that with the same equations the phonon frequencies +ω(i)(k, j) and group velocities v(i) +gr (k, j) have been found in ab +initio procedure via the Hellman-Feynman force30 created by +displacement of atoms fixed already in DP (i)’s. These devia- +tions of DP (i) phonon frequencies from the relevant harmonic +frequency contain information concerning the anharmonicity, +in terms of frequency and eigenvectors. +Collecting the expressions of Eqs (9, 11, 13) the thermal +conductivity tensor reads +κLTC +α,β = +¯h2 +VpuckBT 2 +1 +Ni +Ni +∑ +i=1 +� ∞ +0 dt 1 +Nr < ∑ +k, j +(ω(i)(k, j))2v(i)α +gr (k, j)v(i)β +gr (k, j) +× < Q(i)(k, j | t) +• +Q(i)(k, j | t) +• +Q(i)(k, j | 0)Q(i)(k, j | 0 > +(14) +where r is a number of atoms in primitive unit cell, Vpuc vol- +ume of primitive unit cell. The appeared fourth order phonon +correlation function < Q(t) +• +Q(t) +• +Q(0)Q(0) > needs some com- +ments. +It is the only function under the Laplace integral, +which depends on time t. If the integrated function would be a +constant C = const ̸= 0 then the Laplace integral +� ∞ +0 Cdt = ∞. +This would be the mechanism to make a harmonic crystal hav- +ing infinite thermal conductivity. +To considered the value of the fourth-order correlation +function we need to express the normal mode amplitudes of +phonons by Bose annihilation b and creation b+ operators +Q(i)(k, j) = 1 +√ +2 +1 +� +ω(i)(k, j) +� +b(k, j)+b+(−k, j) +� +• +Q(i)(k, j) = 1 +√ +2 +� +ω(i)(k, j) +� +b(k, j)−b+(−k, j) +� +(15) +Now, the pair time-dependent correlation functions of b, b+ +are found from the solution of the Heisenberg time dependent +equations3, in which the anharmonic hamiltonian HA Eq.(5) +and (7) has been used. Then +< b(k, j | t)b+(k′, j′) | 0 > = +exp(−iω(i)(k, j)−ω(0)(k, j)]t)(n(i)(k, j)+1)δk,k′δ j,j′ +< b+(k, j | t)b(k′, j′) | 0 > = +exp(+iω(i)(k, j)−ω(0)(k, j)]t)n(i)(k, j)δk,k′δj, j′ +(16) +Here, the mean number of phonons in the vibrational mode +(k, j) of DP (i) at temperature T, is +n(i)(k, j) = +1 +eβ ¯hω(i)(k, j) −1 +(17) +and β = +1 +kBT . +Applying Eqs (15, 16), the fourth order correlation func- +tion < Q(t) +• +Q(t) +• +Q(0)Q(0) > can be evaluated, with the +Wick’pairing theorem6,29, to 16 correlation functions of prod- +ucts of averages consisting of four b, b+ operators each. In 10 +functions, out of the mentioned 16, the 4 operator terms vanish +due to averages build from pairs of the same kind of operators. +The remaining 6 correlation functions do not vanish from the + +7 +mentioned reasons. However, 4 functions arrived from the last +6 non-zero terms, mutually cancel, due to averages build from +pairs constructed from the same kind of operators and 6 terms +are not vanishing from these reasons. However, the 2 last +terms < b(t)b(t)b+(0)b+(0) > and < b+(t)b+(t)b(0)b(0) > +remain non-zero and can be written as +< b(k, j | t)b(k, j | t)b+(k,j| 0)b+(k, j | 0) >= +2(n(i)(k, j)+1)2 e−2i[ω(i)(k, j)−ω(0)(k,j)t] +< b+(k, j | t)b+(k, j | t)b(k,j| 0)b(k, j | 0) >= +2(n(i)(k, j))2 e+2i[ω(i)(k, j)−ω(0)(k,j)t] +(18) +Applying the time dependence of the surviving pairs in +Eq.(16), the non-zero fourth-order correlation functions are +< Q(i)(k, j | t) +• +Q(i)(k, j | t) +• +Q(i)(k, j | 0)Q(i)(k, j | 0 >= +(n(i)(k, j)+1)n(i)(k, j)+1/2)× +{(cos2(ω(i)(k, j)−ω(0)(k, j))t} +−i(n(i)(k,j)+1/2)× +{(sin2(ω(i)(k, j)−ω(0)(k, j))t} +(19) +The above correlation function shows real and imaginary +components. The time dependence of the real one is gov- +erned by the cosine functions, which always have a maximum +at t = 0. At increased time t the integrated functions, being +the sum of many cosines with different periods will shrink +to a bundle, which by increasing t finally converges to zero. +Moreover, one may neglect 1/2 because its value appears to +be negligible in comparison to (n + 1)n in ranges of typical +studied temperature. +The imaginary term contains periodic sine functions, which +start from zero at t = 0. At finite values of t the sum of many +sines with different signs and periods will lead the function +shrinking around zero axis to zero bundle, making its contri- +bution small. Consequently, we neglect the imaginary term as +well. +Then, the final thermal factor appears to be equal to +(n(i)(k, j)+1)n(i)(k, j), being the standard form occurring in +the thermal conductivity expressions. It is the occupation co- +efficient responsible for the thermal distribution. Finally, the +fourth-order phonon correlation function reads +< Q(i)(k, j | t) +• +Q(i)(k, j | t) +• +Q(i)(k, j | 0)Q(i)(k, j | 0 >= +1 +2 +� +(n(i)(k, j)+1)n(i)(k, j){cos2(ω(i)(k, j)−ω(0)(k, j))t} +� +(20) +where the time dependence appears in a cosine function +only. Then, the cosine argument consists of difference of two +phonon frequencies. The first one come from DP(i), which +is the partial information of properties of the width of anhar- +monic peak (k, j) in form of phonon frequency. The second +is the reference frequency of the harmonic phonon from the +same harmonic mode ω(0)(k, j). Below, we discuss the proce- +dure to derive analytically the relaxation times for the thermal +conductivity of anharmonic crystals directly from anharmonic +theory30. +Let us insert the fourth-orders correlation function Eq.(20) +into relation of the thermal conductivity Eq.(14). Within the +current non-perturbative anharmonic approach30 this would +be the finale form of the general Green Kubo relation for ther- +mal conductivity in crystals. +κLTC +α,β = +¯h2 +NrVpuckBT 2 +1 +Ni +Ni +∑ +i=1∑ +k, j +(ω(i)(k, j))2v(i)α +gr (k, j)v(i)β +gr (k, j)×(n(i)(k, j)+1))(n(i)(k, j)× 1 +2cos[2(ω(i)(k, j)−ω(0)(k, j))t] +(21) +B. +Relaxation Times of Phonon Modes +The last term of Eq.(21) is related to the relaxation function +of the DP (i) +τ(i) +0 (k, j | t) = 1 +2cos[2(ω(i)(k, j)−ω(0)(k, j))t] +(22) +After time-dependent Laplace integration one obtains a partial +relaxation times (PRT) +τ(i)(k, j) = +� ∞ +0 τ(i) +0 (k, j | t) dt +(23) +Above partial relaxation time is labeled by phonon wavevector +k, phonon branch j and index (i) of DP (i). +The conventional relaxation time (CRT)), characterises the +complete (k, j) anharmonic phonon mode being a result of an +average of all contributions from DP(i). It reads +τcon(k, j) = 1 +Ni +Ni +∑ +i=1 +τ(i)(k, j) = += 1 +2Ni +Ni +∑ +i=1 +� ∞ +0 dt cos(2(ω(i)(k, j)−ω(0(k, j))t) +(24) +This is the relaxation time, which generally is used to calcu- +late the thermal conductivity.The inverse of CRT is related to +the width of the phonon anharmonic mode in frequency space. +To present the computed results of thermal conductivity we + +8 +selected out from general relations of Eq.(21), two auxiliary +expressions (i) the time-independent amplitude function, +Z(i) +α,β(k, j) = +� kB +Vpuc +�� +¯hω(i)(k, j) +kBT +�2 +× +v(i)α +gr (k, j)v(i)β +gr (k, j)(n(i)(k, j)+1)n(i)(k, j) +(25) +and (ii) time-dependent Kubo-Green function being the tensor +of thermal conductivity function. This Kubo-Green functions +have been plotted on many Figures of this article. Notice, that +here averaging over index (i), have not yet been applied.) +G(i) +α,β(t) = 1 +Nr ∑ +k,j +Z(i) +α,β(k, j)τ(i) +0 (k, j | t) +(26) +The above quantity averaged over DP(i) leads to averaged +Green-Kubo components of the thermal conductivity tensor +κα,β(t) = 1 +Ni +Ni +∑ +i=1 +G(i) +α,β(t) +(27) +The Laplace integral over components of thermal conduc- +tivity tensor, Eq.(27) gives final tensor of the thermal conduc- +tivity. These quantity are given in Tables +κα,β = +� ∞ +0 dt κα,β(t) +(28) +Finally, we define the global relaxation time (GRT). It is +a common average relaxation time to characterize the whole +studied system. It includes the Laplace integration, summa- +tion over all DP (i)) and phonons (k, j) +τgl = +1 +2NiNr +Ni +∑ +i=1∑ +k, j +� ∞ +0 dt cos(2(ω(i)(k, j)−ω(0(k, j))t) +(29) +III. +LATTICE THERMAL CONDUCTIVITY +A. +Displacement patterns for LTC +The above described theory will be applied to silicon Si and +magnesium oxide MgO. These crystals belong to cubic struc- +ture with space groups Fd3m and Fm3m, respectively, and +each with r = 2 atoms per primitive unit cells. All the ab ini- +tio calculations have been performed using VASP software47 +on 2×2×2 supercells with periodic boundary conditions and +64 atoms. Using software PHONONA60, and calculating the +Hellmann-Feynman forces created by VASP47 the harmonic +phonon dispersion curves ω(0) = ω(0)(k, j), were established +and plotted. Next, these harmonic curves were used to create +Ni number of DP (i) configurations with additionally randomly +displaced atoms. All these DP have been used to create sets +of the Hellman-Feynman forces, specific for each DP, created +by relaxing the single electronic loop on VASP. +T=200K +=97W/mK +Aver 50x +T=200K +=97W/mK +50x +T=1000K +=62W/mK +50x +T=1000K +=62W/mK +Aver 50x +Si +(a) +(b) +(c) +(d) +GREEN-KUBO FUNCTION (W / m K ps) +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +GREEN-KUBO FUNCTION (W / m K ps) +GREEN-KUBO FUNCTION (W / m K ps) +GREEN-KUBO FUNCTION (W / m K ps) +0 +20 +40 +60 +80 +100 +-20 +20 +40 +60 +80 +100 +0 +120 +20 +40 +60 +80 +100 +0 +120 +-20 +-40 +140 +0 +0.5 +1 +1.5 +2.5 +3.5 +2 +3 +4 +0 +0.5 +1 +1.5 +2.5 +3.5 +2 +3 +4 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +TIME (ps) +TIME (ps) +TIME (ps) +TIME (ps) +FIG. 4. Silicon, Si. Green-Kubo functions, (b,d), Eqs (21) (26), and +averaged Green-Kubo functions, (a,c), Eq.(27). For lattice thermal +conductivity and 50 DP. + +psJ) +100 +K +90 +E +80 +70 +FUNCTION +60 +50 +40 +GREEN-KUBO +30 +20 +10 +0 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +TIME (ps)psJ) +100 +K +] +80 +A) +60 +FUNCTION +40 +20 +GREEN-KUBO I +门 +20 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +TIME (ps)ps]) +K +120 +E +100 +A) +FUNCTION +80 +60 +GREEN-KUBO +40 +20 +0 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +TIME (ps)psJ) +140 +120 +100 +1M) +80 +FUNCTION +60 +40 +20 +GREEN-KUBO I +0 +-20 +40 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +TIME (ps)9 +Si +0 +40 +100 +120 +THERMAL CONDUCTIVITY (W / mK) +20 +80 +60 +TEMPERATURE (K) +0 +600 +800 +1000 +200 +400 +1200 +1400 +FIG. 5. Silicon Si. Lattice thermal conductivity, Eq.(21) +B. +Silicon LTC +At this point we may proceed a calculation of the LTC for +Si using the DP determined in previous section. According to +Eq.(21) these DP are used to average over i = 1,...Ni the ther- +mal conductivity. Phonon frequencies arising from Ni eigen- +value solutions, Eq.(12), can be plotted as phonon dispersion +curves. The averages of the plotted curves reflect the magni- +tude of anharmonicity in any point of the Brillouin zone. They +are just plotted on Fig.3 being results of 500 DP. One sees the +broadening of anharmonic peaks with rising temperature. No- +tice that the same 500 DP could have been used to plot phonon +dispersion curve, density of states, thermodynamic in anhar- +monic state and group velocities and LTC. +Using the formulae for thermal conductivity Eq.(21), +the +LTC +has +been +calculated +for +temperatures +T = +40,70,200,400,600,1000,1500K. The 50 DP were created +for each temperature. During these procedures the lattice con- +stant of a = 5.3847 Å was recorded, and we observed that the +accompanied pressure was stabilized at about 5.6kbar. How- +ever, due to somewhat lower accuracy of determined acoustic +modes, the phonons having frequency below 0.5THz were re- +moved from contributing to atomic configuration, while creat- +ing DP patterns. Hence, low frequency phonons could be not +sufficiently well represented at the low temperature region of +thermal conductivity. +Fig.4b,d shows the detailed behaviour of the Green-Kubo +functions, Eq.(26), for T = 200K and T = 1000K and for +DP i = 1,...50 diagonal components (α,α) = (xx,yy,zz), plot +from bottom to top in color order red, black, green. It is seen +that the Green-Kubo functions are presenting some scatter, +which at t = 0 starts from value +1 +Nr ∑k, j Z(i) +α,β(k, j), then af- +ter a several ps becomes wide and tends at large t to a single +line approaching infinity at G(i) +α,α(t)=0. +Fig.4a,c demonstrates that the global relaxation time for +T = 200K and T = 1000K are about 5.43ps and 2.73ps, +respectively, meaning that longer relaxation occurs at lower +temperature. This figure, Fig.4a,c shows the global relaxation +times plots, Eq.(29), being the averaged of Green-Kubo func- +tions, Gα,β(t), Eq.(26), shows that the function really van- +ishes at larger time t values. In other words, the time depen- +dent cosines periodic functions, which depend on different +TABLE I. Silicon Si and Magnesium Oxide MgO. Calculated aver- +ages of the sum 1 +3(κ1,1 + κ2,2 + κ3,3) of the lattice thermal conduc- +tivities Eq.(21), and the global relaxation times, Eq.(29). +Si T(K) +40 +70 +200 +400 +600 +1000 +1500 +κ(W/mK) +15.8 +43.1 +96.8 +92.4 +79.8 +61.6 +47.5 +τ(ps) +5.94 +5.89 +5.43 +4.40 +3.66 +2.73 +2.11 +MgO T(K) +20 +100 +300 +600 +1000 +1500 +κ(W/mK) +3.14 +49.3 +110.2 +92,2 +68.3 +50.9 +τ(ps) +5,56 +3.63 +2.67 +1.82 +1.28 +0.88 +phonon frequencies, will progressively overlap all terms so +that at long t Gα,β(t) = 0. All diagonal xx,yy,zz components +are computed, and the off-diagonal ones yz,xz,xy vanish due +crystal symmetry and numerically. +Fig.4b,d shows, that the global relaxation time given in +Table I for Si, diminishes with increased T. Moreover, the +amplitude function Z(i) +α,β(k, j) , Eq.(25), has been obtained in +the same computational process as the global relaxation time +τgl. The κα,β and τgl decrease with increased temperature. +The plot of computation silicon thermal behaviour of LTC is +shown on Fig.5. +C. +Magnesium Oxide LTC +At this point we may proceed a derivation of the LTC for +MgO with method described above. Applying the Eq.(21), us- +ing data of DP and related Hellman-Feynman forces coming +from VASP, we determined the LTC coefficients for temper- +atures T = 20,100,300,600,1000,1500K. The 50 DP were +created for each mentioned temperature. +The relaxing lat- +tice constant was a = 4.2462 Å . The LO/TO splitting effect, +present in MgO, was taken into account substituting values of +ionic charges divided by electronic dielectric constant. The +optical phonon branches, which depend on effective charges, +are located at high frequencies and, as one should expect, have +little influence on the thermal conductivity. The remaining +calculation were performed in analogy to procedures used in +Si, and described above. +Fig.6b,d shows examples of the detailed behaviour of the +Green-Kubo functions, Eq.(26) for DP i = 1,...50 for diago- +nal components (α,α) = (xx,yy,zz), plot from bottom to top +in color order red, black, green. It is seen that the Green- +Kubo functions are somehow scattered. At t = 0 they start +from value +1 +Nr ∑k,j Z(i) +α,β(k, j), then after a time of several ps +become wide and tends, at large t, to gather to a single line +approaching infinity at G(i) +α,α(t = 0)=0. +The +relaxing +lattice +constant +was +a = 4.2462 +Å . +The thermal conductivity analysis were run for T = +20,100,300,600,1000,1500K. The structure was stabilized +at about 10kbar. All Ni = 50 DP were created with removed +acoustic phonon modes below 0.5THz. There was 192 exact +wavevectors with the same exact point as for Si. The optical +phonon branches, which dependent on effective charges, are +located at high frequencies and, as one should expect, have + +10 +T=100K +=49W/mK +Aver 50x +MgO +T=100K +=49W/mK +50x +T=1000K +=68W/mK +Aver 50x +T=1000K +=68W/mK +50x +(a) +(c) +(b) +(d) +GREEN-KUBO FUNCTION (W / m K ps) +0 +10 +20 +30 +40 +50 +60 +70 +80 +GREEN-KUBO FUNCTION (W / m K ps) +GREEN-KUBO FUNCTION (W / m K ps) +GREEN-KUBO FUNCTION (W / m K ps) +0 +20 +40 +60 +80 +-20 +50 +100 +150 +200 +250 +0 +300 +50 +100 +150 +200 +250 +0 +300 +-50 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +TIME (ps) +TIME (ps) +TIME (ps) +TIME (ps) +5 +0 +1 +2 +3 +4 +5 +FIG. 6. Magnesium Oxide, MgO. Green-Kubo functions, (b,d), Eqs +(21), (26), and averaged Green-Kubo functions, (a,c), Eq.(27). For +lattice thermal conductivity and 50 DP. +little influence on the thermal conductivity. The remaining +calculation were performed in analogy to procedures used in +Si, and described above. +MgO +0 +40 +100 +120 +THERMAL CONDUCTIVITY (W / mK) +20 +80 +60 +TEMPERATURE (K) +0 +600 +800 +1000 +200 +400 +1200 +1400 +FIG. 7. +Magnesium oxide, MgO. Lattice thermal conductivity, +Eq.(21) +The Figs 6b,d and 6a,c show bundles of Green-Kubo +G(i) +α,α(t), and averaged Green-Kubo functions the Gα,α(t), re- +spectively, Eq.(26). The figures for MgO look very similar +to respective figures of Si. They also present the diagonal el- +ements of the thermal conductivity tensors. The global re- +laxation times were estimated to be 3.63 ps and 1.28 ps for +T=100K, 1000K, respectively. The temperature behaviour of +LTC is shown on Fig.7. Similarly to Si, an abrupt decrease of +LTC is observed below T = 200K. +IV. +HIGH THERMAL CONDUCTIVITY +A. +Elastic Tensor and Equation of Motion +A material with room-temperature thermal conductivity +value larger than 100W/mK is regarded as a high thermal +conductivity material1,21. Such an effect can be achieved ei- +ther by extending the relaxation time τ(i)(k, j), or/and increas- +ing of the amplitude function value Z(i) +α,β(k, j), Eq.(25). Here, +we propose to justify the following approach, which might be +able to provide high values of HTC. +It is well known that material temperature is mainly gov- +erned by its atomic vibrations, which perform vibrations with +amplitudes of order 0.01 − 0.20 Å . Such vibrations in form +of the phonons occur in the whole crystal. Cooling/heating +crystal causes to decrease/increase the atomic vibration ampli- +tudes. For HTC, we are tempting to consider mainly the long, +even very long vibrational waves, which do not care much on +the atomistic details of the material structure. Then, the best +is to use the elastic theory. Within the elastic theory it is con- +venient to determine supercell lattice as element of the crystal +space, where the elastic waves would propagate. So elastic +theory would allow to deform the supercell with no need to +specified the atoms. In particular, one may study properties +through the elastic tensor, which changes with deformations +of supercell. One should, however remember that the elas- +tic tensor is determined by the atomic interactions and atomic +configurations. +There is another reason to pay attention to elastic wave + +80 +K +E +70 +M) +60 +FUNCTION +50 +40 +30 +GREEN-KUBO +20 +10 +0 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +TIME (ps)psJ) +80 +K +[m +70 +60 +A) +50 +FUNCTION +40 +30 +20 +GREEN-KUBO I +10 +0 +-10 +-20 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +TIME (ps)ps]) +300 ++ FUNCTION (W + [m K +250 +200 +150 +GREEN-KUBO +100 +50 +0 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +TIME (ps)psJ) +300 +K +[m +250 +200 +FUNCTION +150 +100 +GREEN-KUBO +50 +0 +50 +0 +.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +TIME (ps)11 +kx +kz +ky +Brilloun zone used for homogeneous +distribution of wavevectors #k for lattice +PHONONS. +Cuboid arround G point provides +homogeneous distribution of +wavevectors #k for ELASTIC waves. +Box in center of Cuboid remains +EMPTY, since the wavevectors #k +there, would have wavelength longer +then typical separations between the +boundaries media. +Reciprocal wavevector space #k +FIG. 8. (Color online) The cuboid volume to which are limited the +wavevectors K influencing the deformations of elastic constants ten- +sors due to phonons displaced in the supercell by the presence of +deformed elastic tensor (DET). +method. The HTC is determined by very long waves, hav- +ing length even above microns. Such long acoustic phonons +are usually represented as acoustic phonons. The frequen- +cies of such acoustic phonons should be of very small and +of very high accuracy, what frequently is difficult to achieve +within lattice dynamics method alone, in particular in com- +plex and low symmetry structures. +A selection of elastic +waves guaranties the correct values of elastic wave frequen- +cies in vicinity of the Γ point. Indeed, sometimes in more +complex crystals, the long acoustic phonons break the crystal +symmetry due to their vibrations, generally guarantied by the +translation-rotation invariances and dynamically violate the +acoustic phonon properties. Therefore, it seems to be rea- +sonable to replace the acoustic phonon modes by the elastic +waves. The elastic waves diminish these effects. The elastic +theory approach would operate with waves characterized by +wavevectors of order k = 0.5 − 0.000001 Å−1, which cover +the object sizes from nanometers- to microns. In this way one +also may introduce the influence of material imperfections on +the HTC thermal conductivity. +The equation of motion for elastic medium is formulated +for elastic plane waves, which are written as +U(X,t) = AJexp(Ω(K,J)t −KX) +(30) +where UJ(X,t) is the supercell material deformation at point +X of the material, and J = 1,2,3 is index of elastic mode. The +AJ is a component of vibration amplitude, Ω(K,J) - angular +frequency for elastic wave mode, t time, and K the wavevector +of monochromatic wave. The equation of motion of the elastic +waves is called CHRISTOFFEL equation10,48–54 +The Hooke’s low relates the elastic strain εKL with the stress +σIJ. +σIJ = +3 +∑ +K.L=1 +CIJKL εKL +(31) +where I,J,K,L are indices each from 1, 2 up to 3. The elastic +properties of the material are described by a fourth-rank tensor +CIJKL with 34 = 81 elements. They can be arranged in a 6×6 +TABLE II. Contraction scheme: indices I,J,K,L in CI,J,K,L are re- +placed by indices α,β in Cα,β . The same rules works in reverse +direction. Scheme used by Voigt and VASP. The PHONONA uses +VASP notation +I,J or K,L +11 +22 +33 +23/32 +13/31 +12/21 +α or β +1 +2 +3 +4 +5 +6 +Voigt +XX +YY +ZZ +YZ/ZY +XZ/ZX +XY/YX +VASP +XX +YY +ZZ +XY/YX +YZ/ZY +ZX/XZ +matrix that is symmetric, with elements Cα,β = Cβ,α. The +elastic tensor in the form of 6 × 6 matrix should usually be +available from external program like VASP. +Cα,β = +� +� +� +� +� +� +� +C11 C12 C13 C14 C15 C16 +C21 C22 C23 C24 C25 C26 +C31 C32 C33 C34 C35 C36 +C41 C42 C43 C44 C45 C46 +C51 C52 C53 C54 C55 C56 +C61 C62 C63 C64 C65 C66 +� +� +� +� +� +� +� +The relations between Cα,β and CJILM are shown in Table II. +The stiffness tensor Cα,β not only contains information +about static materials deformation, but also about the elastic +waves traveling through the material. The equation of mo- +tion for the elastic waves can be obtained from solution of the +Christoffel equation48 +ρΩ2(K,J)·E(K,J) = ∑ +I,L +KICJILMKL ·E(K,M) +(32) +where ρ represents the mass density. +The solution of the +Cristoffel equation for each wavevector K provides three solu- +tions corresponding to elastic waves with definite frequencies. +The equation combines the Cristoffel 3 × 3 square matrix M +with elements +MJM = ∑ +I,L +KICJILMKL +(33) +Now, Eqs(32), and (33) form an eigenvalue problem that can +be routinely solved at arbitrary K. The result is a set of three +frequencies Ω2(K,J) and polarization vectors E(K), Since +M is real and symmetric matrix, the eigenvalues are real and +eigenvectors E(K,J) constitute an orthogonal basis. Further- +more, the property that M is a symmetric matrix involves that +the Ω2(K,J) is real and positive. +It is convenient to introduce auxiliary matrix +GJ,M = ∑ +I,L +KICJILMKL +(34) +This expression represents the core part of the Cristoffel +Eq.(32). The derivatives of GJ,M are needed to specify the +group velocity. They could be derived from three matrices of + +12 +order 3×3, which are wavevector derivatives of matrix GJ,M +Ω2 +0(K,J) = Diag +� +∑ +I,L +KICJILMKL +� +∂Ω2 +0 +∂Kx +(K,J) = Diag +� +∑ +L +(CJ1LMKL +∑ +I +KICJI1M +� +∂Ω2 +0 +∂Ky +(K,J) = Diag +� +∑ +L +(CJ2LMKL +∑ +I +KICJI2M +� +∂Ω2 +0 +∂Kz +(K,J) = Diag +� +∑ +L +(CJ3LMKL +∑ +I +KICJI3M +� +(35) +All matrices in Eqs (35) can be numerically diagonalized, +which is marked by "Diag". The inputs are the right hand ma- +trices, while the outputs constitute of following eigenvalues +Ω2 +0, ∂Ω2 +0 +∂Kx , ∂Ω2 +0 +∂Ky and ∂Ω2 +0 +∂Kz . The diagonalization of the above ma- +trices gives their eigenvalues. Moreover, one must also diago- +nalize the matrix GJ,M. Ratio of these data divided by 2 leads +to the group velocities of the elastic waves. Furthermore, one +might find this derivative differentiating the Cristoffel equa- +tion (32). From Eq.(32) one finds the group velocity of elastic +waves +Vgr(K,J) = +1 +2·Ω0(K,J) +� +i∂Ω2 +0 +∂Kx +,j∂Ω2 +0 +∂Ky +,k∂Ω2 +0 +∂Kz +� +(36) +where i, j, k are versors along x,y,z directions. +Max Born developed in his book55 a method which corre- +lates the elastic constants with the slopes of acoustic phonon +modes at a particularly small wavevectors. As a matter of fact, +the elastic waves have been identified as the acoustic waves at +small K. Therefore, expression for LTC thermal conductivity +should hold for HTC, with only a few differences listed below. +The lattice thermal conductivity calculations introduced +above for phonons can also be used for elastic waves. Al- +though their wavelengths are evidently longer than the size of +the supercell used in ab initio calculations, one may account +the long wavelength doing the following: (i) create phonon +displacement patterns DP in conventional supercells, for ex- +ample the same as for phonons, (ii) see that the displaced +atoms also cause changes of the elastic constants, (iii) call de- +formed elastic tensor (DET), which conventionally will be the +supercell deformed itself, what results in symmetry lowering +to DET. +By solving now the Cristoffel equation, Eq.(32),60 with +DET, one calculates the elastic wave frequencies, finding their +frequency changes with respect of ideal supercell tensor. One +may say that the Cristoffel equation rebuilds the ideal elastic +wave to whole space from the crystal segment belonging to +DET limited to studied supercell. Moreover, the group veloc- +ities Eq.(36), can also change. These changes influence the +relaxation time of conducting objects acting in thermal con- +ductivity. +The relation for high thermal conductivity (HTC) for- +mulated in analogy with lattice thermal conductivity LTC, +Eq.(21), for the simulation with the deformed elastic tensor +DET reads +κHTC +α,β = +¯h2 +NrVpuckBT 2 +1 +Ni +Ni +∑ +i=1 +� ∞ +0 dt +Cuboid +∑ +K,J +(Ω(i)(K,J))2Vpuc(i)α +gr (K,J)V(i)β +gr (K,J) +× (n(i)(K,J)+1))(n(i)(K,J)× 1 +2 cos[2(Ω(i)(K,J)−Ω(0)(K,J))t] +(37) +where Ni is the number of DET - deformed elastic tensors, +Vpuc volume of the primitive unit cell. The Cuboid, see Fig. +8 is a volume in the reciprocal zone, with center point at +K = 0. In LTC Eq.(21), the summations ∑k,j run homoge- +neously over the whole Brillouin zone. In HTC Eq.(37), the +summations ∑K,J should run over small wavevector volume +around K = 0 as indicated in Cuboid, ∑Cuboid +K,J +. The central +green box should always remain empty (it is direct space be- +yond volume of the sample), so there no wavevectors should +be positioned. In the volume of the larger inner box (between +green and red boxes), the wavevectors K for elastic waves +should be placed. Notice, that a lot of wavevectors can deter- +mine crystal phonons, much less wavevectors are indexing the +elastic waves. The central box excludes such long wavevec- +tors K, which surpass the sample macroscopic size, or the +mean distance between boundaries existing in the media. In +general, the wavevectors should be places at random, unless +the wavevectors amplitudes and positions express some super- +structure being a new object of the study. Then, the positions +and amplitudes of wavevectors K could be derived from the +object in the direct space and then transformed to the cuboid +by three-dimensional Fourier transform. Of course, the shape +of the Cuboid may change to adapt to the studied object. Re- +member that sets of wavevectors for phonons k and elastic +waves K are needed for LTC Eq.(21) and HTC Eq.(37) ex- +pressions, respectively. And that the HTC must be described +by wavevectors from the elastic wave region. + +13 +T=20K +T=40K +T=70K +T=200K +T=600K +20x T=600K +(b) +(a) +(c) +(e) +(d) +(f) +Si elastic +=2107W/mK +=16W/mK +=260W/mK +=2197W/mK +=3700W/mK +GREEN-KUBO FUNCTION (W / m K ps) +-100 +0 +0 +2 +6 +8 +4 +10 +TIME (ps) +TIME (ps) +TIME (ps) +TIME (ps) +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +70 +80 +12 +14 +0 +2 +6 +8 +4 +10 +TIME (ps) +12 +14 +-200 +100 +200 +300 +400 +500 +600 +700 +800 +TIME (ps) +GREEN-KUBO FUNCTION (W / m K ps) +-100 +0 +-200 +100 +200 +300 +400 +500 +600 +700 +800 +GREEN-KUBO FUNCTION (W / m K ps) +-100 +0 +100 +200 +300 +400 +500 +600 +700 +800 +GREEN-KUBO FUNCTION (W / m K ps) +-100 +0 +-200 +100 +200 +300 +400 +500 +600 +700 +800 +-300 +GREEN-KUBO FUNCTION (W / [m K ps) +-100 +0 +-200 +100 +200 +300 +400 +500 +600 +700 +800 +-300 +GREEN-KUBO FUNCTION (W / m K ps) +-400 +0 +-200 +200 +400 +600 +800 +0 +5 +10 +15 +20 +25 +30 +35 +40 +FIG. 9. Silicon Si. (a,e) HTC averaged Green-Kubo functions calculated from Eqs(37) and (26), (f) Green-Kubo functions for HTC (elastic +waves) calculated directly from Eq.(37). Each run used 20 DET. +B. +Silicon HTC +To calculate the HTC of silicon the formulae for thermal +conductivity Eq.(37) was used. The 20 DP were prepared for +each T = 7,20,40,70,200 and 600K, and then 20 DET’s were +created in each case. During heating the lattice constants and +pressure stayed constant as observed for LTC of Si. The elas- +tic tensors, called also elastic modulus, were calculated on +VASP,47,56. One must also add that the cpu calculation time +of this process is long in comparison to cpu run for elastic +tensor possessing some symmetry elements. It is a result of +the fact that the DET’s do not have any symmetry, hence, it +requires to calculate much more iterations. +Moving back to HTC, one should supply some informa- +tion on the crystal microstructure, and include it to Eq.(37). +The minimum information should indicate the possible range +of the wavelengths of the elastic waves before they reach +the boundaries, which hinder their travel and then determine +the expected HTC. In the present stage of the current theory +we may propose to select the proper wavelengths, or rather +wavevectors of the elastic waves only, and check whether they +lead to correct results observed in experiments. The idea is +that the shortest wavelengths λmin = 2π/Kmax of the elastic +waves start from a distance just above the active wavelength +of low frequency of acoustic phonons, and spreads to longest +distances to ”boundaries” λmax = 2π/Kmin. +It is expected +that elastic waves characterized by wavevectors K from the +interval Kmin < K < Kmin could propagate in the crystal with- +out obstacles. Generally, such precise information is missing. +Moreover, the crystal microstructure may depend on distribu- +tions of the boundaries within the crystal, microcracks, kind +of defects and impurities, etc. and as such they ought to be a +topic of separate study. Here, the volume of the perfect crystal +FIG. 10. Silicon Si. (red) Calculated HTC data using Eq.(37) making +use of elastic waves. (blue) Single point of LTC data at T = 600K, +imported from Fig. 5. (black) Measured thermal conductivity from +Ref.25. +is represented by a cuboid, Fig. 8 filled with the wavevectors +K with an exception of the inner box, which should be empty. +In the direct space the empty box represents the outer part be- +yond the crystal and the cuboid represents surface layers of +the real crystal, expected to suppress propagation of elastic + +10 000 +Si +(W/ mK) + elastic +1000 +THERMAL CONDUCTIVIT +100 +10 +2 +10 +100 +1000 +TEMPERATURE (K)psl) +800 +GREEN-KUBO FUNCTION (W 1 [m K +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +0 +5 +10 +15 +20 +25 +30 +35 +40 +TIME (ps)GREEN-KUBO FUNCTION (W 1 [m K ps]) +800 +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +300 +0 +10 +20 +30 +40 +50 +60 +70 +80 +TIME (ps)psJ) +800 +K +] +600 +GREEN-KUBO FUNCTION (W 1 [ +400 +200 +0 +200 +400 +0 +N +4 +6 +8 +10 +12 +14 +TIME (ps)ps) +800 + K +700 +u +600 +GREEN-KUBO FUNCTION (W [ +500 +400 +300 +200 +100 +0 +-100 +200 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60 +TIME (ps)GREEN-KUBO FUNCTION (W 1 [m K ps]) +800 +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +300 +0 +N +4 +6 +8 +10 +12 +14 +TIME (ps)ps) +800 +GREEN-KUBO FUNCTION (W 1 [m K +700 +600 +500 +400 +300 +200 +100 +0 +-100 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60 +TIME (ps)14 +waves. +In the present stage of the theory the microstructure in- +formation could be accounted in a primitive way, namely by +limiting the boundary to select the wavevectors K belonging +to cuboid in the summation ∑Cuboid +K,J +of the formula Eq.(37). +For the present study of Si, the random wavevectors were se- +lected out the volume of the cuboid confined by minimal value +Kmin = 0.00001Å−1 to maximal value of Kmax = 0.030Å−1 +The inner box inside Kmin = 0.00001 Å−1 was left empty. +The wavevectors Kmax and Kmin correspond to the lengths of +elastic waves from 0.021µm to 63µm, respectively. Since in +the cuboid the wavevectors distribution is homogeneous the +amount of elastic waves taken in derivation of the HTC close +to 63µm is less than in vicinity of 0.021µm. +The above mentioned summation in the cuboid was used +to compute the Green-Kubo functions. The selected function +are shown on Fig.9. The vertical axes of plots are mainly de- +termined by the group velocities and temperature occupation +distributions and they stay almost constant. But the horizontal +axes cover changeable time intervals. Thus, time seems to de- +cide about the magnitude of the relaxation times and thermal +conductivities. In particular after initial maximum at t = 0, the +Green-Kubo function diminishes to negative minimum. Such +a decrease is caused by the difference of two cosines occur- +ring in the relaxation time expression, Eq.(37). Namely, small +differences between the elastic wave frequencies of deformed +and perfect supercells lead to minimum at longer times, in +contrary to opposite situation with large frequency difference +and the minima occurring at shorter times. The Green-Kubo +functions accompanied the elastic waves can be seen on the +plots. The minima of the (a-f) plots are drawn from 20 runs of +DET’s. On (f) all 20 plot are seen and some smearing of the +data are observed. +Relation Eq.(37) gives also the numerical value of ther- +mal conductivity as a function of temperature. For silicon +the low temperature HTC results are plotted on Fig.10 The +LTC data reaches maximum 96.8W/mK, while the HTC is el- +evated to 3700W/mK. This high difference is mainly caused +by the increase of the global relaxation times in HTC mech- +anism. +Longer relaxation times result in higher values of +κα,β. The silicon single crystal, for the HTC measurements +by Glassbrenner and Slack reported in25 and used for the low- +temperature measurements, was grown from high-purity sili- +con. The growth process was made with care in order to make +the crystal to be oxygen and dislocation free. Then, any va- +cancy clusters were less than one micron in diameter. The +sample boundary sizes were not reported. +It is worth to mention that at low temperature the occupa- +tion factor ( ω +T )2(n( ω +T ) + 1)n( ω +T ) in Eq.(37) reduces the in- +tensity of the Green-Kubo function. +These properties also +force to decrease HTC close to T = 0K to a very small value. +With increasing temperature the mentioned thermal factors +approach 1. Similar effect has been seen for MgO. +TABLE III. Silicon Si and Magnesium Oxide MgO. Calculated av- +eraged sums 1 +3(κ1,1 +κ2,2 +κ3,3) of the high thermal conductivities +Eq.(37), and the global relaxation times, Eq.(29). +Si T(K) +7 +20 +40 +70 +200 +600 +κ(W/mK) +628 +2107 +3700 +2197 +260 +21.2 +τ(ps) +29.6 +21.6 +20.7 +10.4 +1.63 +0.11 +MgO T(K) +5 +25 +40 +100 +300 +κ(W/mK) +514 +3225 +2932 +218 +0.14 +τ(ps) +1.15 +9.94 +8.81 +8.36 +0.13 +C. +Magnesium Oxide HTC +To calculate the HTC of magnesium oxide, MgO atomic +displacements DP were prepared for each T = 5,25,40,100 +and 300K and next 20 DET’s tensors were created for each T +using VASP. During heating the lattice constants and pressure +behaved as observed for LTC of MgO. +The below mentioned summation within the cuboid was +used to compute the Green-Kubo functions. +The selected +functions are shown on Fig.11. +The vertical plot axes are +mainly determined by the group velocities and temperature +occupation distributions and they stay almost constant. But +the horizontal axes cover changeable time intervals. Thus, +time seems to decide about the magnitude of the relaxation +times and thermal conductivities. +To calculate HTC of MgO one should select data for the +cuboid. The following wavevectors have been proposed: max- +imal values of Kmax = 0.0030 Å−1 and minimal value of +Kmin = 0.00001 Å−1. The selected cuboid wavevectors for +MgO correspond to boundaries of the elastic waves being in +the range from 0.21µm to about 63µm, respectively. This +information was used in Eq.(37) to compute the time depen- +dent Green-Kubo HTC functions, and later plotted on Fig. +11. The HTC data reaches maximum 3700W/mK, The vari- +ation of the HTC in MgO are presented on Fig.12. +The +T = 5,25,40,100K fit to measured data, but the contribution +of HTC at T = 300K practically vanishes. At T = 300K only +LTC contributes. +The two approaches provide thermal conductivity LTC and +HTC. Therefore, it rises a question what happens in the tem- +perature interval between the LTC and HTC regions. +The +present results give for Si at T = 600K: HTC: 21,2W/mK, +LTC: 84W/mK Exp:84.5W/mK, and for MgO at T = 300K: +HTC: 0.14W/mK, LTC: 110W/mK Exp: 75W/mK. +This +means that LTC mechanism is used at higher T and then at +lower T it became replaced by HTC processes. Probably, this +effect occurs in special materials only. +D. +Microstructure in HTC +The current approach relates the HTC and microstructure in +the region of low temperatures. The conventional process to +carry on the HTC calculations would be to look into cuboid for +the wavevectors data to get an agreement between calculated + +15 +MgO elastic +T=5K +=514W/mK +(a) +T=40K +T=300K +T=25K +T=100K +T=300K +=218W/mK +=2932W/mK +=3225W/mK +=0.16W/mK +(b) +(c) +(d) +(e) +(f) +GREEN-KUBO FUNCTION (W / [M K PS]) +-800 +0 +TIME (ps) +TIME (ps) +TIME (ps) +TIME (ps) +0 +100 +20 +30 40 +50 60 +0 +10 +20 +30 +40 +50 +60 +0 +160 +120 +40 +200 +240 +80 +0 +TIME (ps) +-400 +800 +400 +1600 +GREEN-KUBO FUNCTION (W / [M K PS]) +-500 +0 +500 +1000 +1500 +GREEN-KUBO FUNCTION (W / [M K PS]) +GREEN-KUBO FUNCTION (W / [M K PS]) +GREEN-KUBO FUNCTION (W / [M K PS]) +0 +GREEN-KUBO FUNCTION (W / [M K PS]) +10 +70 +90 +80 +1200 +-500 +0 +500 +1000 +1500 +-400 +800 +400 +1600 +1200 +0 +70 +80 +TIME (ps) +0 +200 +400 +600 +800 +1000 +1200 +-400 +800 +400 +1600 +1200 +100 +20 +40 +60 +120 +80 +0 +100 +20 +40 +60 +120 +80 +-500 +0 +500 +1000 +1500 +FIG. 11. Magnesium oxide, MgO. (a,e) HTC averaged Green-Kubo functions calculated from Eqs(37) and (26), (f) Green-Kubo functions for +HTC (elastic waves) calculated directly from Eq.(37). Each run used 20 DET. +10 +FIG. 12. Magnesium oxide, MgO. (red) Contributions of only elas- +tic waves to high thermal conductivity. (green) Single point of LTC +data at T = 300K, imported from Fig.7. (black) Experimental points +from22 +and measured κHTC +α,β . It would mean to find right values of the +wavevectors inserted to cuboid. However, the reverse process +would be more valuable. First the microstructure features are +foreseen and used to modify the cuboid, and next to collate the +calculated data with the behaviour of the measured HTC. In +this case the method could have some predictive power, which +might help to design the required properties of the material, +say microstructure. +As a test we have inserted to the cuboid of Si crystal be- +ing at T = 40K, the wavevectors of Kmin = 0.00001 Å−1 and +Kmax = 0.00015Å−1, corresponding to Si boundary distances +from 63 and 4.2 µm, respectively. For these wavevectors HTC +tremendously increases the global relaxation time to 1700 ps +and thermal conductivity to κHTC +α,β += 320000W/mK. Other +obstacles might diminish/change this value. +Presently the cuboid volume is filled with wavevectors of +the same amplitude. However, having a concept of the mi- +crostructure of considered crystal one might convert this infor- +mation to the amplitudes of wavevectors placed in the cuboid. +Such a project is still waiting for realization. +There are many studies, which need to combine the mi- +crostructure of the sample with its thermal conductivity. Here, +follows some example (1) The thermal energy transport in +actinide oxide nuclear fuel materials57, thorium dioxide and +uranium dioxide. The first has a characteristic maximum of +thermal conductivity below 40 K. The second has a reduced +thermal conductivity, in spite of similar crystal symmetries. It +happens due to the presence of elastic phase transition. (2) +The dislocation impact on thermal conductivity58. Disloca- +tions induce the stress field, which might lead to anisotropy +of thermal transport. Such a contribution can be estimated. +Another goal would be to analyze the collection of disloca- +tions on thermal conductivity. (3) Thermal properties of the +superelastic, which consist of many crystal variant of shape +memory alloys (as NiTi). The complex microstructure exists +due to well-known compatible equations for pair of crystals +variants, which require to identify the interfaces59. + +psJ) +k1500 +] +M) +1000 +GREEN-KUBO FUNCTION +500 +0 +500 +0 +TIME(pS)K 1400 +E1200 +GREEN-KUBO FUNCTION (W +1000 +800 +600 +400 +200 +0 +200 +400 +600 +800 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +TIME (ps)1400 +1200 +GREEN-KUBO FUNCTION +800 +600 +400 +200 +0 +200 +400 +600 +0 +10 +20 +30 +40 +50 +60 +70 +80 +TIME (ps)ps]) +K1500 +E +GREEN-KUBO FUNCTION +500 +0 +500 +0 +20 +40 +60 +80 +TIME (ps)K +1400 +1200 +E +1000 +GREEN-KUBO FUNCTION +800 +600 +400 +200 +0 +200 +400 +600 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 110 120 +TIME (ps)ps]) +K 1500 +E +M) +1000 +GREEN-KUBO FUNCTION +500 +0 +500 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 110 120 +TIME (ps)10.000 +5 +MgO + elastic +1000 +THERMAL CONDUCTVITY +100 +10 +1 +3 +10 +30 +100 +300 +1000 +TEMPERATURE (K)16 +The MD simulations of simple crystal models have shown +that realistic microstructures of YBa2Cu3O7 superconductor61 +and LaNbO4 ferroelastic62 could be obtained starting from +a simple crystal model at relatively high temperature, and +next quenching. The simulated miscrostructures and those +obtained from TEM observations are very similar. It is also +worthwhile to mention the effort to increase effectiveness of +thermoelectric CaCd2Sb263. It has been proposed to replace +Cd by Mg, as point defect, to induce significantly phonon +scattering, but maintaining the carrier concentration, lower es- +sentially their LTC and increase figure of merit ZT. +V. +CONCLUSIONS AND DISCUSSIONS +The mechanisms of thermal conductivities discussed in this +article is based on the anharmonic phonons formalism han- +dled within the non-perturbative approach for crystals,30 . Due +to it, the formulated theories of LTC and HTC have been +retrieved starting from different and not conventional anhar- +monic approaches. In the LTC case phonons play a role of +the heat transport media realized by anharmonic phonons of +the crystals. The anharmonic vibrations of atoms determine +the crystals temperature. Next, one prepares several displace- +ment pattern DP of atoms in the crystal supercells, with ampli- +tudes displaced corresponding to studied temperature. Then, +the forces induced by the displaced atoms are calculated with +the ab initio software, which permits to solve the set of lat- +tice dynamics equations, and find information about the har- +monic and anharmonic interatomic potentials. This technique +has been successfully used to create positions, shifts, widths +and shapes of anharmonic peaks and determine the analytical +expression for the mode relaxation times for all anharmonic +modes without performing expansion of the interaction po- +tential over anharmonic terms and without using the Boltzman +equation. Specially, the relaxation times could be calculated +analytically, and this process needs only to know the differ- +ences of anharmonic and harmonic frequencies for each seg- +ment of the anharmonic phonon mode belonging to the same +(k, j) phonon mode. +Some crystals require to take into account also the elastic +waves, which need to be considered within different mathe- +matical formalism. The elastic waves, travel in the crystal and +form a strain variation of predefined units of crystal, usually +supercells. Atomic displacement patterns, similar to those of +phonons, create some strains, deform supercells, and hence +create the elastic waves. Finally, the elastic waves can be +found by solving the Cristoffel equation, being entirely de- +fined by the elastic constant tensors. +We have shown that the crystal thermal conductivity is de- +termined by the Green-Kubo relationship being the correlation +function of the heat flux. The high thermal conductivity can be +calculated from products of elastic wave frequencies, elastic +wave group velocities, and phonon relaxation times specific +for elastic waves. +Second essential difference between lattice and high ther- +mal conductivity is related with the wavevectors summa- +tion within the correlation functions. In the phonon part all +wavevectors need to be used in the sum over the Brillouin +zone. The elastic waves are characterized by very long wave- +lengths, so only the short wavevectors around k = 0 and lower +then 1 − 4THz participate in the thermal conductivity. This +effect is applied to fix the longest wavevectors as being able +to reach the sample sizes, or other obstacles which limit the +transport of the elastic waves. This criterion leads to statement +that the calculated HTC may agree with the measurements. 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Zhang, Appl.Phys.Lett.120, 041901 (2022). + diff --git a/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/load_file.txt b/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60f68b98113d922103c0f8d2b3c1d1795a9f4732 --- /dev/null +++ b/d9AzT4oBgHgl3EQf3f6y/content/tmp_files/load_file.txt @@ -0,0 +1,1809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf,len=1808 +page_content='Ab initio determination of thermal conductivity in crystals Krzysztof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Parlinski1, 2 1)Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland 2)Computing for Materials, Kraków, Poland (*Electronic mail: Krzysztof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Parlinski@ifj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='pl) (Dated: 04 January 2023) The calculations of thermal conductivity requires to know anharmonic properties of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For this purpose a non-perturbative anharmonic theory is applied, which do not make use of the potential energy expansion over atomic displacements, but instead, runs ab initio calculations of Hellmann-Feynman forces for atomic patterns of atoms with specific displacements to rebuild the anharmonic phonon frequencies, and group velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' see [K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Parlinski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' B 98, 054305 (2018),] The Green-Kubo equation for the thermal conductivity needs to know the above quantities and the phonon relaxation times, which are related to the 4th-order phonon correlation function expressed in terms of phonon anihilation and creation Bose operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In currect formulation of anharmonic theory the relaxation times can be derived as analitical expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The Green-Kubo formulae was succesfully applied to find thermal conductivity of Si and conductivities, related to the phonon and elastic waves, respectivily, were computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' INTRODUCTION The understanding of thermal conductivity in solids is needed for applications of technically relevant materials to nanofabrication technology, to manufacture electronic devices for nanoscale demands, to understand the mechanisms, pre- dict the properties of solid thermal condunctivities and to be able to run related computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Similarly, handle of ther- mal conductivity describes partly the behaviour of thermo- electrics, electron-mediate superconductors and thermal con- ductivity materials, which govern the heat transfer processes in the Earth’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The heat transport properties of solids are usually divided into two mechanisms: First kind is called Lattice thermal conductivity (LTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is calculated applying phonon anhar- monicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The method seems to be rather well known, and in this case the Green-Kubo linear-response theory1 is mainly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There are some variants in formulating this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In one of them the harmonic phonon frequencies, the group velocities for phonon modes, and some relaxation times are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Another way is to find the input data from the anhar- monic perturbation method, usually with help of the triple and quatric order terms2–10, where relaxation time comes from solving the Boltzmann equation11,12, generally using third, or third and fourth order anharmonic terms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Next method is to run molecular dynamics (MD)13–18 provided that the po- tential of the studied system is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The last mentioned ap- proach solves the classical equations of motions for the sys- tem, tracing particle’s evolution and then collecting the nec- essary quantities, which are required by Green-Kubo formu- lae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Typically, the Green-Kubo equations describe properly the thermal conductivity of solids for LTC, in the interval from around room to melting temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The anharmonic effects alone can be also studied applying the stochastic self- consistent harmonic approximation method19,20, which ac- cording to the Gibbs – Bogoliubov variational principle re- quires that the true free energy of the system reaches the min- imum of the functional F[ ˜ρ] with respect to all possible trial density matrices [ ˜ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The second kind of heat transport will be in this case called High thermal conductivity (HTC), for which the complete theory is still under construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The HTC typically occurs in simple crystal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At low temperature (below 200 - 300K) HTC materials exhibit usually two, even three order of magnitude higher thermal conductivity values than the same material at high temperature range only (above 300K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To this group of crystals belongs: C (diamond), Si, Ge, AlN, AlP, BAs, BN, BP, BeS, GaN, MgO21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Thermal properties of sev- eral HTC crystals have been measured by Slack et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='21–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These HTC materials attracted special attention and called for relevant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In 1964 Glassbrenner and Slack25 proposed a mechanism of HTC for silicon Si and Germanium Ge, based on phenomenological approach24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Later, a similar consid- eration on ab initio level was published by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ward et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Recently, Esfarjani et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='26, studying Si, have discussed HTC mechanism as arising from large mean free path of phonons, determined by size of sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It was shown that HTC of Si arises for more than order of magnitude, if mean free path spans from about nanometers to 100 microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is generally accepted that the LTC is totally described by the acoustic and optic phonon modes, and therefore the LTC heat transfer is described by Green-Kubo formulae, where usually the relaxation time is found by Boltzmann equation or MD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In articles27 and28 it was shown how the Green function of the anharmonic perturbation theory may lead to the typical Lorentzian term, with shift and width of anharmonic peak28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the article29 it was attempted to decouple the fourth-order correlation function, responsible for the relaxation times and needed for thermal conductivity, using the pairing Wick’s the- orem, but finally MD runs validated the results for silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the present article we reformulate the Green-Kubo ap- proach to use so called displacement patterns (DPs) of atomic configurations to derive the LTC directly from phonon disper- sion curves created from DP and simultaneously determine phonon relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, derivation of the relax- ation time from solution of the Boltzmann equation or MD calculations is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='01831v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 2 In next sections the method has been extended to handle also the HTC phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this case the low frequency and very long elastic waves are used to govern the HTC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To calculate such long wavelength states we compute ab ini- tio the elastic constant tensors with the equilibrium atoms in the supercell and for series of similar supercells with atoms displaced from equilibrium positions due to presence of tem- perature, like in DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' From these elastic supercells one calcu- lates the frequencies and group velocities of the elastic waves, and apply them to the Green-Kubo expression to find HTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this case it is obvious that the accounted wavelength of elas- tic waves could be considerably longer than wavelength of ordinary phonons, therefore one must introduce limit to the longest active wavevector which could be accommodated in the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The LTC and HTC calculation of Green- Kubo relations, as derived in this paper, have the same formal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Anharmonicity The thermal conductivity in solids is determined by anhar- monicity of the system, therefore, one should start from dis- cussion how to handle anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In present article a pro- cedure, which takes also advantage of the ab initio calcula- tions considers anharmonic properties of crystals within a new non-perturbative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It would be much much easier to understand the current article first looking to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30 and glance at the examples presented there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There, the procedure begins from selecting the supercell of the studied crystal and calculating the harmonic phonons, using PHONON software31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At equilibrium every atom of the crystal resides near the potential energy minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Displacing an atom from its equi- librium position by a vector u one creates so called Hellmann- Feynman (HF) forces computed using VASP33, and acting on the surrounding atoms, in particular atoms of the super- cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' PHONON computes in this way the harmonic phonon frequencies ω(0)(k, j) and eigenvectors e(0)(k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The HF forces could be calculated with the ab initio program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The same HF forces are used to build all force constants and dy- namical matrix elements, which are the essential quantities in lattice dynamics theory since more as a century3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One should only keep in mind that the used atomic displacements am- plitudes u should probe only small interval of the harmonic potential around the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' From these data the mentioned software calculates harmonic phonon dispersion curves in the whole Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the harmonic calculations the used atomic displacements u are small, of order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='03 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='04 Å, which is close to zero-temperature phonon vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this harmonic theory30 the method uses first the exact wavevectors k, with wavelengths, being commensurate with the supercell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At such exact wavevectors30–32 the peri- odic structure of the crystals ensures that the harmonic fre- quencies ω(0)(k, j) and eigenvectors e(0)(k, j) are calculated exactly, independent on the size of the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Unfor- tunately, the list of exact wavevectors diminishes with de- creasing size of the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Of course, certain balance between computational time and accuracy of the result will determine the selected supercell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The phonon frequen- cies and eigenvectors beyond exact wavevectors are interpola- tions between exact wavevectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The interpolations are sup- ported by a traditional analytical derivation of dynamical ma- trix elements, which must be solved, what in practice leads to the valid results in the whole Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The interpo- lated procedure uses the singular value decomposition (SVD) method32,34, which simply assures that the finale phonon dis- persion curves are the best fit in the mean square sens to the exact phonons frequencies of the exact points within the con- strains of classical phonon dispersion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' As a matter of fact this approach32 to phonon theory was already equipped in 1996 with the procedure similar to the machine learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The PHONON software30 is also able to calculate the phonon dispersion curves from supercell with many atoms, which are displaced simultaneously out from their equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, if the atomic displacements stay small, it means they do not enter the non-parabolic part of the potential, then the resulting phonon dispersion curves look like in the harmonic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Indeed, the force constants are determined by proportionality coefficient between atomic displacement and HF force and in harmonic regime do not depend on the ampli- tude of displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' However, if in the above procedure the displacements are larger, some deviation of the phonon frequencies might be ob- served because in reality atoms during vibrations visit the non- parabolic parts of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These changes of frequencies and eigenvectors manifest the anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Hence, the deviation of the particular phonon frequency (ω(anh)(k, j) − ω(0)(k, j)), for the same (k, j), could be considered as a mea- sure of the anharmonicity Of course, it is well known that atoms vibrate in the crystal sites due to finite temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For a given T, one should displace the atoms from their equilibrium positions and create the displacements pattern (DP) next used to find the phonon vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At a given T, the DP could be represented as a snap- shot of supercell with many atoms displaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One would like to create sets of Ni atomic displacement patterns DP(i), i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ni, which might arise in the crystal at a given T, and in different moments and locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The proposition given in30 is as follows: Each supercell DP should be filled with the phonon waves, determined by the well known expression of atomic displacements u(m,µ,γ) and supplemented by the phase factor φ(k, j) of traveled phonon waves, where meaning of indices is later given before Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' u(m,µ,γ) = Q(k, j) �Mµ eγ(k, j | µ)exp[2πi(k·R(m,µ)−φ(k, j)] (1) The phase φ(k, j) of the phonon wave could be taken at ran- dom from the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0) to mimic different atomic displacement pattern labelled by the same (k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The mean square displacement amplitude < Q2(k, j) > of the phonon wave was determined in35,36 by 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' A schematic set of DP(i) in single anharmonic phonon peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Si Si (a) (b) k=X k=K T=1000K T=1000K PEAK INTENSITY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 0 2 4 6 2 8 10 12 14 16 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50 PEAK INTENSITY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='25 0 2 4 6 2 8 10 12 14 16 18 FREQUENCY (THz) FREQUENCY (THz) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Anharmonic phonon peaks calculated for crystal at T = 1000K and wavevectors (a) X = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0) and (b) K = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='375,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='375,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='725).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The plots arrived from DP(i) i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' < Q2(k, j) >= ¯h 2ω(k, j)coth � ¯hω(k, j) 2kBT � (2) In the harmonic approximation the above relation is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Si T= 200K T=1000K (a) (b) WAVE VECTOR WAVE VECTOR FREQUENCY (THz) FREQUENCY (THz) X K L Γ Γ X K L Γ Γ 18 0 2 2 16 14 12 10 8 6 4 0 2 16 14 12 10 8 6 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The maps of anharmonic phonon dispersion curves along the line of wavevector Γ−X −K−Γ−L for temperature (a) T = 200K and (b) T = 1000K calculated from DP(i) 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='500 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Blue-green-red colours indicates intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='overleaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='com/project/63668958d1320f3c7c9a7540 The Si and MgO the 2 × 2 × 2 supercell contains 64 atoms, 32 exact wavevectors each with 6 degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' More- over, the phonon waves may still be supplemented by random number of phase φ(k, j) from the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For Si and MgO, the atomic displacement changes with T, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (2), it follows Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='05 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='16 Å in temperature range T = 40 − 1500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='07%, respectively, of the nearest neighbor interatomic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Using the above method it is rather easy to obtain the an- harmonic peaks for any wavevector k and phonon branch j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These can be any wavevectors, although those which do not belong to list of exact wavevectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One needs to create dis- placement patterns DP (i), i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ni in the range from Ni = 20 to 500, depending on the requested precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For conventional anharmonic peaks it could be limited to about Ni = 50 DP, but to study a peculiar form of the anharmonic peak, such as asymmetric shape, particularly high background under the peak of non - Lorentzian shape, or even splitting of the single anharmonic peak, the value of Ni should be larger Ni = 200−500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The amplitudes of the vibrating atoms caus- ing anharmonic effects and estimated above occur in real crys- tals and create many HF forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These multiplicity of forces create multiplicity of force constants, which in turn, are used to solve the equations pf classical lattice dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Schemat- ically the construction of anharmonic phonon mode can be 0 10110 Reference 9 12 Frequency 13 8 14 15 16 6 17 5 18 19 20 FREQUENCY (i)(k,j) (THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='15 PEAK 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='05 2 0 2 6 8 10 12 14 16 18 FREQUENCY (THz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50 45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='40 INTENSITY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='25 PBAK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='05 0 2 0 2 6 8 10 12 14 16 18 FREQUENCY (THz)r x K T 16 14 (zHL) 12 FREQUENCY 10 8 6 4 2 0 WAVE VECTORr x K r 18 16 14 FREQUENCY (THz) 12 10 8 6 4 2 0 2 WAVE VECTOR4 performed as shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Examples of calculated an- harmonic phonon peaks are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is a set of δ(ω(i)) functions of DP(i)1,2,i = ···20, Every segment i rep- resents single snapshot of atomic displacements for the same anharmonic phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The phonon waves have differ- ent phases counting against the fixed sites of the atoms, hence the frequencies and intensities may vary a little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The δ(ω(i)) frequencies together with intensities (amplitude) are solutions of the lattice dynamic equations for the selected wavevector k and accompanied displacements corresponding to tempera- ture T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the above scheme a set of 20 δs mimic envelope of single anharmonic phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In further one calculation of anharmonic phonon mode with wavevector k being located in between the already plotted one can be added to increase statistic and precision of phonon peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The envelope of the delta set should give the form of the anharmonic peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Refer- ence frequency on the plot corresponds to harmonic frequency used letter in the conductivity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There appear more profits, following this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Namely, in this theory the symmetry of each obtained anharmonic peak is uniquely labeled by the irreducible representation of the crystal space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Normally, it is done only for the har- monic phonon δ-kind peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, however, the calculated area under the anharmonic phonon peaks is characterized by the same irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' From the same DP(i), i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ni, with value Ni as dis- cussed above, one may construct histograms for the phonon dispersion curves along any path of the reciprocal space, which next can be plotted as a map of the phonon dispersion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such maps for Si at T = 200K and 1000K are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Harmonic and anharmonic hamiltonians The vibrational hamiltonian for a crystal in harmonic approximation3 can be written as H(0) = ∑ m,µ,γ P2,(0)(m,µ,γ) 2Mµ + 1 2 ∑ m,µ,γ ∑ n,ν,δ Φ(0)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) × (U(0)(m,µ,γ)(U(0)n,nu,δ) (3) where the harmonic force constants Φ(0) have been calculated from the Hellman-Feynman forces of the perfect crystal with atoms preserving the crystal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The H(0) hamilto- nian describes the harmonic phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Solving the eigenvalue equation for H(0) one arrives to harmonic phonon frequencies ω(0)(k, j) and polarization vectors e(0) µ (k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These collection of harmonic phonons are used as a reference set of data when analysing the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The current method requires also to find phonon frequen- cies from the hamiltonians H(i),where "anharmonic" force constants Φ(i), i > 0, lead to larger/smaller displacement amplitudes, then in harmonic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Now, one creates the Hellmann-Feynman forces for all displaced atoms collected in DP (i), Eqs (1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Solution of these eigenvector equations leads to little different phonon frequencies and one may write H(i) = ∑ m,µ,γ P2,(i)(m,µ,γ) 2Mm,µ + 1 2 ∑ m,µ,γ ∑ n,ν,δ Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) × (U(i)(m,µ,γ)(U(i)n,nu,δ) (4) If the anharmonic system converts to the harmonic one, then the force constants converge Φ(i) → Φ(0), and the forces are reduced to harmonic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' From the relations given above we conclude that in similar conditions as proclaimed above occurs H(i) → H(0), and therefore the anharmonic hamiltoni- ans disappears HA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Anharmonic hamiltonian vanishes if the phonons of crystal become harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the thermal conductivity becomes infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='. Above, the two body anharmonic force constants, Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ), are labelled also by index (i) of DP(i), which indicates that the anharmonic force constant acting on the atom (m,µ,γ) arises not only due to displacing a single atom (n,ν,δ) (as was in the harmonic case), but it really senses also forces coming from all other displaced atoms of supercell according to the configuration imposed by DP(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This suggests that all atoms affects the anharmonic force constant Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This means that Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) is in some sens a many body force con- stant, which feels simultaneous displacements of all other atoms in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In other words all anharmonic force constants are computed not in the perfect crystal, but in the crystal being represented by a series of i = ···, supercells , having atoms shifted out from equilibrium positions, due to finite temperature, and from that configuration one computes the contributions to anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The hamiltonian H(0) provides harmonic phonon frequen- cies only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The harmonic potential for perfect insulator should lead to infinity thermal conductivity of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This state- ment has been expressed in the textbook of Ashcroft and Mermin37, in Callaway’s11 and Maradudin38 papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Ashcroft and Mermin says that ” in perfect harmonic insulator crystal the phonon scattering does not occur, so such a crystal should have infinite thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Scattering of phonons from lattice imperfections would produce a finite thermal conduc- tivity, but with a wrong temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The only way to explain the realistic thermal conductivity data is to ad- mit that phonons can be scattered by other phonons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Thus, the relevant thermal conductivity should exhibit the following properties: (i) demonstrate infinite thermal conductivity for strictly harmonic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (ii) describe the finite thermal con- ductivity for crystal with anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Consequently, one may propose to treat the thermal conductivity using the fol- lowing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The anharmonic effects are described by the excess of effects arising from H(i) hamiltonians, superim- posed on the harmonic modes coming from H(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Thus, the anharmonicity effects of a crystal can be determined by the 5 following hamiltonian HA = 1 Ni Ni ∑ i=1 � H(i) −H(0)� (5) From the relations given above we may conclude that for vanishing anharmonicity, when H(i) → H(0), the anharmonic hamiltonians disappear HA = 0 and the crystal exhibits infinite thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Because the hamiltonians H(0) and H(i), Eqs(3,4) are sums of two positively definite quadratic forms, one in the com- ponents of the momenta and the other in the components of the atomic displacements, it follows from a theorem of ma- trix algebra39 that it is possible to find principal axes, or nor- mal coordinate transformations which simultaneously diago- nalized the kinetic and potential energies in these hamiltoni- ans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such a principal axis transformations are generated by the conventional expansion of displacements and momenta in terms of plane waves and next Bose annihilation b(k, j) and creation b+(k, j) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In therms of these operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' the hamiltonians Eqs(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4) take the simple forms H(0) = ∑k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j ¯hω(0)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)[b+(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)b(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)+ 1 2] H(i) = ∑k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j ¯hω(i)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)[b+(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)b(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)+ 1 2] (6) From Eqs (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 6) the anharmonic hamiltonian HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' with sub- tructed harmonic phonon contribution H0 reads HA = ∑ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j Ni ∑ i=0 � ¯hω(i)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)− ¯hω(0)(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j) � × b+(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j)b(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j) (7) where it has been assumed that the Bose operators b+(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j) and b(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j) for the same mode (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' j) with close frequen- cies should be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' very similar and further we assume that they remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Indeed, in this ap- proach the anharmonicity is determined by the differences of � ¯hω(i)(k, j)− ¯hω(0)(k, j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These frequencies could be systematized and collected to histograms, labeled by a wavevector and phonon branch (k, j) and finally to present as a Lorenzian-kind anharmonic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such peaks could be measured by inelastic neutron scattering, Raman spectra, or infrared absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Below we shall use this method to model the thermal conductivity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is essential to remind that the path from the DP (i) to phonon frequencies is performed by the solution of lattice dynamics equation of motion only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, a single DP (i) for fixed i can be treated as an anhar- monic perturbation cluster, arising from simultaneously dis- placements of many atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In traditional perturbation the- ory, DP(i) is typically limited to triple or quatric interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, a crystal with supercell of 64 atoms provides single DP (i) data for all wavevectors (k, j) of the Brillouin zone, so some cross interaction therms are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' FORMULAE FOR THERMAL CONDUCTIVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Phonons The Green-Kubo approach is based on statistical thermodynamics40–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' A derivation of basic formulae can be found in references8,26,44–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The heat flux J(t), for simplicity, is usually determined without contribution from diffusion and convection, (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here also, we adapt the formalism of the anharmonic theory described in previous section, to apply the set of anharmonic hamiltonians H(i) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The mentioned method expects the crystal to be presented as a set of Ni supercell’s subsystems with atoms randomly displaced patterns DP (i), i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ni, corresponding to studied temperature T, J(i) α (t) = 1 2 ∑ m,µ,γ ∑ n,ν,δ (Rα(m,µ,γ)−Rα(n,ν,δ)) × � U(i)(m,µ,γ | t)·Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) 1 Mν P(i)(n,ν,δ | t) � (8) Here, we use indexing of atoms: first atom: (m,µ,γ), sec- ond atom:(n,ν,δ), where m,n are coordinates of primi- tive unit cells, µ,ν are atomic indices within primitive unit cells, and γ,δ stay for coordinate x,y,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The force constants Φ(i)(m,µ,γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='n,ν,δ) may have contributions from harmonic and/or anharmonic regions of the interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this sens the force constants may contain contributions from any higher order anharmonic therms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, the force constants might also have contributions from other displaced atoms of used DP (i), and not shown explicitly in the now dis- cussed form of Φ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The same force constant may also repre- sent harmonic force constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' As argued in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='I B the thermal conductivity should be cal- culated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (9), over thermal fluctuations repre- sented by the harmonic and anharmonic hamiltonians Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (5), determined by the components DP (i) (i = 1,···Ni), all gen- erated for the same T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The Green Kubo expression is then written as κα,β = 1 VkBT 2 1 Ni Ni ∑ i=1 � ∞ 0 < J(i) α (t)J(i) β (0) > dt (9) Averaging the above correlation function over DP (i) one may use it to study also anharmonic phonon peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Using the expansions of atom displacements and momenta over plane 6 waves Q(i)(k, j) and Q (i) (k, j), respectively,3, one has U(i)(m,µ,γ | t) = � ¯h NMµ ∑ k,j e(i) γ (k, j | µ) × exp[2πi(k·R(m,µ)]Q(i)(k, j | t) P(i)(n,ν,δ | t) = 1 i � ¯hMν N ∑ k,j e(i) δ (k, j | ν) × exp[2πi(k·R(n,ν)] Q (i) (k, j | t), (10) where (bold i = √−1), N is the number of wavevectors k used in the summation of Eqs (10), and j is the index of phonon branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Now, recalculating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (8) one can rewrite it in the form J(i) α (t) = ¯h iN ∑ k, j ω(i)(k, j)v(i)α gr (k, j) ×Q(i)(k, j | t) Q (i) (k, j | t) (11) Here, imaginary unit i appears since it was added to the expo- nent of the dynamical matrix D(i)(k), when used to define the group velocity, Eq(13) In next steps one finds the phonon frequencies and eigen- vectors for perfect crystal (i = 0) and for crystal modified with DP(i), (i > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Both are lattice dynamic solutions of the eigenvalue phonon equation ω(i)2(k, j) = e(i)T(k, j)D(i)(k)·e(i)(k, j) (12) Of course they need different values of the elements of dy- namical matrix D(i)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Further, the group velocity vectors can be found from rele- vant dynamical matrices using v(i) gr (k, j) = 1 2ω(i)(k, j) � e(i)T(k, j) � ∂ ∂kD(i)(k) � e(i)k, j) � (13) Notice that with the same equations the phonon frequencies ω(i)(k, j) and group velocities v(i) gr (k, j) have been found in ab initio procedure via the Hellman-Feynman force30 created by displacement of atoms fixed already in DP (i)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These devia- tions of DP (i) phonon frequencies from the relevant harmonic frequency contain information concerning the anharmonicity, in terms of frequency and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Collecting the expressions of Eqs (9, 11, 13) the thermal conductivity tensor reads κLTC α,β = ¯h2 VpuckBT 2 1 Ni Ni ∑ i=1 � ∞ 0 dt 1 Nr < ∑ k, j (ω(i)(k, j))2v(i)α gr (k, j)v(i)β gr (k, j) × < Q(i)(k, j | t) Q(i)(k, j | t) Q(i)(k, j | 0)Q(i)(k, j | 0 > (14) where r is a number of atoms in primitive unit cell, Vpuc vol- ume of primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The appeared fourth order phonon correlation function < Q(t) Q(t) Q(0)Q(0) > needs some com- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is the only function under the Laplace integral, which depends on time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' If the integrated function would be a constant C = const ̸= 0 then the Laplace integral � ∞ 0 Cdt = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This would be the mechanism to make a harmonic crystal hav- ing infinite thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To considered the value of the fourth-order correlation function we need to express the normal mode amplitudes of phonons by Bose annihilation b and creation b+ operators Q(i)(k, j) = 1 √ 2 1 � ω(i)(k, j) � b(k, j)+b+(−k, j) � Q(i)(k, j) = 1 √ 2 � ω(i)(k, j) � b(k, j)−b+(−k, j) � (15) Now, the pair time-dependent correlation functions of b, b+ are found from the solution of the Heisenberg time dependent equations3, in which the anharmonic hamiltonian HA Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (5) and (7) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then < b(k, j | t)b+(k′, j′) | 0 > = exp(−iω(i)(k, j)−ω(0)(k, j)]t)(n(i)(k, j)+1)δk,k′δ j,j′ < b+(k, j | t)b(k′, j′) | 0 > = exp(+iω(i)(k, j)−ω(0)(k, j)]t)n(i)(k, j)δk,k′δj, j′ (16) Here, the mean number of phonons in the vibrational mode (k, j) of DP (i) at temperature T, is n(i)(k, j) = 1 eβ ¯hω(i)(k, j) −1 (17) and β = 1 kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Applying Eqs (15, 16), the fourth order correlation func- tion < Q(t) Q(t) Q(0)Q(0) > can be evaluated, with the Wick’pairing theorem6,29, to 16 correlation functions of prod- ucts of averages consisting of four b, b+ operators each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In 10 functions, out of the mentioned 16, the 4 operator terms vanish due to averages build from pairs of the same kind of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The remaining 6 correlation functions do not vanish from the 7 mentioned reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' However, 4 functions arrived from the last 6 non-zero terms, mutually cancel, due to averages build from pairs constructed from the same kind of operators and 6 terms are not vanishing from these reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' However, the 2 last terms < b(t)b(t)b+(0)b+(0) > and < b+(t)b+(t)b(0)b(0) > remain non-zero and can be written as < b(k, j | t)b(k, j | t)b+(k,j| 0)b+(k, j | 0) >= 2(n(i)(k, j)+1)2 e−2i[ω(i)(k, j)−ω(0)(k,j)t] < b+(k, j | t)b+(k, j | t)b(k,j| 0)b(k, j | 0) >= 2(n(i)(k, j))2 e+2i[ω(i)(k, j)−ω(0)(k,j)t] (18) Applying the time dependence of the surviving pairs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (16), the non-zero fourth-order correlation functions are < Q(i)(k, j | t) Q(i)(k, j | t) Q(i)(k, j | 0)Q(i)(k, j | 0 >= (n(i)(k, j)+1)n(i)(k, j)+1/2)× {(cos2(ω(i)(k, j)−ω(0)(k, j))t} −i(n(i)(k,j)+1/2)× {(sin2(ω(i)(k, j)−ω(0)(k, j))t} (19) The above correlation function shows real and imaginary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The time dependence of the real one is gov- erned by the cosine functions, which always have a maximum at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At increased time t the integrated functions, being the sum of many cosines with different periods will shrink to a bundle, which by increasing t finally converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, one may neglect 1/2 because its value appears to be negligible in comparison to (n + 1)n in ranges of typical studied temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The imaginary term contains periodic sine functions, which start from zero at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At finite values of t the sum of many sines with different signs and periods will lead the function shrinking around zero axis to zero bundle, making its contri- bution small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Consequently, we neglect the imaginary term as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the final thermal factor appears to be equal to (n(i)(k, j)+1)n(i)(k, j), being the standard form occurring in the thermal conductivity expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is the occupation co- efficient responsible for the thermal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Finally, the fourth-order phonon correlation function reads < Q(i)(k, j | t) Q(i)(k, j | t) Q(i)(k, j | 0)Q(i)(k, j | 0 >= 1 2 � (n(i)(k, j)+1)n(i)(k, j){cos2(ω(i)(k, j)−ω(0)(k, j))t} � (20) where the time dependence appears in a cosine function only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the cosine argument consists of difference of two phonon frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The first one come from DP(i), which is the partial information of properties of the width of anhar- monic peak (k, j) in form of phonon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The second is the reference frequency of the harmonic phonon from the same harmonic mode ω(0)(k, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Below, we discuss the proce- dure to derive analytically the relaxation times for the thermal conductivity of anharmonic crystals directly from anharmonic theory30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Let us insert the fourth-orders correlation function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (20) into relation of the thermal conductivity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Within the current non-perturbative anharmonic approach30 this would be the finale form of the general Green Kubo relation for ther- mal conductivity in crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' κLTC α,β = ¯h2 NrVpuckBT 2 1 Ni Ni ∑ i=1∑ k, j (ω(i)(k, j))2v(i)α gr (k, j)v(i)β gr (k, j)×(n(i)(k, j)+1))(n(i)(k, j)× 1 2cos[2(ω(i)(k, j)−ω(0)(k, j))t] (21) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Relaxation Times of Phonon Modes The last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21) is related to the relaxation function of the DP (i) τ(i) 0 (k, j | t) = 1 2cos[2(ω(i)(k, j)−ω(0)(k, j))t] (22) After time-dependent Laplace integration one obtains a partial relaxation times (PRT) τ(i)(k, j) = � ∞ 0 τ(i) 0 (k, j | t) dt (23) Above partial relaxation time is labeled by phonon wavevector k, phonon branch j and index (i) of DP (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The conventional relaxation time (CRT)), characterises the complete (k, j) anharmonic phonon mode being a result of an average of all contributions from DP(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It reads τcon(k, j) = 1 Ni Ni ∑ i=1 τ(i)(k, j) = = 1 2Ni Ni ∑ i=1 � ∞ 0 dt cos(2(ω(i)(k, j)−ω(0(k, j))t) (24) This is the relaxation time, which generally is used to calcu- late the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='The inverse of CRT is related to the width of the phonon anharmonic mode in frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To present the computed results of thermal conductivity we 8 selected out from general relations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), two auxiliary expressions (i) the time-independent amplitude function, Z(i) α,β(k, j) = � kB Vpuc �� ¯hω(i)(k, j) kBT �2 × v(i)α gr (k, j)v(i)β gr (k, j)(n(i)(k, j)+1)n(i)(k, j) (25) and (ii) time-dependent Kubo-Green function being the tensor of thermal conductivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This Kubo-Green functions have been plotted on many Figures of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Notice, that here averaging over index (i), have not yet been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=') G(i) α,β(t) = 1 Nr ∑ k,j Z(i) α,β(k, j)τ(i) 0 (k, j | t) (26) The above quantity averaged over DP(i) leads to averaged Green-Kubo components of the thermal conductivity tensor κα,β(t) = 1 Ni Ni ∑ i=1 G(i) α,β(t) (27) The Laplace integral over components of thermal conduc- tivity tensor, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (27) gives final tensor of the thermal conduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These quantity are given in Tables κα,β = � ∞ 0 dt κα,β(t) (28) Finally, we define the global relaxation time (GRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is a common average relaxation time to characterize the whole studied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It includes the Laplace integration, summa- tion over all DP (i)) and phonons (k, j) τgl = 1 2NiNr Ni ∑ i=1∑ k, j � ∞ 0 dt cos(2(ω(i)(k, j)−ω(0(k, j))t) (29) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' LATTICE THERMAL CONDUCTIVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Displacement patterns for LTC The above described theory will be applied to silicon Si and magnesium oxide MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These crystals belong to cubic struc- ture with space groups Fd3m and Fm3m, respectively, and each with r = 2 atoms per primitive unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' All the ab ini- tio calculations have been performed using VASP software47 on 2×2×2 supercells with periodic boundary conditions and 64 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Using software PHONONA60, and calculating the Hellmann-Feynman forces created by VASP47 the harmonic phonon dispersion curves ω(0) = ω(0)(k, j), were established and plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Next, these harmonic curves were used to create Ni number of DP (i) configurations with additionally randomly displaced atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' All these DP have been used to create sets of the Hellman-Feynman forces, specific for each DP, created by relaxing the single electronic loop on VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' T=200K \uf06b=97W/mK Aver 50x T=200K \uf06b=97W/mK 50x T=1000K \uf06b=62W/mK 50x T=1000K \uf06b=62W/mK Aver 50x Si (a) (b) (c) (d) GREEN-KUBO FUNCTION (W / m K ps) 0 10 20 30 40 50 60 70 80 90 100 GREEN-KUBO FUNCTION (W / m K ps) GREEN-KUBO FUNCTION (W / m K ps) GREEN-KUBO FUNCTION (W / m K ps) 0 20 40 60 80 100 20 20 40 60 80 100 0 120 20 40 60 80 100 0 120 20 40 140 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 3 4 0 1 2 3 4 5 6 0 1 2 3 4 5 6 TIME (ps) TIME (ps) TIME (ps) TIME (ps) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon, Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Green-Kubo functions, (b,d), Eqs (21) (26), and averaged Green-Kubo functions, (a,c), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For lattice thermal conductivity and 50 DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' psJ) 100 K 90 E 80 70 FUNCTION 60 50 40 GREEN-KUBO 30 20 10 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 6 TIME (ps)psJ) 100 K ] 80 A) 60 FUNCTION 40 20 GREEN-KUBO I 门 20 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 6 TIME (ps)ps]) K 120 E 100 A) FUNCTION 80 60 GREEN-KUBO 40 20 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 TIME (ps)psJ) 140 120 100 1M) 80 FUNCTION 60 40 20 GREEN-KUBO I 0 20 40 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 TIME (ps)9 Si 0 40 100 120 THERMAL CONDUCTIVITY (W / mK) 20 80 60 TEMPERATURE (K) 0 600 800 1000 200 400 1200 1400 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Lattice thermal conductivity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon LTC At this point we may proceed a calculation of the LTC for Si using the DP determined in previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21) these DP are used to average over i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Ni the ther- mal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Phonon frequencies arising from Ni eigen- value solutions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (12), can be plotted as phonon dispersion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The averages of the plotted curves reflect the magni- tude of anharmonicity in any point of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' They are just plotted on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3 being results of 500 DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One sees the broadening of anharmonic peaks with rising temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' No- tice that the same 500 DP could have been used to plot phonon dispersion curve, density of states, thermodynamic in anhar- monic state and group velocities and LTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Using the formulae for thermal conductivity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), the LTC has been calculated for temperatures T = 40,70,200,400,600,1000,1500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The 50 DP were created for each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' During these procedures the lattice con- stant of a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3847 Å was recorded, and we observed that the accompanied pressure was stabilized at about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='6kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' How- ever, due to somewhat lower accuracy of determined acoustic modes, the phonons having frequency below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5THz were re- moved from contributing to atomic configuration, while creat- ing DP patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Hence, low frequency phonons could be not sufficiently well represented at the low temperature region of thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4b,d shows the detailed behaviour of the Green-Kubo functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (26), for T = 200K and T = 1000K and for DP i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50 diagonal components (α,α) = (xx,yy,zz), plot from bottom to top in color order red, black, green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is seen that the Green-Kubo functions are presenting some scatter, which at t = 0 starts from value 1 Nr ∑k, j Z(i) α,β(k, j), then af- ter a several ps becomes wide and tends at large t to a single line approaching infinity at G(i) α,α(t)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4a,c demonstrates that the global relaxation time for T = 200K and T = 1000K are about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='43ps and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='73ps, respectively, meaning that longer relaxation occurs at lower temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This figure, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4a,c shows the global relaxation times plots, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (29), being the averaged of Green-Kubo func- tions, Gα,β(t), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (26), shows that the function really van- ishes at larger time t values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In other words, the time depen- dent cosines periodic functions, which depend on different TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si and Magnesium Oxide MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Calculated aver- ages of the sum 1 3(κ1,1 + κ2,2 + κ3,3) of the lattice thermal conduc- tivities Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), and the global relaxation times, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Si T(K) 40 70 200 400 600 1000 1500 κ(W/mK) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 τ(ps) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='11 MgO T(K) 20 100 300 600 1000 1500 κ(W/mK) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='14 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2 92,2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='9 τ(ps) 5,56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='88 phonon frequencies, will progressively overlap all terms so that at long t Gα,β(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' All diagonal xx,yy,zz components are computed, and the off-diagonal ones yz,xz,xy vanish due crystal symmetry and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4b,d shows, that the global relaxation time given in Table I for Si, diminishes with increased T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, the amplitude function Z(i) α,β(k, j) , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (25), has been obtained in the same computational process as the global relaxation time τgl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The κα,β and τgl decrease with increased temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The plot of computation silicon thermal behaviour of LTC is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium Oxide LTC At this point we may proceed a derivation of the LTC for MgO with method described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Applying the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), us- ing data of DP and related Hellman-Feynman forces coming from VASP, we determined the LTC coefficients for temper- atures T = 20,100,300,600,1000,1500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The 50 DP were created for each mentioned temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The relaxing lat- tice constant was a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2462 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The LO/TO splitting effect, present in MgO, was taken into account substituting values of ionic charges divided by electronic dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The optical phonon branches, which depend on effective charges, are located at high frequencies and, as one should expect, have little influence on the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The remaining calculation were performed in analogy to procedures used in Si, and described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='6b,d shows examples of the detailed behaviour of the Green-Kubo functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (26) for DP i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50 for diago- nal components (α,α) = (xx,yy,zz), plot from bottom to top in color order red, black, green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is seen that the Green- Kubo functions are somehow scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At t = 0 they start from value 1 Nr ∑k,j Z(i) α,β(k, j), then after a time of several ps become wide and tends, at large t, to gather to a single line approaching infinity at G(i) α,α(t = 0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The relaxing lattice constant was a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2462 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The thermal conductivity analysis were run for T = 20,100,300,600,1000,1500K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The structure was stabilized at about 10kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' All Ni = 50 DP were created with removed acoustic phonon modes below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There was 192 exact wavevectors with the same exact point as for Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The optical phonon branches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' which dependent on effective charges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' are located at high frequencies and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' as one should expect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='T=100K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='\uf06b=49W/mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Aver 50x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='MgO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='T=100K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='\uf06b=49W/mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='T=1000K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='\uf06b=68W/mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Aver 50x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='T=1000K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='\uf06b=68W/mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='GREEN-KUBO FUNCTION (W / m K ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='GREEN-KUBO FUNCTION (W / m K ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='GREEN-KUBO FUNCTION (W / m K ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='TIME (ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='TIME (ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='TIME (ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='TIME (ps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium Oxide, MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Green-Kubo functions, (b,d), Eqs (21), (26), and averaged Green-Kubo functions, (a,c), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For lattice thermal conductivity and 50 DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' little influence on the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The remaining calculation were performed in analogy to procedures used in Si, and described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' MgO 0 40 100 120 THERMAL CONDUCTIVITY (W / mK) 20 80 60 TEMPERATURE (K) 0 600 800 1000 200 400 1200 1400 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium oxide, MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Lattice thermal conductivity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21) The Figs 6b,d and 6a,c show bundles of Green-Kubo G(i) α,α(t), and averaged Green-Kubo functions the Gα,α(t), re- spectively, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The figures for MgO look very similar to respective figures of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' They also present the diagonal el- ements of the thermal conductivity tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The global re- laxation times were estimated to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='63 ps and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='28 ps for T=100K, 1000K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The temperature behaviour of LTC is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Similarly to Si, an abrupt decrease of LTC is observed below T = 200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' HIGH THERMAL CONDUCTIVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Elastic Tensor and Equation of Motion A material with room-temperature thermal conductivity value larger than 100W/mK is regarded as a high thermal conductivity material1,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such an effect can be achieved ei- ther by extending the relaxation time τ(i)(k, j), or/and increas- ing of the amplitude function value Z(i) α,β(k, j), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, we propose to justify the following approach, which might be able to provide high values of HTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is well known that material temperature is mainly gov- erned by its atomic vibrations, which perform vibrations with amplitudes of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='20 Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such vibrations in form of the phonons occur in the whole crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Cooling/heating crystal causes to decrease/increase the atomic vibration ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For HTC, we are tempting to consider mainly the long, even very long vibrational waves, which do not care much on the atomistic details of the material structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the best is to use the elastic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Within the elastic theory it is con- venient to determine supercell lattice as element of the crystal space, where the elastic waves would propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' So elastic theory would allow to deform the supercell with no need to specified the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In particular, one may study properties through the elastic tensor, which changes with deformations of supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One should, however remember that the elas- tic tensor is determined by the atomic interactions and atomic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There is another reason to pay attention to elastic wave 80 K E 70 M) 60 FUNCTION 50 40 30 GREEN-KUBO 20 10 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 TIME (ps)psJ) 80 K [m 70 60 A) 50 FUNCTION 40 30 20 GREEN-KUBO I 10 0 10 20 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 TIME (ps)ps]) 300 + FUNCTION (W + [m K 250 200 150 GREEN-KUBO 100 50 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 TIME (ps)psJ) 300 K [m 250 200 FUNCTION 150 100 GREEN-KUBO 50 0 50 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 5 TIME (ps)11 kx kz ky Brilloun zone used for homogeneous distribution of wavevectors #k for lattice PHONONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Cuboid arround G point provides homogeneous distribution of wavevectors #k for ELASTIC waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Box in center of Cuboid remains EMPTY, since the wavevectors #k there, would have wavelength longer then typical separations between the boundaries media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Reciprocal wavevector space #k FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (Color online) The cuboid volume to which are limited the wavevectors K influencing the deformations of elastic constants ten- sors due to phonons displaced in the supercell by the presence of deformed elastic tensor (DET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The HTC is determined by very long waves, hav- ing length even above microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such long acoustic phonons are usually represented as acoustic phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The frequen- cies of such acoustic phonons should be of very small and of very high accuracy, what frequently is difficult to achieve within lattice dynamics method alone, in particular in com- plex and low symmetry structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' A selection of elastic waves guaranties the correct values of elastic wave frequen- cies in vicinity of the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Indeed, sometimes in more complex crystals, the long acoustic phonons break the crystal symmetry due to their vibrations, generally guarantied by the translation-rotation invariances and dynamically violate the acoustic phonon properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Therefore, it seems to be rea- sonable to replace the acoustic phonon modes by the elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic waves diminish these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic theory approach would operate with waves characterized by wavevectors of order k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='000001 Å−1, which cover the object sizes from nanometers- to microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this way one also may introduce the influence of material imperfections on the HTC thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The equation of motion for elastic medium is formulated for elastic plane waves, which are written as U(X,t) = AJexp(Ω(K,J)t −KX) (30) where UJ(X,t) is the supercell material deformation at point X of the material, and J = 1,2,3 is index of elastic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The AJ is a component of vibration amplitude, Ω(K,J) - angular frequency for elastic wave mode, t time, and K the wavevector of monochromatic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The equation of motion of the elastic waves is called CHRISTOFFEL equation10,48–54 The Hooke’s low relates the elastic strain εKL with the stress σIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' σIJ = 3 ∑ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='L=1 CIJKL εKL (31) where I,J,K,L are indices each from 1, 2 up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic properties of the material are described by a fourth-rank tensor CIJKL with 34 = 81 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' They can be arranged in a 6×6 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Contraction scheme: indices I,J,K,L in CI,J,K,L are re- placed by indices α,β in Cα,β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The same rules works in reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Scheme used by Voigt and VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The PHONONA uses VASP notation I,J or K,L 11 22 33 23/32 13/31 12/21 α or β 1 2 3 4 5 6 Voigt XX YY ZZ YZ/ZY XZ/ZX XY/YX VASP XX YY ZZ XY/YX YZ/ZY ZX/XZ matrix that is symmetric, with elements Cα,β = Cβ,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic tensor in the form of 6 × 6 matrix should usually be available from external program like VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Cα,β = � � � � � � � C11 C12 C13 C14 C15 C16 C21 C22 C23 C24 C25 C26 C31 C32 C33 C34 C35 C36 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 C55 C56 C61 C62 C63 C64 C65 C66 � � � � � � � The relations between Cα,β and CJILM are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The stiffness tensor Cα,β not only contains information about static materials deformation, but also about the elastic waves traveling through the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The equation of mo- tion for the elastic waves can be obtained from solution of the Christoffel equation48 ρΩ2(K,J)·E(K,J) = ∑ I,L KICJILMKL ·E(K,M) (32) where ρ represents the mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The solution of the Cristoffel equation for each wavevector K provides three solu- tions corresponding to elastic waves with definite frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The equation combines the Cristoffel 3 × 3 square matrix M with elements MJM = ∑ I,L KICJILMKL (33) Now, Eqs(32), and (33) form an eigenvalue problem that can be routinely solved at arbitrary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The result is a set of three frequencies Ω2(K,J) and polarization vectors E(K), Since M is real and symmetric matrix, the eigenvalues are real and eigenvectors E(K,J) constitute an orthogonal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Further- more, the property that M is a symmetric matrix involves that the Ω2(K,J) is real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is convenient to introduce auxiliary matrix GJ,M = ∑ I,L KICJILMKL (34) This expression represents the core part of the Cristoffel Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The derivatives of GJ,M are needed to specify the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' They could be derived from three matrices of 12 order 3×3, which are wavevector derivatives of matrix GJ,M Ω2 0(K,J) = Diag � ∑ I,L KICJILMKL � ∂Ω2 0 ∂Kx (K,J) = Diag � ∑ L (CJ1LMKL +∑ I KICJI1M � ∂Ω2 0 ∂Ky (K,J) = Diag � ∑ L (CJ2LMKL +∑ I KICJI2M � ∂Ω2 0 ∂Kz (K,J) = Diag � ∑ L (CJ3LMKL +∑ I KICJI3M � (35) All matrices in Eqs (35) can be numerically diagonalized, which is marked by "Diag".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The inputs are the right hand ma- trices, while the outputs constitute of following eigenvalues Ω2 0, ∂Ω2 0 ∂Kx , ∂Ω2 0 ∂Ky and ∂Ω2 0 ∂Kz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The diagonalization of the above ma- trices gives their eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, one must also diago- nalize the matrix GJ,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Ratio of these data divided by 2 leads to the group velocities of the elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Furthermore, one might find this derivative differentiating the Cristoffel equa- tion (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (32) one finds the group velocity of elastic waves Vgr(K,J) = 1 2·Ω0(K,J) � i∂Ω2 0 ∂Kx ,j∂Ω2 0 ∂Ky ,k∂Ω2 0 ∂Kz � (36) where i, j, k are versors along x,y,z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Max Born developed in his book55 a method which corre- lates the elastic constants with the slopes of acoustic phonon modes at a particularly small wavevectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' As a matter of fact, the elastic waves have been identified as the acoustic waves at small K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Therefore, expression for LTC thermal conductivity should hold for HTC, with only a few differences listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The lattice thermal conductivity calculations introduced above for phonons can also be used for elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Al- though their wavelengths are evidently longer than the size of the supercell used in ab initio calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' one may account the long wavelength doing the following: (i) create phonon displacement patterns DP in conventional supercells,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' for ex- ample the same as for phonons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (ii) see that the displaced atoms also cause changes of the elastic constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (iii) call de- formed elastic tensor (DET),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' which conventionally will be the supercell deformed itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' what results in symmetry lowering to DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' By solving now the Cristoffel equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (32),60 with DET, one calculates the elastic wave frequencies, finding their frequency changes with respect of ideal supercell tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One may say that the Cristoffel equation rebuilds the ideal elastic wave to whole space from the crystal segment belonging to DET limited to studied supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, the group veloc- ities Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (36), can also change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These changes influence the relaxation time of conducting objects acting in thermal con- ductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The relation for high thermal conductivity (HTC) for- mulated in analogy with lattice thermal conductivity LTC, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), for the simulation with the deformed elastic tensor DET reads κHTC α,β = ¯h2 NrVpuckBT 2 1 Ni Ni ∑ i=1 � ∞ 0 dt Cuboid ∑ K,J (Ω(i)(K,J))2Vpuc(i)α gr (K,J)V(i)β gr (K,J) × (n(i)(K,J)+1))(n(i)(K,J)× 1 2 cos[2(Ω(i)(K,J)−Ω(0)(K,J))t] (37) where Ni is the number of DET - deformed elastic tensors, Vpuc volume of the primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The Cuboid, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 8 is a volume in the reciprocal zone, with center point at K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In LTC Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21), the summations ∑k,j run homoge- neously over the whole Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In HTC Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37), the summations ∑K,J should run over small wavevector volume around K = 0 as indicated in Cuboid, ∑Cuboid K,J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The central green box should always remain empty (it is direct space be- yond volume of the sample), so there no wavevectors should be positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the volume of the larger inner box (between green and red boxes), the wavevectors K for elastic waves should be placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Notice, that a lot of wavevectors can deter- mine crystal phonons, much less wavevectors are indexing the elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The central box excludes such long wavevec- tors K, which surpass the sample macroscopic size, or the mean distance between boundaries existing in the media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In general, the wavevectors should be places at random, unless the wavevectors amplitudes and positions express some super- structure being a new object of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the positions and amplitudes of wavevectors K could be derived from the object in the direct space and then transformed to the cuboid by three-dimensional Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Of course, the shape of the Cuboid may change to adapt to the studied object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Re- member that sets of wavevectors for phonons k and elastic waves K are needed for LTC Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (21) and HTC Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) ex- pressions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' And that the HTC must be described by wavevectors from the elastic wave region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (a,e) HTC averaged Green-Kubo functions calculated from Eqs(37) and (26), (f) Green-Kubo functions for HTC (elastic waves) calculated directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Each run used 20 DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon HTC To calculate the HTC of silicon the formulae for thermal conductivity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The 20 DP were prepared for each T = 7,20,40,70,200 and 600K, and then 20 DET’s were created in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' During heating the lattice constants and pressure stayed constant as observed for LTC of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elas- tic tensors, called also elastic modulus, were calculated on VASP,47,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' One must also add that the cpu calculation time of this process is long in comparison to cpu run for elastic tensor possessing some symmetry elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is a result of the fact that the DET’s do not have any symmetry, hence, it requires to calculate much more iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moving back to HTC, one should supply some informa- tion on the crystal microstructure, and include it to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The minimum information should indicate the possible range of the wavelengths of the elastic waves before they reach the boundaries, which hinder their travel and then determine the expected HTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the present stage of the current theory we may propose to select the proper wavelengths, or rather wavevectors of the elastic waves only, and check whether they lead to correct results observed in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The idea is that the shortest wavelengths λmin = 2π/Kmax of the elastic waves start from a distance just above the active wavelength of low frequency of acoustic phonons, and spreads to longest distances to ”boundaries” λmax = 2π/Kmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is expected that elastic waves characterized by wavevectors K from the interval Kmin < K < Kmin could propagate in the crystal with- out obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Generally, such precise information is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Moreover, the crystal microstructure may depend on distribu- tions of the boundaries within the crystal, microcracks, kind of defects and impurities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' and as such they ought to be a topic of separate study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, the volume of the perfect crystal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (red) Calculated HTC data using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) making use of elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (blue) Single point of LTC data at T = 600K, imported from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (black) Measured thermal conductivity from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' is represented by a cuboid, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 8 filled with the wavevectors K with an exception of the inner box, which should be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the direct space the empty box represents the outer part be- yond the crystal and the cuboid represents surface layers of the real crystal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' expected to suppress propagation of elastic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Si ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(W/ mK) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='elastic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='THERMAL CONDUCTIVIT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='GREEN-KUBO FUNCTION (W 1 [m K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='200 ' metadata={'source': 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the summation ∑Cuboid K,J of the formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For the present study of Si, the random wavevectors were se- lected out the volume of the cuboid confined by minimal value Kmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='00001Å−1 to maximal value of Kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='030Å−1 The inner box inside Kmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='00001 Å−1 was left empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The wavevectors Kmax and Kmin correspond to the lengths of elastic waves from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='021µm to 63µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Since in the cuboid the wavevectors distribution is homogeneous the amount of elastic waves taken in derivation of the HTC close to 63µm is less than in vicinity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='021µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The above mentioned summation in the cuboid was used to compute the Green-Kubo functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The selected function are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The vertical axes of plots are mainly de- termined by the group velocities and temperature occupation distributions and they stay almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' But the horizontal axes cover changeable time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Thus, time seems to de- cide about the magnitude of the relaxation times and thermal conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In particular after initial maximum at t = 0, the Green-Kubo function diminishes to negative minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such a decrease is caused by the difference of two cosines occur- ring in the relaxation time expression, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Namely, small differences between the elastic wave frequencies of deformed and perfect supercells lead to minimum at longer times, in contrary to opposite situation with large frequency difference and the minima occurring at shorter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The Green-Kubo functions accompanied the elastic waves can be seen on the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The minima of the (a-f) plots are drawn from 20 runs of DET’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' On (f) all 20 plot are seen and some smearing of the data are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) gives also the numerical value of ther- mal conductivity as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For silicon the low temperature HTC results are plotted on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='10 The LTC data reaches maximum 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='8W/mK, while the HTC is el- evated to 3700W/mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This high difference is mainly caused by the increase of the global relaxation times in HTC mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Longer relaxation times result in higher values of κα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The silicon single crystal, for the HTC measurements by Glassbrenner and Slack reported in25 and used for the low- temperature measurements, was grown from high-purity sili- con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The growth process was made with care in order to make the crystal to be oxygen and dislocation free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, any va- cancy clusters were less than one micron in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The sample boundary sizes were not reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is worth to mention that at low temperature the occupa- tion factor ( ω T )2(n( ω T ) + 1)n( ω T ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) reduces the in- tensity of the Green-Kubo function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' These properties also force to decrease HTC close to T = 0K to a very small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' With increasing temperature the mentioned thermal factors approach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Similar effect has been seen for MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Silicon Si and Magnesium Oxide MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Calculated av- eraged sums 1 3(κ1,1 +κ2,2 +κ3,3) of the high thermal conductivities Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37), and the global relaxation times, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Si T(K) 7 20 40 70 200 600 κ(W/mK) 628 2107 3700 2197 260 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2 τ(ps) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='11 MgO T(K) 5 25 40 100 300 κ(W/mK) 514 3225 2932 218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='14 τ(ps) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='94 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='81 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium Oxide HTC To calculate the HTC of magnesium oxide, MgO atomic displacements DP were prepared for each T = 5,25,40,100 and 300K and next 20 DET’s tensors were created for each T using VASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' During heating the lattice constants and pressure behaved as observed for LTC of MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The below mentioned summation within the cuboid was used to compute the Green-Kubo functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The selected functions are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The vertical plot axes are mainly determined by the group velocities and temperature occupation distributions and they stay almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' But the horizontal axes cover changeable time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Thus, time seems to decide about the magnitude of the relaxation times and thermal conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To calculate HTC of MgO one should select data for the cuboid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The following wavevectors have been proposed: max- imal values of Kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0030 Å−1 and minimal value of Kmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='00001 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The selected cuboid wavevectors for MgO correspond to boundaries of the elastic waves being in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='21µm to about 63µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This information was used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (37) to compute the time depen- dent Green-Kubo HTC functions, and later plotted on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The HTC data reaches maximum 3700W/mK, The vari- ation of the HTC in MgO are presented on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The T = 5,25,40,100K fit to measured data, but the contribution of HTC at T = 300K practically vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' At T = 300K only LTC contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The two approaches provide thermal conductivity LTC and HTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Therefore, it rises a question what happens in the tem- perature interval between the LTC and HTC regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The present results give for Si at T = 600K: HTC: 21,2W/mK, LTC: 84W/mK Exp:84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='5W/mK, and for MgO at T = 300K: HTC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='14W/mK, LTC: 110W/mK Exp: 75W/mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This means that LTC mechanism is used at higher T and then at lower T it became replaced by HTC processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Probably, this effect occurs in special materials only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Microstructure in HTC The current approach relates the HTC and microstructure in the region of low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The conventional process to carry on the HTC calculations would be to look into cuboid for the wavevectors data to get an agreement between calculated 15 MgO elastic T=5K \uf06b=514W/mK (a) T=40K T=300K T=25K T=100K T=300K \uf06b=218W/mK \uf06b=2932W/mK \uf06b=3225W/mK \uf06b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='16W/mK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='GREEN-KUBO FUNCTION (W / [M K PS]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium oxide, MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (a,e) HTC averaged Green-Kubo functions calculated from Eqs(37) and (26), (f) Green-Kubo functions for HTC (elastic waves) calculated directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Each run used 20 DET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Magnesium oxide, MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (red) Contributions of only elas- tic waves to high thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (green) Single point of LTC data at T = 300K, imported from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (black) Experimental points from22 and measured κHTC α,β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It would mean to find right values of the wavevectors inserted to cuboid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' However, the reverse process would be more valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' First the microstructure features are foreseen and used to modify the cuboid, and next to collate the calculated data with the behaviour of the measured HTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In this case the method could have some predictive power, which might help to design the required properties of the material, say microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' As a test we have inserted to the cuboid of Si crystal be- ing at T = 40K, the wavevectors of Kmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='00001 Å−1 and Kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='00015Å−1, corresponding to Si boundary distances from 63 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='2 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' For these wavevectors HTC tremendously increases the global relaxation time to 1700 ps and thermal conductivity to κHTC α,β = 320000W/mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Other obstacles might diminish/change this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Presently the cuboid volume is filled with wavevectors of the same amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' However, having a concept of the mi- crostructure of considered crystal one might convert this infor- mation to the amplitudes of wavevectors placed in the cuboid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such a project is still waiting for realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' There are many studies, which need to combine the mi- crostructure of the sample with its thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Here, follows some example (1) The thermal energy transport in actinide oxide nuclear fuel materials57, thorium dioxide and uranium dioxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The first has a characteristic maximum of thermal conductivity below 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The second has a reduced thermal conductivity, in spite of similar crystal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It happens due to the presence of elastic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (2) The dislocation impact on thermal conductivity58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Disloca- tions induce the stress field, which might lead to anisotropy of thermal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Such a contribution can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Another goal would be to analyze the collection of disloca- tions on thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' (3) Thermal properties of the superelastic, which consist of many crystal variant of shape memory alloys (as NiTi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The complex microstructure exists due to well-known compatible equations for pair of crystals variants, which require to identify the interfaces59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='psJ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='k1500 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='100 110 120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='TIME (ps)10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='000 5 MgO elastic 1000 THERMAL CONDUCTVITY 100 10 1 3 10 30 100 300 1000 TEMPERATURE (K)16 The MD simulations of simple crystal models have shown that realistic microstructures of YBa2Cu3O7 superconductor61 and LaNbO4 ferroelastic62 could be obtained starting from a simple crystal model at relatively high temperature, and next quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The simulated miscrostructures and those obtained from TEM observations are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It is also worthwhile to mention the effort to increase effectiveness of thermoelectric CaCd2Sb263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' It has been proposed to replace Cd by Mg, as point defect, to induce significantly phonon scattering, but maintaining the carrier concentration, lower es- sentially their LTC and increase figure of merit ZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSIONS The mechanisms of thermal conductivities discussed in this article is based on the anharmonic phonons formalism han- dled within the non-perturbative approach for crystals,30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Due to it, the formulated theories of LTC and HTC have been retrieved starting from different and not conventional anhar- monic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the LTC case phonons play a role of the heat transport media realized by anharmonic phonons of the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The anharmonic vibrations of atoms determine the crystals temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Next, one prepares several displace- ment pattern DP of atoms in the crystal supercells, with ampli- tudes displaced corresponding to studied temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Then, the forces induced by the displaced atoms are calculated with the ab initio software, which permits to solve the set of lat- tice dynamics equations, and find information about the har- monic and anharmonic interatomic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This technique has been successfully used to create positions, shifts, widths and shapes of anharmonic peaks and determine the analytical expression for the mode relaxation times for all anharmonic modes without performing expansion of the interaction po- tential over anharmonic terms and without using the Boltzman equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Specially, the relaxation times could be calculated analytically, and this process needs only to know the differ- ences of anharmonic and harmonic frequencies for each seg- ment of the anharmonic phonon mode belonging to the same (k, j) phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Some crystals require to take into account also the elastic waves, which need to be considered within different mathe- matical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic waves, travel in the crystal and form a strain variation of predefined units of crystal, usually supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Atomic displacement patterns, similar to those of phonons, create some strains, deform supercells, and hence create the elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Finally, the elastic waves can be found by solving the Cristoffel equation, being entirely de- fined by the elastic constant tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' We have shown that the crystal thermal conductivity is de- termined by the Green-Kubo relationship being the correlation function of the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The high thermal conductivity can be calculated from products of elastic wave frequencies, elastic wave group velocities, and phonon relaxation times specific for elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' Second essential difference between lattice and high ther- mal conductivity is related with the wavevectors summa- tion within the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In the phonon part all wavevectors need to be used in the sum over the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' The elastic waves are characterized by very long wave- lengths, so only the short wavevectors around k = 0 and lower then 1 − 4THz participate in the thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This effect is applied to fix the longest wavevectors as being able to reach the sample sizes, or other obstacles which limit the transport of the elastic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' This criterion leads to statement that the calculated HTC may agree with the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' To facilitate the procedure to limit the used wavevectors for the elastic wavevectors, the cuboid of 3d figure was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' In future the cuboid might also allow to study thermal con- ductivity of crystals with defects, microstructure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author would like to acknowledge Dr Erich Wimmer and Dr Walter Wolf from Materials Design Inc for suggestions and fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' AUTHOR DECLARATIONS Conflict of Interests The author has no conflict to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available within the article.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Broido, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Stewart, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Deinzer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' B, 80, 125203 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 13D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Kim, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Hellman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Shulumba, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Saunders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Herriman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Niedziela, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Abernathy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Li, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Fultz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' B 84, 180301 (R) (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 15O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Hellman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Abrikosov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 7, 034030-1 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 18BinWei, Qiyang Sun, Chen Li, and Jiawang Hong, Science China, Physics, Mechanics & Astronomy, 64, 117001 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 19I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Errea, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Cherubini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Calandra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Errea and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Mauri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQf3f6y/content/2301.01831v1.pdf'} +page_content=' 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distillation +with undetected light +Jorge Fuenzalida,1,2,†,⋆ Marta Gilaberte Basset,1,3,⋆ Sebastian +T¨opfer,1,2,⋆ Juan P. Torres,4,‡ and Markus Gr¨afe1,2,3,§ +1Fraunhofer Institute for Applied Optics and Precision +Engineering IOF, Albert-Einstein-Str. 7, 07745 Jena, Germany. +2Technical University of Darmstadt, Department of Physics, Institute +of Applied Physics, Schloßgartenstraße 7, 64289 Darmstadt, Germany. +3Friedrich-Schiller-University Jena, Abbe Center of Photonics, +Albert-Einstein-Str. +6, 07745 Jena, Germany. +4ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, +08860 Castelldefels, Spain, and Dept. Signal Theory and Communications, +Universitat Politecnica de Catalunya, 08034 Barcelona, Spain. +† jorge.fuenzalida@tu-darmstadt.de +‡ juanp.torres@icfo.eu +§ markus.graefe@tu-darmstadt.de +⋆These authors contributed equally. +(Dated: January 9, 2023) +Abstract +Imaging based on the induced coherence effect makes use of photon pairs to obtain informa- +tion of an object without detecting the light that probes it. While one photon illuminates +the object, only its partner is detected, so no measurement of coincidence events are needed. +The sought-after object’s information is revealed observing a certain interference pattern +on the detected photon. Here we demonstrate experimentally that this imaging technique +can be made resilient to noise. We introduce an imaging distillation approach based on the +interferometric modulation of the signal of interest. We show that our scheme can generate +arXiv:2301.02529v1 [quant-ph] 6 Jan 2023 + +2 +a high-quality image of an object even against noise levels up to 250 times the actual signal +of interest. We also include a detailed theoretical explanation of our findings. +Introduction +Quantum imaging [1] is an emerging and promising field in quantum technologies with +certified advantages over classical protocols. This has been demonstrated in different sce- +narios: in schemes that work in the low-photon flux regime [2, 3], in schemes that make +use of undetected probing photons [4, 5], for super-resolution imaging [6–9], sub-shot-noise +imaging [10–12], or enhanced two-photon absorption processes [13]. Moreover, protocols +in quantum imaging with no classical counterpart has been developed based on quantum +interference [14], and entanglement [15, 16]. In recent years, it has also been proven that +quantum imaging protocols can be resilient to noise [17–19]. +Distillation (also known as purification) is the process wherein the decoherence introduced +in a quantum system by the environment can be removed [20]. In quantum imaging, the +effect of the environment can be modeled through classical illumination superimposed over +a quantum image on the camera. Since most cameras only detect intensity, quantum and +classical images seems to be indistinguishable. However, quantum correlations of photon +pairs can be employed to differentiate the quantum image from a classical one. Quantum +imaging distillation has been implemented with one and several photon pair degrees of +freedom [21–25]. +To the best of our knowledge, every implementation to date has used +the joint detection of photon pairs. In this work, we introduce and experimentally verify a +quantum imaging distillation technique that employs the detection of single photons only. +Quantum imaging with undetected light (QIUL) [4, 26–28] is a two-photon wide-field inter- +ferometric imaging technique. In QIUL, one photon illuminates an object and its partner +photon is detected on the camera. The photon that illuminates the object remains un- +detected. Using an interferometric configuration, the object information is transferred to +the detected photon interference pattern. Due to its unique detection advantage, QIUL +has been employed to probe samples with unconventional wavelengths while visible light is +detected [29–32]. Up to date, the effects of noise in QIUL have not yet been explored. + +3 +Here we introduce a source of noise in a QIUL scheme and study the resilience of the quantum +imaging technique. +The properties of the noise, its intensity and variance, are changed +during this study. We perform a quantum imaging distillation technique based on quantum +phase-shifting digital holography [33]. Our distillation technique uses phase modulation of +the undetected photon to vary the interference pattern detected on the camera. We notice +that if the intensity difference of the interference patterns is bigger than the variance of the +noise, the quantum image can be distilled. We also observe that the noise variance affects +the distilled quantum images linearly in their phase estimation. Our technique shows a good +performance, even for noise intensities above 250 times the quantum signal intensity. +Distillation principle +Quantum imaging distillation is a process whereby a quantum image is cleaned from noise. +To explain our distillation technique let us consider two images: a quantum image, which is +acquired by illuminating the sample with with non-classical light, and a noise image, which is +an image detected at the camera and generated with classical illumination. These two images +are shown in Figs. 1(a) and 1(b), respectively. A noise image is an unwanted signal that is +superimposed over a quantum image on the camera. This image superposition is shown in +Fig. 1(c). To distill an image, different photon pair degrees of freedom can be employed, +e.g., frequency, time, or spatial correlations. We use the amplitude modulation of quantum +holography with undetected light (QHUL) [33], which is an interferometric quantum imaging +technique [4]. In QHUL, the object information is carried into a single-photon interference +pattern. When the noise reaches the camera, if the intensity difference of QHUL is bigger +than the intensity variance of the noise, the quantum image can be distilled. The resulting +distilled image is shown in Fig. 1(d). +Theory +Spontaneous parametric down-conversion (SPDC) [34, 35] is a well-known nonlinear process +that generates photon pairs (signal and idler) mediated by the interaction of an intense +pump beam with the atoms of a nonlinear crystal [36]. Our imaging scheme makes use on +a SU(1,1) interferometer, wherein a pair of signal-idler photons can be generated in one of + +4 +0 +1 +2 +3 +4 +5 +6 +y in mm +quantum image +0 +50 +100 +150 +200 +250 +Intensity in arb. unit +noise image +0 +25 +50 +75 +100 +125 +Intensity in arb. unit +0 +1 +2 +3 +4 +5 +6 +x in mm +0 +1 +2 +3 +4 +5 +6 +y in mm +superposition +0 +100 +200 +300 +Intensity in arb. unit +0 +1 +2 +3 +4 +5 +6 +x in mm +distilled image +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Visibility +a +b +c +d +Figure 1: Principle of quantum imaging distillation. We employ a quantum imaging distilla- +tion protocol to remove a noisy image from a quantum image. (a) is the quantum image we aim to +distill. (b) is the noise image that is superimposed on the quantum image. (c) is the superposition +of noise and quantum images.(d) is the resulting distilled image from noise. +two propagation modes, forward and backward. The probability to generate paired photons +in both modes (forward or backward) simultaneously is negligible [37, 38]. In the forward +propagation mode, the pump, signal, and idler beams are spatially separated, and later, +back-reflected into the nonlinear crystal with the help of 4f systems. Before back reflection +the idler photon is reflected from an object, with complex reflectivity R = |R| exp(iϕR), +placed in front of its end-mirror. In the backward propagation mode, the idler photon do +not interact with the sample. The signal photons are collected by a camera, and the idler +photon remains undetected. +The mean value of signal photons detected in time TD at one pixel of the camera of area SD + +5 +(see supplementary material for details) is +⟨NS⟩δ = 2 S0 [1 + |R| γ cos(δ + ϕR)]. +(1) +δ is an interferometric spatially invariant phase and S0 is the number of signal photons +generated in single-pass SPDC (in a time window TD and area SD). The parameter γ is +related to the effective bandwidth of the signal-idler photon pairs, determined essentially by +the bandwidth of the filters located in front of the camera [33]. From Eq. (1), we see that +⟨NS⟩ changes when δ is varied. In particular, using δ = 0, π/2, π, and 3π/2, the object’s +information can be retrieved by means of QHUL as follows: +|R| = 2 × +�� +⟨NS⟩3π/2 − ⟨NS⟩π/2 +�2 ++ +� +⟨NS⟩0 − ⟨NS⟩π +�2�1/2 +⟨NS⟩0 + ⟨NS⟩π/2 + ⟨NS⟩π + ⟨NS⟩3π/2 +, +(2) +ϕR = arctan +� +⟨NS⟩3π/2 − ⟨NS⟩π/2 +⟨NS⟩0 − ⟨NS⟩π +� +. +(3) +Equations (2) and (3) are not unique representations of |R| and ϕR and, in general, these +quantities can be extracted using different number of phases [33]. We emphasize that in this +technique paired photon coincidences are not needed and only signal photons are measured. +For the important case of phase estimation, we evaluate the sensitivity of QHUL obtaining +the variance of ϕR given by Eq. (3). We first calculate the variance of the signal-photon +flux ⟨(∆NS)2⟩ = ⟨N 2 +S⟩ − ⟨NS⟩2. Since the coherence time TC of signal-idler photon pairs +(TC ∼ 1/B, B is the effective bandwidth of SPDC) is much smaller than the detection +time, we can approximate the variance of the signal-photon flux to as (see Supplementary +Material), +� +(∆NS)2� += ⟨NS⟩ , +(4) +which is equivalent to considering Poissonian statistics. This result is well-known to be +valid when considering a multimode signal where each mode has the same non-Poissonian +statistics. +The result in Eq. (4) corresponds to the minimum signal variance achievable in QHUL. +However, this variance can rapidly increase by electronic noise, e.g., camera signal-to-noise +ratio, or external sources of noise such as temperature fluctuations, airflow, and external + +6 +illumination, among others. In this work, we study the effect of an external classical illumi- +nation impinging on the camera, overlapping the quantum image of interest. Since QHUL +detects only the single-photon stream of signal photons, the decoherence produced by an +external source of light seems extremely harmful and, therefore, the image distillation highly +improbable. However, we will demonstrate experimentally that the quantum imaging dis- +tillation is possible even in scenarios with considerable high levels of noise in comparison to +the quantum signal intensity. +Quantum holography with undetected light employs interferometric modulation of the signal +photon to retrieve the object information, as shown in Eqs. (2) and (3). We show that this +modulation can also be used for distillation purposes, which is depicted in Fig. 2. +For +each value of the interference phase δ, the signal photon has a well-defined intensity and +variance, given by Eqs. (1) and (4), respectively. In Fig. 2(a), signal intensity (variance) +is represented with pink bar charts (error bars). In contrast, the intensity of a stochastic +noise ⟨NT⟩ fluctuates randomly with a variance ⟨(∆NT)2⟩. For the sake of simplicity, we +consider (but are not restricted to) to the case where the noise has the same total mean +intensity and variance than the signal photon. In Fig. 2(b), the noise intensity (variance) +is represented with blue bar charts (error bars). As a result of adding these two intensities, +see Fig. 2(c), the background increases up to the noise intensity, while the signal intensity +varies on top of it. The noise variance contributes to the signal intensity variance, i.e., the +shot noise increases. In this way, one can infer that if the difference of the signal intensity is +higher than the noise variance, the quantum image can be distilled. Additionally, the shot +noise of QHUL always increases if the noise intensity and/or its variance increases. More +details are given in the Supplementary Material. +Let us analyze the performance of our distillation technique by considering a generalization +of Eq. (2). For QHUL with M phase steps, we have that +ϕR = − tan−1 +�� +j +⟨NS⟩j sin δj/ +� +j +⟨NS⟩j cos δj +� +, +(5) +with δj = j 2π/M, and j = 0, 1, ..., M − 1. The phase variance, including noise, it reads +� +(∆ϕR)2� += +� +j +� +∂ϕR +∂ ⟨NS⟩j +�2 �� +(∆NS)2� +j + +� +(∆NT)2�� +, +(6) + +7 +0 +0 +0.2 +0.4 +0.6 +0.8 +Signal photon intensity +Phase +3π/2 +π/2 +π +0 +0 +0.2 +0.4 +0.6 +0.8 +Noise intensity +Phase +3π/2 +π/2 +π +Intensity superposition +0 +0 +0.2 +0.4 +0.6 +Phase +3π/2 +π/2 +π +0.8 +} +} +} +a +b +c +Figure 2: Intensity detected in one camera pixel. For visualization purposes, we have con- +sidered the same intensities (chart bars) and variances (error bars) for the signal photon and the +noise. In (a) is presented the signal photon intensity (in pink) for different values of δ. This inten- +sity fluctuation allows us to compute the object information using QHUL [33]. In (b) is presented +the noise intensity (in blue) for the same phases δ. In contrast to the signal intensity, the noise +intensity is not affected by the value of δ. In (c) are added the noise and signal intensities. Since +noise intensity does not change, its contribution just sets a higher background. On top of it, signal +photon intensity still changes, and the total variance is its previous variance plus the noise variance. +Thus, an external source of noise increases the shot noise of QHUL. +where +� +∂ϕR +∂ ⟨NS⟩j +�2 += +1 +M 2 γ2 |R|2 S2 +0 +sin2(ϕR + δj). +(7) +Replacing Eqs. (4) and (7) into Eq. (6), and considering n measurements, we obtain that + +8 +the variance of phase estimation is +� +(∆ϕR)2� += +1 +n S0 +1 +M V 2 +� +1 + ⟨(∆NT)2⟩ +2 S0 +� +. +(8) +S0 is the number of idler photons that illuminate the object whose phase we want to estimate, +⟨(∆NT)2⟩ is the variance of the number of background photons that illuminate the detector, +and V = |R|γ is the visibility of the signal-photon flux interference pattern as a function of +the phase δ [43]. Equation (8) contains two contributions to the variance of the phase: the +first term comes from the quantum illumination, and the second term comes from the noise +illumination. While the former can be shown to be well described by Poissonian statistics +(see Supplementary Material), the latter depends on the specific characteristics of the noise +illumination. +Results +Quantum imaging +A sketch of our experimental implementation is depicted in Fig. 3. For our quantum im- +age, we used a SU(1,1) nonlinear interferometer [39] in a Michelson configuration, where its +input/output is a nonlinear medium. Our crystal is a periodically poled potassium titanyl +phosphate (ppKTP) of 2 × 2 × 1 mm3 (length × width × height), which is pumped bidirec- +tionally (a and f directions) by a CW laser at 405 nm and with avarage power of 90 mW. +Due to its strong χ(2)–nonlinearity, a photon pair (signal and idler photons), is generated +through SPDC into the paths a or f, but never simultaneously. Signal (idler) photons have +a central wavelength of λS = 910 nm (λI = 730 nm). +In the forward propagation direction a, signal, idler, and pump beams are spatially sepa- +rated with dichroic mirrors DM2 and DM3 into the paths b, c, and d, and reflected back +into the crystal with mirrors M1, M2, and M3. In front of mirror M2, an object with a +complex amplitude R = |R| exp(iϕR) is placed. In b, c, and d, lenses of focal length f= 150 +mm transform transverse position (source plane) into transverse momentum (mirrors plane). +Therefore, in path c, a wave vector kI representing a plane wave of the idler photon is focused +to a point on the object [26, 27]. The interaction of idlers being absorbed or reflected by the +object can be modeled with the help of a beam splitter [28]. In the backward propagation f, + +9 +L1 +L2 +L3 +DM1 +DM2 +DM3 +PPKTP +L4 +LASER +OBJECT +PIEZO +CAMERA +M1 +M2 +M3 +L5 +LASER +LIGHT +DIFFUSER +NOISY +OBJECT +10:90 BS +L6 +L7 +LINEAR +POLARIZER +Figure 3: Setup. The signal and idler beams (in paths b and c, respectively) are generated by +the pump beam in path a interacting with the nonlinear crystal (ppKTP) in the forward direction, +while paths e and f represent propagations of the down converted beams generated after the pump +beam is reflected back into the crystal by mirror M3 in path d. An object in path c is illuminated +with the idler beam in the Fourier plane of the crystal. To create the noise, a laser diode of the +same wavelength as the signal photon (910 nm) is used. The signal beam in path e is merged with +the noise in path g before reaching the camera with a 10:90 beam splitter. On the detector plane +we obtain the quantum image with lenses L2, L4 and L5, and the noise image with lenses L6 and +L7. Different type of noise are created with a linear polarizer and a light diffuser in path g. The +linear polarizer controls the pump power of the diode laser. The diffuser that consists on a rotating +ground glass plate produces a speckle pattern of the noise source. The speed of the rotation is +controlled through the glass plate motor interface. +signals are collected by a camera and idlers remain undetected. We ensured that by placing +a 800 nm long-pass filter and a 910 ± 1.5 nm interference filter in front of the camera. Our +camera is the Prime BSI Scientific CMOS from Teledyne Photometrics with a pixel size of +6.5 µm. Signal photon’s transverse momentum is obtained with the lens L4 of focal length +f= 100 mm performing a Fourier transform of the source plane. This plane is later imaged + +10 +on camera with the lens L5 of focal length f= 150 mm. Thus, a wave vector kS of the signal +photon is focused to a point on the camera. If a and f propagation are perfectly aligned, the +photon pair emission (which-source) information is erased. Consequently, on the camera, +an interference pattern of signal photons is observed [40]. Moreover, the object information +obtained herein by the idler photon, is transferred to the signal photon interference pat- +tern [4, 5], see Eq. (1). The interferometric phase δ is changed with a piezo placed below +mirror M2. The object information is retrieved by employing QHUL of 12 steps [33] with +an acquisition time of t=1 s per image. +Noise source +A CW diode laser of λN = 910 nm and variable pump power is employed to introduce noise in +the system. The laser illuminates an object, which is imaged on the camera with a 4f system +using the lenses L6 and L7 of focal lengths f= 150 mm and f= 125 mm, respectively. This +classical image is superimposed on top of the quantum image on the camera using a 10:90 +beam splitter, see Fig. 3. Properties of classical illumination, intensity and variance, are +changed in order to evaluate the effects of noise in QHUL and our distillation performance. +Experimental details about the noise properties can be found in the Supplemental Material. +Distillation performance to different noise intensities +In the first experiment, while having superimposed classical and quantum images, QHUL +is performed to distill the quantum image under different intensities of noise. +We first +characterized the signal flux rate emission measuring its mean intensity of an illuminated +area on the camera. Signal intensity does not changed during experiments. In a similar way +but independently measured, different noise intensities are characterized, which are obtained +by changing the angle of a linear polarizer (LP) in front of the laser in path g. The experiment +starts superimposing the quantum and classical images on the camera. Experimental results +are shown in Fig. 4. Its first row shows the superposition of classical and quantum images; +noise intensity increases from left to right with the following ratios (r = signal intensity:noise +intensity), r ≈ 1 : 8, r ≈ 1 : 37, r ≈ 1 : 50, r ≈ 1 : 104, and r ≈ 1 : 252. Second row +in Fig. 4 shows distilled images by QHUL of the corresponding upper superposed images. +Imaging distillation through QHUL is successfully achieved in every case, even with a noise +intensity 250 times higher than the signal intensity. However, we notice that while noise + +11 +r + 1:8 +r + 1:37 +r + 1:50 +r + 1:104 +r + 1:252 +0 +1 +2 +3 +x in mm +0 +1 +2 +Phase +R in rad +0 +1 +2 +3 +x in mm +0 +1 +2 +3 +x in mm +0 +1 +2 +3 +x in mm +0 +1 +2 +3 +x in mm +0.0 +0.5 +1.0 +Norm. Intensity +0 +1 +2 +Phase +R in rad +Figure 4: Resilience to different noise intensities. In the upper row are shown superpositions +of quantum (IOF letters) and classical (square shape) images. The ratio between their intensities +is stated on top of each image. In the middle row are presented the experimental results for our +distillation technique through QHUL. In the last row are presented a transverse cut of the distilled +images. We observe that while the noise intensity increases, the phase estimation diminishes. +intensity increases, sharpness of distilled images decreases. One can observe this in detail in +the third row that shows a transverse cut of the distilled images (represented by a dotted +red line). It is clear from our experimental results that phase accuracy diminishes as the +noise intensity increases, which also corresponds to prediction given in Eq. (8). +Induced variance by noise +In a second experiment, we quantified the effects of noise variances on the phase accuracy +of distilled images. The same configurations of noise intensities are employed. Additionally, +a light diffuser mounted on a rotational motor with four angular frequencies of 0, 1, 2, +and 3 Hz changed the noise variance. The noise variance is characterized considering the +intensity variation of one pixel over 12 consecutive images. Experimental results are shown +in Fig. 5. We plot the experimental measured values for the phase variance of distilled +images against the noise variance for different angular frequencies (data point with error +bars; purple circle = 0 Hz; rose star = 1 Hz; green triangle = 2 Hz; yellow square = + +IOFIOFIOFIOFIOF12 +3 Hz). +We also provide a theoretical prediction for comparison (solid black line). +The +results show that an increment of the noise variance increases proportional with the phase +variance. Also it can be observed that in every case a linear dependence appears between +these two variances. However, the noise variance introduces a slightly higher phase variance +than expected. A reasonable explanation for this is that additional sources of noise were +involved during the measurement process, such as airflow or temperature fluctuations. The +experimental behaviour of variances is in good agreement to theoretical predictions presented +above in Eq. (8). +Discussion and conclusion +Our work explores for the first time the effects of noise in Quantum imaging with undetected +light. We have also introduced a technique to distill the quantum image from that noise. Our +quantum imaging distillation technique is based on quantum holography with undetected +light [33]. This technique uses a photon pair, signal and idler, where idler illuminates the +object and signal is detected on the camera. The idler photon remains undetected and its +phase modulation is employed in the imaging distillation procedure. +To prove our technique, we superimposed partially or completely a classical source of noise +on top of our quantum image on the camera. Our technique worked in every occasion, even +for noise intensities 250 times higher than our signal intensity. However, the noise variance +does affect the phase accuracy of our distillation technique. A higher noise variance produces +a higher phase inaccuracy, where in general, these two quantities are linearly dependent. +Although, in our experiment we employed a classical source of noise, this distillation tech- +nique should also work for a quantum source of noise. Furthermore, since the underlying +principle of this distillation technique is the phase modulation, our technique should be +applicable to QIUL based on position correlations [41, 42]. +Our experiment is a step forward for quantum imaging in open systems and could be useful +to the understand the limitations of a (quantum) LIDAR with undetected light. + +13 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0x105 +Noise Variance ( NT)2 in photons +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Phase Variance ( +R)2 in rad +theory +0 Hz +1 Hz +2 Hz +3 Hz +Figure 5: Distillation phase variance affected by noise variance. The noise variance was +measured with S0 = 134 signal photons generated in a single pass through the crystal. +The +detection window was set to be TD = 1 s and the detection area SD was ≈ 32.5 × 32.5 µm2. A +light diffuser with four different rotation speeds is employed to change the properties of the noise +illumination. In all configurations we observed that the phase sensitivity is linearly dependent +with the noise variance. The noise variance in dependence to the signal intensity shows a linear +dependence as well (see Supplemental Material). +A higher noise variance increases the phase +inaccuracy in QHUL. +Acknowledgments +The authors thank Fabian Steinlechner for fruitful discussions. This work was supported +as a Fraunhofer LIGHTHOUSE PROJECT (QUILT). Furthermore, funding support is ac- +knowledged from the German Federal Ministry of Education and Research (BMBF) within +the funding program Photonics Research Germany with contract number 13N15088. In ad- +dition, funding is acknowledged from the Th¨uringer Aufbaubank via the projects Multi-Use + +14 +(2020FGI0023), SPEQTRA (2021FE9016), and Quantum Hub Thuringia (2021FGI0041). +J.P.T. acknowledges the funding by the R&D project CEX2019-000910-S, funded by the +Ministry of Science and innovation (MCIN/ AEI/10.13039/501100011033/), from Fundaci´o +Cellex, Fundaci´o Mir-Puig, and from Generalitat de Catalunya through the CERCA pro- +gram. J.P.T. also acknowledges support from the project 20FUN02 “POLight”, that has +received funding from the EMPIR programme co-financed by the Participating States and +from the European Union’s Horizon 2020 research and innovation programme, and finan- +cial support from project QUISPAMOL (PID2020-112670GB-I00) funded by MCIN/AEI +/10.13039/501100011033. +References +[1] M. Gilaberte Basset, F. Setzpfandt, F. Steinlechner, E. Beckert, T. Pertsch, and M. Gr¨afe, +Laser Photonics Rev. 13, 1900097 (2019). +[2] R. S. Aspden, D. S. Tasca, R. W. 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Ramelow, Optics Express 30, 5916 (2022). +[43] The visibility is defined by V = (⟨N⟩max−⟨N⟩min)/(⟨N⟩max+⟨N⟩min), where ⟨N⟩max (⟨N⟩min) +is the maximum (minimum) photon count rate. + +Supplementary Material of “Experimental quantum imaging +distillation with undetected light” +Jorge Fuenzalida,1,2,†,⋆, Marta Gilaberte Basset,1,3,⋆ Sebastian +T¨opfer,1,2,⋆ Juan P. Torres,4,‡ and Markus Gr¨afe1,2,3,§ +1Fraunhofer Institute for Applied Optics and Precision +Engineering IOF, Albert-Einstein-Str. 7, 07745 Jena, Germany. +2Technical University of Darmstadt, Department of Physics, Institute +of Applied Physics, Schloßgartenstraße 7, 64289 Darmstadt, Germany. +3Friedrich-Schiller-University Jena, Abbe Center of Photonics, +Albert-Einstein-Str. +6, 07745 Jena, Germany. +4ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, +08860 Castelldefels, Spain, and Dept. Signal Theory and Communications, +Universitat Politecnica de Catalunya, 08034 Barcelona, Spain. +† jorge.fuenzalida@tu-darmstadt.de +‡ juanp.torres@icfo.eu +§ markus.graefe@tu-darmstadt.de +⋆These authors contributed equally. +(Dated: January 9, 2023) +A. The quantum state of signal photons. +In the undepleted pump beam approximation, if the bandwidth of parametric down- +conversion (in the frequency and transverse wavenumber domains) is much larger than the +corresponding bandwidth of the pump beam, the relationship between the quantum opera- +tors aS(Ω, q) and aI(Ω, q) (output face of the nonlinear crystal) and the operators bS(Ω, q) +and bI(Ω, q) (input face of the nonlinear crystal) can be written in the low parametric gain +arXiv:2301.02529v1 [quant-ph] 6 Jan 2023 + +2 +approximation as [1] +aS(Ω, q) =bS(Ω, q) exp +� +ikS(Ω, q)L +� ++ FS(Ω, q) +� +dωP dqP EP(ΩP, qP) b† +I(ΩP − Ω, qP − q), +aI(Ω, q) =bI(Ω, q) exp +� +ikI(Ω, q)L +� ++ FI(Ω, q) +� +dωP dqP EP(ΩP, qP) b† +S(ΩP − Ω, qP − q), +(1) +where +FS,I(Ω, q) = −i(βL) sinc +�∆S,I(Ω, q)L +2 +� +exp +� +i +� +k0 +P + kS,I(Ω, q) − kI,S(−Ω, −q) +� L +2 +� +. (2) +The nonlinear coefficient β is +β = +� ¯h ωP ωS ωI [χ(2)]2 +64 π3 ϵ0 c3 nP nS nI +�1/2 +, +(3) +nS,I,P are refractive indexes, χ(2) is the second-order nonlinear coefficient of the crystal, L +is the crystal length, and ωP,S,I are central frequencies. The phase mismatch functions are +∆S,I = k0 +P − kS,I(Ω, q) − kI,S(−Ω, −q), kS,I are the wavenumbers of signal and idler waves, +and k0 +P is the wavenumber of the pump beam. The total number of photons NP carried by +the pump beam is +NP = +� +dΩ dq +��EP(Ω, q) +��2. +(4) +If we consider that signal and idler photons S1 and I1 (forward propagation modes) are re- +injected back into the nonlinear crystal with the help of 4f systems, the operator associated +to signal photon S2 (backward propagation mode) is +aS2(Ω, q) = FS(Ω, q) +� +exp +� +ikS(Ω, q)L + iϕS(Ω) + iϕP1 +� ++R∗ exp +� +− ikI(−Ω, −q)L + iϕP2 − iϕI(−Ω) +�� � +dωP dqP EP(ΩP, qP) b† +I(ΩP − Ω, qP − q) ++FS(Ω, q) exp +� +iϕP2 +� � +dΩP dqP EP(ΩP, qP) f †(ΩP − Ω, qP − q). +(5) +Notice that in Eq. (5), for the sake of simplicity, we have omitted the terms that depend +on the operator bS since they will yield a zero contribution to the flux density of signal +S2 photons. We have made use of the approximation kI(ΩP − Ω, qP − q) ∼ kI(−Ω, −q), +since the bandwidth of the pump beam is assumed to be very small when compared with + +3 +the bandwidth of parametric down-conversion. +We also consider that the phases of the +pump beams, ϕP1 and ϕP2, that illuminate the nonlinear crystal might be different. The +phases ϕS(Ω) and ϕI(Ω) are phases acquired by the signal S1 and idler I1 photons that +traverse 4f systems before being re-injected back into the nonlinear crystal. The reflection +coefficient of the object located in the idler I1 path is R = |R| exp(iϕR). The operator f +takes into account [2] the presence of the object with reflectivity R. These operators fulfill +the relationships +� +f(Ω, q), f †(Ω′, q′) +� += +� +1 − |R|2� +δ(Ω − Ω′) δ(q − q′). +B. Flux density of signal photons detected in a 2f system +We detect the flux density of signal photons S2 with the help of a 2f system with focal length +f. The input-output relationship between operators is similar to the classical relationship [3] +aS2(t, x2) = +1 +λS f +� +dx1 aS2(t, x1) exp +� +−i 2π +λS f x1 · x2 +� +, +(6) +where x1 and x2 are the corresponding transverse coordinates in the input and output planes +of the 2f system. For the sake of simplicity, we ignore any global phase. If we introduce the +Fourier transform of the operator aS2(t, x1) as aS2(t, x) = (2π)−3/2 � +dΩ dq aS2(Ω, q) exp +� +iq· +x − iΩt +� +, we can write Eq. (6) as +aS2(t, x2) = (2π)1/2 +λS f +� +dΩ aS2 +� +Ω, 2π +λS f x2 +� +exp (−i Ω t) , +(7) +where we have made use of the identity +� +dx exp +� +i +� +q − q′� +· x +� += (2π)2δ(q − q′). The flux +density (photons/m2) of signal photons detected at position x2 is +N(x2) = (2π)2 +(λSf)2 +� +dΩ +� +a† +S +� +Ω, 2π +λSf x2 +� +aS +� +Ω, 2π +λSf x2 +�� +. +(8) +If we make use of Eq. (5) into Eq. (8) we obtain +N(x2) = (2π)2 +(λSf)2 Np +� +dΩ +����FS +� +Ω, 2π +λSf x2 +����� +2 � +1 − |R|2 ++ +��� exp +� +ikS(Ω, q)L + iϕS(Ω) +� ++ R∗ exp +� +− ikI(−Ω, −q)L − iϕI(−Ω) +���� +2� +, +(9) +where NP = IPSPTP. IP is the peak intensity of the pump beam, SP its area, and TP its +time duration. If we make the substitution β2 = σ/[(2π)3IP], define VS(Ω) ≡ VS(Ω, x2 = 0), + +4 +and associate a large value of TP with the detection time TD, the number of photons detected +in a small area SD centered around x2 = 0 is +N0 = SPSD +(λSf)2 +TD +π +� +dΩ +���VS(Ω) +��� +2� +1 + |R| cos(θ + ϕR) +� +. +(10) +VS is the same functions as FS in Eq. (2) by substituting the nonlinear coefficient β by σ. +We should notice that similar expressions to Eq. (10) for describing flux rates with detection +systems based on 2f systems has been derived using other methods [4, 5]. +If we approximate the phase matching function as ∆S = DΩ, where D is the difference of +inverse group velocities at the central frequencies between signal and idler photons, +θ = k0 +PL + ωSLS + ωILI +c ++ Ω +� +DL + LS − LI +c +� ++ ϕP1 − ϕP2. +(11) +LS,I are the lengths of the 4f systems traversed by signal and idler photons, respectively, +before being re-injected back into the nonlinear crystal. For the sake of simplicity, we write +|V (Ω)|2 = exp +� +−π Ω2 +B2 +� +. +(12) +where B is the bandwidth of parametric down-conversion. +Making the integration over +frequency in Eq. (10), we obtain +⟨N0⟩ = 2S0 [1 + |R|γ cos(δ + ϕR)] , +(13) +with δ = k0 +PL + (ωSLS + ωILI)/c, +S0 = SP SD +(λSf)2 +TDB +2π (σL)2, +(14) +and +γ = exp +� +−B2 +4π +� +DL + Ls − Li +c +�2 � +. +(15) +Notice that the visibility of signal ⟨N0⟩ as function of phase δ is +V = |R|γ. +(16) +N0 is the variable that we designate as NS in the main text. + +5 +C. Sensitivity of phase estimation with phase shifting digital holog- +raphy under the presence of external noise +For the sake of simplicity, we write nelow NS = N0 ≡ N. We aim at estimating ϕR using +M phases (δj = j 2π/M with j = 0, 1, ..., M − 1). The expression for the estimation of the +phase is +ϕR = − tan−1 +� +j ⟨N⟩j sin(δj) +� +j ⟨N⟩j cos(δj). +(17) +For M=4 (phases δj = 0, π/2, π, 3π/2) we have +ϕR = tan−1 ⟨N⟩3π/2 − ⟨N⟩π/2 +⟨N⟩0 − ⟨N⟩π +. +(18) +We will make use of the error propagation formula +⟨(∆ϕR)2⟩ = +� +j +� +∂ϕR +∂ ⟨N⟩j +�2 � +⟨(∆N)2⟩j + +� +(∆NT)2� � +. +(19) +where ⟨NT⟩ is the background signal (independent of the signal of interest) that reaches the +detector with a variance of ⟨(∆NT)2⟩. If the signal Nj has a bandwidth B, and the detection +time TD is large, i.e., TD ≫ 1/B, a condition that applies in the experiment, we can safely +write that ⟨(∆N)2⟩j = ⟨N⟩j. Therefore, we can write +⟨(∆ϕR)2⟩ = +� +j +� +∂ϕR +∂ ⟨N⟩j +�2 � +⟨N⟩j + +� +(∆NT)2� � +. +(20) +The derivatives are +∂ϕR +∂ ⟨N⟩i += +� +j ⟨N⟩j sin δj cos δi − � +j ⟨N⟩j cos δj sin δi +�� +j ⟨N⟩j sin δj +�2 ++ +�� +j ⟨N⟩j cos δj +�2 +. +(21) +After some calculations, we obtain +� +j +⟨N⟩j sin(δj) = −M|R|γ S0 sin(ϕR), +� +j +⟨N⟩j cos(δj) = M|R|γ S0 cos(ϕR), +(22) +where we have made use of +� +i +sin2(ϕR + δj) = M +2 , +(23) + +6 +and +� +j +sin2(ϕR − δj) cos(ϕR + δj) = 0. +(24) +We can easily verify numerically the validity of these expressions for several values of M. +Finally, we obtain +� +∂ϕR +∂ ⟨N⟩j +�2 += +1 +M 2γ2|R|2 +1 +S2 +0 +sin2(ϕR + δj). +(25) +The sensitivity is +⟨(∆ϕR)2⟩ = +� +j +� +∂ϕR +∂ ⟨N⟩j +�2 � +⟨N⟩j + +� +(∆NT)2� � += +1 +M 2γ2|R|2 +1 +S2 +0 +× +� +2S0 +� +j +sin2(ϕR + δj) + 2|R|γ S0 +� +j +sin2(ϕR + δj) cos(ϕR + δj) ++ +� +(∆NT)2� � +j +sin2(ϕR + δj) +� += +1 +M 2γ2|R|2 +1 +S2 +0 +M +� +S0 + ⟨(∆NT)2⟩ +2 +� +. +(26) +If we make the measurement n times, the phase sensitivity is +⟨(∆ϕR)2⟩ = +1 +Mγ2|R|2 n S0 +� +1 + ⟨(∆NT)2⟩ +2S0 +� +. +(27) +If we make use of the visibility V given by Eq. (16), we can write Eq. (27) as +⟨(∆ϕR)2⟩ = +1 +MV 2 n S0 +� +1 + ⟨(∆NT)2⟩ +2S0 +� +. +(28) +D. Characterization of the noise +We characterized the intensity and variance of the noise. In front of the noise source, we +placed a linear polarizer to vary its intensity. We also placed a light diffuser in the path to the +camera that rotates at different angular frequencies. In Fig. 1, we plot experimental data +for the noise variances against noise intensities. The purple circle represents an angular +frequency of 0 Hz. Similarly, the pink start, green triangle, and yellow square represent +angular frequencies of 1, 2, and 3 Hz. A theoretical black line representing ⟨(∆NT)2⟩ = +⟨NT⟩, is also provided. All configurations of noise employed in the experiments show super- +Poissonian statistics. + +7 +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +8.0 +9.0x104 +Noise Intensity NT in photons +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0x105 +Noise Variance ( NT)2 in photons +Poissonian +0 Hz +1 Hz +2 Hz +3 Hz +Figure 1: Noise variance against noise intensity. The noise properties were characterized +for different configurations. Experimental data points of the noise variances are plotted against +the noise intensities. The noise variance was measured with S0 = 134 signal photons generated +in a single pass through the crystal. +The detection window was set to be TD = 1 s and the +detection area SD was 32.5 × 32.5 µm2. A theoretical black line represents the Poissonian case, +e.g., +� +(∆NT )2� += ⟨NT ⟩. In all the experiments, the noise exhibit super-Poissonian statistics. +E. Signal intensity affected by noise +In quantum holography with undetected light (QHUL), the signal intensity varies depending +on the phase value. If an additional source of noise is superimposed on the camera, the +signal variance increases. Figure 2 shows five QHUL measurements of 12 steps for the signal +intensity collected by one pixel. We have also superimposed a noise in different ratios to +the signal intensity. Solid-shaded areas represent the obtained signal variances. The blue +area represents a ratio of r ≈ 1 : 8 and resulted in a small signal variance. The orange area + +8 +represents a ratio of r ≈ 1 : 91 and shows an increment in the signal variance with respect +to the blue one. Finally, the green area represents a ratio of r ≈ 1 : 252, and we obtained +the biggest variance for the signal photon. +0 +2 +3 +phase +400 +300 +200 +100 +0 +100 +200 +300 +average centered intensity in arb. units +r + 1:8 +r + 1:91 +r + 1:252 +Figure 2: Signal intensity in QHUL affected by noise. Applying a QHUL of 12 steps, we +obtained the signal intensity against the phase. The experimental signal intensities are shown with +dotted lines. The solid-shaded areas show the signal variances for different noise intensities. A +higher noise intensity increases the signal variance. More details in the main text. +[1] B. Dayan, Phys. Rev. A 76, 043813 (2007). +[2] R. W. Boyd, G. S. Agarwal, K. W. C. Chan, A. K. Jha, and M. N. O’Sullivan, Opt. Commun. +281, 3732 (2008). +[3] J. W. Goodman, Introduction to Fourier optics, Chap. 5 (2005). +[4] E. Brambilla, A. Gatti, L. A. Lugiato, and M. I. Kolobov, Eur. Phys. J. D 15, 127–135 (2001). +[5] E. Brambilla, A. Gatti, M. Bache, and L. A. Lugiato, Phys. Rev. A 69, 023802 (2004). + diff --git a/fNE0T4oBgHgl3EQfowEk/content/tmp_files/load_file.txt b/fNE0T4oBgHgl3EQfowEk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8e9933b852bb23daa36e686d27d7889eba76b48 --- /dev/null +++ b/fNE0T4oBgHgl3EQfowEk/content/tmp_files/load_file.txt @@ -0,0 +1,888 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf,len=887 +page_content='Experimental quantum imaging distillation with undetected light Jorge Fuenzalida,1,2,†,⋆ Marta Gilaberte Basset,1,3,⋆ Sebastian T¨opfer,1,2,⋆ Juan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Torres,4,‡ and Markus Gr¨afe1,2,3,§ 1Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 7, 07745 Jena, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2Technical University of Darmstadt, Department of Physics, Institute of Applied Physics, Schloßgartenstraße 7, 64289 Darmstadt, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 3Friedrich-Schiller-University Jena, Abbe Center of Photonics, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 6, 07745 Jena, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 4ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain, and Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' † jorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='fuenzalida@tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='de ‡ juanp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='torres@icfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='eu § markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='graefe@tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='de ⋆These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (Dated: January 9, 2023) Abstract Imaging based on the induced coherence effect makes use of photon pairs to obtain informa- tion of an object without detecting the light that probes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' While one photon illuminates the object, only its partner is detected, so no measurement of coincidence events are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The sought-after object’s information is revealed observing a certain interference pattern on the detected photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Here we demonstrate experimentally that this imaging technique can be made resilient to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We introduce an imaging distillation approach based on the interferometric modulation of the signal of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We show that our scheme can generate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='02529v1 [quant-ph] 6 Jan 2023 2 a high-quality image of an object even against noise levels up to 250 times the actual signal of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We also include a detailed theoretical explanation of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Introduction Quantum imaging [1] is an emerging and promising field in quantum technologies with certified advantages over classical protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This has been demonstrated in different sce- narios: in schemes that work in the low-photon flux regime [2, 3], in schemes that make use of undetected probing photons [4, 5], for super-resolution imaging [6–9], sub-shot-noise imaging [10–12], or enhanced two-photon absorption processes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Moreover, protocols in quantum imaging with no classical counterpart has been developed based on quantum interference [14], and entanglement [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In recent years, it has also been proven that quantum imaging protocols can be resilient to noise [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Distillation (also known as purification) is the process wherein the decoherence introduced in a quantum system by the environment can be removed [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In quantum imaging, the effect of the environment can be modeled through classical illumination superimposed over a quantum image on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Since most cameras only detect intensity, quantum and classical images seems to be indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, quantum correlations of photon pairs can be employed to differentiate the quantum image from a classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Quantum imaging distillation has been implemented with one and several photon pair degrees of freedom [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' To the best of our knowledge, every implementation to date has used the joint detection of photon pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In this work, we introduce and experimentally verify a quantum imaging distillation technique that employs the detection of single photons only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Quantum imaging with undetected light (QIUL) [4, 26–28] is a two-photon wide-field inter- ferometric imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In QIUL, one photon illuminates an object and its partner photon is detected on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The photon that illuminates the object remains un- detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Using an interferometric configuration, the object information is transferred to the detected photon interference pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Due to its unique detection advantage, QIUL has been employed to probe samples with unconventional wavelengths while visible light is detected [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Up to date, the effects of noise in QIUL have not yet been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 3 Here we introduce a source of noise in a QIUL scheme and study the resilience of the quantum imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The properties of the noise, its intensity and variance, are changed during this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We perform a quantum imaging distillation technique based on quantum phase-shifting digital holography [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our distillation technique uses phase modulation of the undetected photon to vary the interference pattern detected on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We notice that if the intensity difference of the interference patterns is bigger than the variance of the noise, the quantum image can be distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We also observe that the noise variance affects the distilled quantum images linearly in their phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our technique shows a good performance, even for noise intensities above 250 times the quantum signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Distillation principle Quantum imaging distillation is a process whereby a quantum image is cleaned from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' To explain our distillation technique let us consider two images: a quantum image, which is acquired by illuminating the sample with with non-classical light, and a noise image, which is an image detected at the camera and generated with classical illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' These two images are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 1(a) and 1(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A noise image is an unwanted signal that is superimposed over a quantum image on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This image superposition is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' To distill an image, different photon pair degrees of freedom can be employed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', frequency, time, or spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We use the amplitude modulation of quantum holography with undetected light (QHUL) [33], which is an interferometric quantum imaging technique [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In QHUL, the object information is carried into a single-photon interference pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' When the noise reaches the camera, if the intensity difference of QHUL is bigger than the intensity variance of the noise, the quantum image can be distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The resulting distilled image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Theory Spontaneous parametric down-conversion (SPDC) [34, 35] is a well-known nonlinear process that generates photon pairs (signal and idler) mediated by the interaction of an intense pump beam with the atoms of a nonlinear crystal [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our imaging scheme makes use on a SU(1,1) interferometer, wherein a pair of signal-idler photons can be generated in one of 4 0 1 2 3 4 5 6 y in mm quantum image 0 50 100 150 200 250 Intensity in arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' unit noise image 0 25 50 75 100 125 Intensity in arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' unit 0 1 2 3 4 5 6 x in mm 0 1 2 3 4 5 6 y in mm superposition 0 100 200 300 Intensity in arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' unit 0 1 2 3 4 5 6 x in mm distilled image 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 Visibility a b c d Figure 1: Principle of quantum imaging distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We employ a quantum imaging distilla- tion protocol to remove a noisy image from a quantum image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (a) is the quantum image we aim to distill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (b) is the noise image that is superimposed on the quantum image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (c) is the superposition of noise and quantum images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (d) is the resulting distilled image from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' two propagation modes, forward and backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The probability to generate paired photons in both modes (forward or backward) simultaneously is negligible [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the forward propagation mode, the pump, signal, and idler beams are spatially separated, and later, back-reflected into the nonlinear crystal with the help of 4f systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Before back reflection the idler photon is reflected from an object, with complex reflectivity R = |R| exp(iϕR), placed in front of its end-mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the backward propagation mode, the idler photon do not interact with the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The signal photons are collected by a camera, and the idler photon remains undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The mean value of signal photons detected in time TD at one pixel of the camera of area SD 5 (see supplementary material for details) is ⟨NS⟩δ = 2 S0 [1 + |R| γ cos(δ + ϕR)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (1) δ is an interferometric spatially invariant phase and S0 is the number of signal photons generated in single-pass SPDC (in a time window TD and area SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The parameter γ is related to the effective bandwidth of the signal-idler photon pairs, determined essentially by the bandwidth of the filters located in front of the camera [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (1), we see that ⟨NS⟩ changes when δ is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In particular, using δ = 0, π/2, π, and 3π/2, the object’s information can be retrieved by means of QHUL as follows: |R| = 2 × �� ⟨NS⟩3π/2 − ⟨NS⟩π/2 �2 + � ⟨NS⟩0 − ⟨NS⟩π �2�1/2 ⟨NS⟩0 + ⟨NS⟩π/2 + ⟨NS⟩π + ⟨NS⟩3π/2 , (2) ϕR = arctan � ⟨NS⟩3π/2 − ⟨NS⟩π/2 ⟨NS⟩0 − ⟨NS⟩π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (3) Equations (2) and (3) are not unique representations of |R| and ϕR and, in general, these quantities can be extracted using different number of phases [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We emphasize that in this technique paired photon coincidences are not needed and only signal photons are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For the important case of phase estimation, we evaluate the sensitivity of QHUL obtaining the variance of ϕR given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We first calculate the variance of the signal-photon flux ⟨(∆NS)2⟩ = ⟨N 2 S⟩ − ⟨NS⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Since the coherence time TC of signal-idler photon pairs (TC ∼ 1/B, B is the effective bandwidth of SPDC) is much smaller than the detection time, we can approximate the variance of the signal-photon flux to as (see Supplementary Material), � (∆NS)2� = ⟨NS⟩ , (4) which is equivalent to considering Poissonian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This result is well-known to be valid when considering a multimode signal where each mode has the same non-Poissonian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (4) corresponds to the minimum signal variance achievable in QHUL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, this variance can rapidly increase by electronic noise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', camera signal-to-noise ratio, or external sources of noise such as temperature fluctuations, airflow, and external 6 illumination, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In this work, we study the effect of an external classical illumi- nation impinging on the camera, overlapping the quantum image of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Since QHUL detects only the single-photon stream of signal photons, the decoherence produced by an external source of light seems extremely harmful and, therefore, the image distillation highly improbable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, we will demonstrate experimentally that the quantum imaging dis- tillation is possible even in scenarios with considerable high levels of noise in comparison to the quantum signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Quantum holography with undetected light employs interferometric modulation of the signal photon to retrieve the object information, as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We show that this modulation can also be used for distillation purposes, which is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For each value of the interference phase δ, the signal photon has a well-defined intensity and variance, given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (1) and (4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2(a), signal intensity (variance) is represented with pink bar charts (error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In contrast, the intensity of a stochastic noise ⟨NT⟩ fluctuates randomly with a variance ⟨(∆NT)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For the sake of simplicity, we consider (but are not restricted to) to the case where the noise has the same total mean intensity and variance than the signal photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2(b), the noise intensity (variance) is represented with blue bar charts (error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' As a result of adding these two intensities, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2(c), the background increases up to the noise intensity, while the signal intensity varies on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise variance contributes to the signal intensity variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', the shot noise increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In this way, one can infer that if the difference of the signal intensity is higher than the noise variance, the quantum image can be distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Additionally, the shot noise of QHUL always increases if the noise intensity and/or its variance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' More details are given in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Let us analyze the performance of our distillation technique by considering a generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For QHUL with M phase steps, we have that ϕR = − tan−1 �� j ⟨NS⟩j sin δj/ � j ⟨NS⟩j cos δj � , (5) with δj = j 2π/M, and j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The phase variance, including noise, it reads � (∆ϕR)2� = � j � ∂ϕR ∂ ⟨NS⟩j �2 �� (∆NS)2� j + � (∆NT)2�� , (6) 7 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='8 Signal photon intensity Phase 3π/2 π/2 π 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='8 Noise intensity Phase 3π/2 π/2 π Intensity superposition 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='6 Phase 3π/2 π/2 π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='8 } } } a b c Figure 2: Intensity detected in one camera pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For visualization purposes, we have con- sidered the same intensities (chart bars) and variances (error bars) for the signal photon and the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In (a) is presented the signal photon intensity (in pink) for different values of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This inten- sity fluctuation allows us to compute the object information using QHUL [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In (b) is presented the noise intensity (in blue) for the same phases δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In contrast to the signal intensity, the noise intensity is not affected by the value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In (c) are added the noise and signal intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Since noise intensity does not change, its contribution just sets a higher background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' On top of it, signal photon intensity still changes, and the total variance is its previous variance plus the noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Thus, an external source of noise increases the shot noise of QHUL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' where � ∂ϕR ∂ ⟨NS⟩j �2 = 1 M 2 γ2 |R|2 S2 0 sin2(ϕR + δj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (7) Replacing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (4) and (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (6), and considering n measurements, we obtain that 8 the variance of phase estimation is � (∆ϕR)2� = 1 n S0 1 M V 2 � 1 + ⟨(∆NT)2⟩ 2 S0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (8) S0 is the number of idler photons that illuminate the object whose phase we want to estimate, ⟨(∆NT)2⟩ is the variance of the number of background photons that illuminate the detector, and V = |R|γ is the visibility of the signal-photon flux interference pattern as a function of the phase δ [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Equation (8) contains two contributions to the variance of the phase: the first term comes from the quantum illumination, and the second term comes from the noise illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' While the former can be shown to be well described by Poissonian statistics (see Supplementary Material), the latter depends on the specific characteristics of the noise illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Results Quantum imaging A sketch of our experimental implementation is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For our quantum im- age, we used a SU(1,1) nonlinear interferometer [39] in a Michelson configuration, where its input/output is a nonlinear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our crystal is a periodically poled potassium titanyl phosphate (ppKTP) of 2 × 2 × 1 mm3 (length × width × height), which is pumped bidirec- tionally (a and f directions) by a CW laser at 405 nm and with avarage power of 90 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Due to its strong χ(2)–nonlinearity, a photon pair (signal and idler photons), is generated through SPDC into the paths a or f, but never simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal (idler) photons have a central wavelength of λS = 910 nm (λI = 730 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the forward propagation direction a, signal, idler, and pump beams are spatially sepa- rated with dichroic mirrors DM2 and DM3 into the paths b, c, and d, and reflected back into the crystal with mirrors M1, M2, and M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In front of mirror M2, an object with a complex amplitude R = |R| exp(iϕR) is placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In b, c, and d, lenses of focal length f= 150 mm transform transverse position (source plane) into transverse momentum (mirrors plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Therefore, in path c, a wave vector kI representing a plane wave of the idler photon is focused to a point on the object [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The interaction of idlers being absorbed or reflected by the object can be modeled with the help of a beam splitter [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the backward propagation f, 9 L1 L2 L3 DM1 DM2 DM3 PPKTP L4 LASER OBJECT PIEZO CAMERA M1 M2 M3 L5 LASER LIGHT DIFFUSER NOISY OBJECT 10:90 BS L6 L7 LINEAR POLARIZER Figure 3: Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The signal and idler beams (in paths b and c, respectively) are generated by the pump beam in path a interacting with the nonlinear crystal (ppKTP) in the forward direction, while paths e and f represent propagations of the down converted beams generated after the pump beam is reflected back into the crystal by mirror M3 in path d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' An object in path c is illuminated with the idler beam in the Fourier plane of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' To create the noise, a laser diode of the same wavelength as the signal photon (910 nm) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The signal beam in path e is merged with the noise in path g before reaching the camera with a 10:90 beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' On the detector plane we obtain the quantum image with lenses L2, L4 and L5, and the noise image with lenses L6 and L7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Different type of noise are created with a linear polarizer and a light diffuser in path g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The linear polarizer controls the pump power of the diode laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The diffuser that consists on a rotating ground glass plate produces a speckle pattern of the noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The speed of the rotation is controlled through the glass plate motor interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' signals are collected by a camera and idlers remain undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We ensured that by placing a 800 nm long-pass filter and a 910 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 nm interference filter in front of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our camera is the Prime BSI Scientific CMOS from Teledyne Photometrics with a pixel size of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal photon’s transverse momentum is obtained with the lens L4 of focal length f= 100 mm performing a Fourier transform of the source plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This plane is later imaged 10 on camera with the lens L5 of focal length f= 150 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Thus, a wave vector kS of the signal photon is focused to a point on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If a and f propagation are perfectly aligned, the photon pair emission (which-source) information is erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Consequently, on the camera, an interference pattern of signal photons is observed [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Moreover, the object information obtained herein by the idler photon, is transferred to the signal photon interference pat- tern [4, 5], see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The interferometric phase δ is changed with a piezo placed below mirror M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The object information is retrieved by employing QHUL of 12 steps [33] with an acquisition time of t=1 s per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Noise source A CW diode laser of λN = 910 nm and variable pump power is employed to introduce noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The laser illuminates an object, which is imaged on the camera with a 4f system using the lenses L6 and L7 of focal lengths f= 150 mm and f= 125 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This classical image is superimposed on top of the quantum image on the camera using a 10:90 beam splitter, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Properties of classical illumination, intensity and variance, are changed in order to evaluate the effects of noise in QHUL and our distillation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Experimental details about the noise properties can be found in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Distillation performance to different noise intensities In the first experiment, while having superimposed classical and quantum images, QHUL is performed to distill the quantum image under different intensities of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We first characterized the signal flux rate emission measuring its mean intensity of an illuminated area on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal intensity does not changed during experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In a similar way but independently measured, different noise intensities are characterized, which are obtained by changing the angle of a linear polarizer (LP) in front of the laser in path g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The experiment starts superimposing the quantum and classical images on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Its first row shows the superposition of classical and quantum images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' noise intensity increases from left to right with the following ratios (r = signal intensity:noise intensity), r ≈ 1 : 8, r ≈ 1 : 37, r ≈ 1 : 50, r ≈ 1 : 104, and r ≈ 1 : 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 4 shows distilled images by QHUL of the corresponding upper superposed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Imaging distillation through QHUL is successfully achieved in every case, even with a noise intensity 250 times higher than the signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, we notice that while noise 11 r 1:8 r 1:37 r 1:50 r 1:104 r 1:252 0 1 2 3 x in mm 0 1 2 Phase R in rad 0 1 2 3 x in mm 0 1 2 3 x in mm 0 1 2 3 x in mm 0 1 2 3 x in mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Intensity 0 1 2 Phase R in rad Figure 4: Resilience to different noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the upper row are shown superpositions of quantum (IOF letters) and classical (square shape) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The ratio between their intensities is stated on top of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the middle row are presented the experimental results for our distillation technique through QHUL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the last row are presented a transverse cut of the distilled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We observe that while the noise intensity increases, the phase estimation diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' intensity increases, sharpness of distilled images decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' One can observe this in detail in the third row that shows a transverse cut of the distilled images (represented by a dotted red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' It is clear from our experimental results that phase accuracy diminishes as the noise intensity increases, which also corresponds to prediction given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Induced variance by noise In a second experiment, we quantified the effects of noise variances on the phase accuracy of distilled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The same configurations of noise intensities are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Additionally, a light diffuser mounted on a rotational motor with four angular frequencies of 0, 1, 2, and 3 Hz changed the noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise variance is characterized considering the intensity variation of one pixel over 12 consecutive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We plot the experimental measured values for the phase variance of distilled images against the noise variance for different angular frequencies (data point with error bars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' purple circle = 0 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' rose star = 1 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' green triangle = 2 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' yellow square = IOFIOFIOFIOFIOF12 3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We also provide a theoretical prediction for comparison (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The results show that an increment of the noise variance increases proportional with the phase variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Also it can be observed that in every case a linear dependence appears between these two variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, the noise variance introduces a slightly higher phase variance than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A reasonable explanation for this is that additional sources of noise were involved during the measurement process, such as airflow or temperature fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The experimental behaviour of variances is in good agreement to theoretical predictions presented above in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Discussion and conclusion Our work explores for the first time the effects of noise in Quantum imaging with undetected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We have also introduced a technique to distill the quantum image from that noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our quantum imaging distillation technique is based on quantum holography with undetected light [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This technique uses a photon pair, signal and idler, where idler illuminates the object and signal is detected on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The idler photon remains undetected and its phase modulation is employed in the imaging distillation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' To prove our technique, we superimposed partially or completely a classical source of noise on top of our quantum image on the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our technique worked in every occasion, even for noise intensities 250 times higher than our signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' However, the noise variance does affect the phase accuracy of our distillation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A higher noise variance produces a higher phase inaccuracy, where in general, these two quantities are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Although, in our experiment we employed a classical source of noise, this distillation tech- nique should also work for a quantum source of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Furthermore, since the underlying principle of this distillation technique is the phase modulation, our technique should be applicable to QIUL based on position correlations [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Our experiment is a step forward for quantum imaging in open systems and could be useful to the understand the limitations of a (quantum) LIDAR with undetected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0x105 Noise Variance ( NT)2 in photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 Phase Variance ( R)2 in rad theory 0 Hz 1 Hz 2 Hz 3 Hz Figure 5: Distillation phase variance affected by noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise variance was measured with S0 = 134 signal photons generated in a single pass through the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The detection window was set to be TD = 1 s and the detection area SD was ≈ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A light diffuser with four different rotation speeds is employed to change the properties of the noise illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In all configurations we observed that the phase sensitivity is linearly dependent with the noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise variance in dependence to the signal intensity shows a linear dependence as well (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A higher noise variance increases the phase inaccuracy in QHUL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Acknowledgments The authors thank Fabian Steinlechner for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' This work was supported as a Fraunhofer LIGHTHOUSE PROJECT (QUILT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Furthermore, funding support is ac- knowledged from the German Federal Ministry of Education and Research (BMBF) within the funding program Photonics Research Germany with contract number 13N15088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In ad- dition, funding is acknowledged from the Th¨uringer Aufbaubank via the projects Multi-Use 14 (2020FGI0023), SPEQTRA (2021FE9016), and Quantum Hub Thuringia (2021FGI0041).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' acknowledges the funding by the R&D project CEX2019-000910-S, funded by the Ministry of Science and innovation (MCIN/ AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='13039/501100011033/), from Fundaci´o Cellex, Fundaci´o Mir-Puig, and from Generalitat de Catalunya through the CERCA pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' also acknowledges support from the project 20FUN02 “POLight”, that has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme, and finan- cial support from project QUISPAMOL (PID2020-112670GB-I00) funded by MCIN/AEI /10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Kuznetsov, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Krivitsky, Nanophotonics 10, 1775 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' T¨opfer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Gilaberte Basset, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [41] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Viswanathan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Lemos, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Lahiri, Optics Express 29, 38185 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Kviatkovsky, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Chrzanowski, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Ramelow, Optics Express 30, 5916 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [43] The visibility is defined by V = (⟨N⟩max−⟨N⟩min)/(⟨N⟩max+⟨N⟩min), where ⟨N⟩max (⟨N⟩min) is the maximum (minimum) photon count rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Supplementary Material of “Experimental quantum imaging distillation with undetected light” Jorge Fuenzalida,1,2,†,⋆, Marta Gilaberte Basset,1,3,⋆ Sebastian T¨opfer,1,2,⋆ Juan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Torres,4,‡ and Markus Gr¨afe1,2,3,§ 1Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 7, 07745 Jena, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 2Technical University of Darmstadt, Department of Physics, Institute of Applied Physics, Schloßgartenstraße 7, 64289 Darmstadt, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 3Friedrich-Schiller-University Jena, Abbe Center of Photonics, Albert-Einstein-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 6, 07745 Jena, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 4ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain, and Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' † jorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='fuenzalida@tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='de ‡ juanp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='torres@icfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='eu § markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='graefe@tu-darmstadt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='de ⋆These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (Dated: January 9, 2023) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The quantum state of signal photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In the undepleted pump beam approximation, if the bandwidth of parametric down- conversion (in the frequency and transverse wavenumber domains) is much larger than the corresponding bandwidth of the pump beam, the relationship between the quantum opera- tors aS(Ω, q) and aI(Ω, q) (output face of the nonlinear crystal) and the operators bS(Ω, q) and bI(Ω, q) (input face of the nonlinear crystal) can be written in the low parametric gain arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='02529v1 [quant-ph] 6 Jan 2023 2 approximation as [1] aS(Ω, q) =bS(Ω, q) exp � ikS(Ω, q)L � + FS(Ω, q) � dωP dqP EP(ΩP, qP) b† I(ΩP − Ω, qP − q), aI(Ω, q) =bI(Ω, q) exp � ikI(Ω, q)L � + FI(Ω, q) � dωP dqP EP(ΩP, qP) b† S(ΩP − Ω, qP − q), (1) where FS,I(Ω, q) = −i(βL) sinc �∆S,I(Ω, q)L 2 � exp � i � k0 P + kS,I(Ω, q) − kI,S(−Ω, −q) � L 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (2) The nonlinear coefficient β is β = � ¯h ωP ωS ωI [χ(2)]2 64 π3 ϵ0 c3 nP nS nI �1/2 , (3) nS,I,P are refractive indexes, χ(2) is the second-order nonlinear coefficient of the crystal, L is the crystal length, and ωP,S,I are central frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The phase mismatch functions are ∆S,I = k0 P − kS,I(Ω, q) − kI,S(−Ω, −q), kS,I are the wavenumbers of signal and idler waves, and k0 P is the wavenumber of the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The total number of photons NP carried by the pump beam is NP = � dΩ dq ��EP(Ω, q) ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (4) If we consider that signal and idler photons S1 and I1 (forward propagation modes) are re- injected back into the nonlinear crystal with the help of 4f systems, the operator associated to signal photon S2 (backward propagation mode) is aS2(Ω, q) = FS(Ω, q) � exp � ikS(Ω, q)L + iϕS(Ω) + iϕP1 � +R∗ exp � − ikI(−Ω, −q)L + iϕP2 − iϕI(−Ω) �� � dωP dqP EP(ΩP, qP) b† I(ΩP − Ω, qP − q) +FS(Ω, q) exp � iϕP2 � � dΩP dqP EP(ΩP, qP) f †(ΩP − Ω, qP − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (5) Notice that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (5), for the sake of simplicity, we have omitted the terms that depend on the operator bS since they will yield a zero contribution to the flux density of signal S2 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We have made use of the approximation kI(ΩP − Ω, qP − q) ∼ kI(−Ω, −q), since the bandwidth of the pump beam is assumed to be very small when compared with 3 the bandwidth of parametric down-conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We also consider that the phases of the pump beams, ϕP1 and ϕP2, that illuminate the nonlinear crystal might be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The phases ϕS(Ω) and ϕI(Ω) are phases acquired by the signal S1 and idler I1 photons that traverse 4f systems before being re-injected back into the nonlinear crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The reflection coefficient of the object located in the idler I1 path is R = |R| exp(iϕR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The operator f takes into account [2] the presence of the object with reflectivity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' These operators fulfill the relationships � f(Ω, q), f †(Ω′, q′) � = � 1 − |R|2� δ(Ω − Ω′) δ(q − q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Flux density of signal photons detected in a 2f system We detect the flux density of signal photons S2 with the help of a 2f system with focal length f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The input-output relationship between operators is similar to the classical relationship [3] aS2(t, x2) = 1 λS f � dx1 aS2(t, x1) exp � −i 2π λS f x1 · x2 � , (6) where x1 and x2 are the corresponding transverse coordinates in the input and output planes of the 2f system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For the sake of simplicity, we ignore any global phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If we introduce the Fourier transform of the operator aS2(t, x1) as aS2(t, x) = (2π)−3/2 � dΩ dq aS2(Ω, q) exp � iq· x − iΩt � , we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (6) as aS2(t, x2) = (2π)1/2 λS f � dΩ aS2 � Ω, 2π λS f x2 � exp (−i Ω t) , (7) where we have made use of the identity � dx exp � i � q − q′� x � = (2π)2δ(q − q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The flux density (photons/m2) of signal photons detected at position x2 is N(x2) = (2π)2 (λSf)2 � dΩ � a† S � Ω, 2π λSf x2 � aS � Ω, 2π λSf x2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (8) If we make use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (8) we obtain N(x2) = (2π)2 (λSf)2 Np � dΩ ����FS � Ω, 2π λSf x2 ����� 2 � 1 − |R|2 + ��� exp � ikS(Ω, q)L + iϕS(Ω) � + R∗ exp � − ikI(−Ω, −q)L − iϕI(−Ω) ���� 2� , (9) where NP = IPSPTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' IP is the peak intensity of the pump beam, SP its area, and TP its time duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If we make the substitution β2 = σ/[(2π)3IP], define VS(Ω) ≡ VS(Ω, x2 = 0), 4 and associate a large value of TP with the detection time TD, the number of photons detected in a small area SD centered around x2 = 0 is N0 = SPSD (λSf)2 TD π � dΩ ���VS(Ω) ��� 2� 1 + |R| cos(θ + ϕR) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (10) VS is the same functions as FS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (2) by substituting the nonlinear coefficient β by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We should notice that similar expressions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (10) for describing flux rates with detection systems based on 2f systems has been derived using other methods [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If we approximate the phase matching function as ∆S = DΩ, where D is the difference of inverse group velocities at the central frequencies between signal and idler photons, θ = k0 PL + ωSLS + ωILI c + Ω � DL + LS − LI c � + ϕP1 − ϕP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (11) LS,I are the lengths of the 4f systems traversed by signal and idler photons, respectively, before being re-injected back into the nonlinear crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' For the sake of simplicity, we write |V (Ω)|2 = exp � −π Ω2 B2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (12) where B is the bandwidth of parametric down-conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Making the integration over frequency in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (10), we obtain ⟨N0⟩ = 2S0 [1 + |R|γ cos(δ + ϕR)] , (13) with δ = k0 PL + (ωSLS + ωILI)/c, S0 = SP SD (λSf)2 TDB 2π (σL)2, (14) and γ = exp � −B2 4π � DL + Ls − Li c �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (15) Notice that the visibility of signal ⟨N0⟩ as function of phase δ is V = |R|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (16) N0 is the variable that we designate as NS in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Sensitivity of phase estimation with phase shifting digital holog- raphy under the presence of external noise For the sake of simplicity, we write nelow NS = N0 ≡ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We aim at estimating ϕR using M phases (δj = j 2π/M with j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', M − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The expression for the estimation of the phase is ϕR = − tan−1 � j ⟨N⟩j sin(δj) � j ⟨N⟩j cos(δj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (17) For M=4 (phases δj = 0, π/2, π, 3π/2) we have ϕR = tan−1 ⟨N⟩3π/2 − ⟨N⟩π/2 ⟨N⟩0 − ⟨N⟩π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (18) We will make use of the error propagation formula ⟨(∆ϕR)2⟩ = � j � ∂ϕR ∂ ⟨N⟩j �2 � ⟨(∆N)2⟩j + � (∆NT)2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (19) where ⟨NT⟩ is the background signal (independent of the signal of interest) that reaches the detector with a variance of ⟨(∆NT)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If the signal Nj has a bandwidth B, and the detection time TD is large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', TD ≫ 1/B, a condition that applies in the experiment, we can safely write that ⟨(∆N)2⟩j = ⟨N⟩j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Therefore, we can write ⟨(∆ϕR)2⟩ = � j � ∂ϕR ∂ ⟨N⟩j �2 � ⟨N⟩j + � (∆NT)2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (20) The derivatives are ∂ϕR ∂ ⟨N⟩i = � j ⟨N⟩j sin δj cos δi − � j ⟨N⟩j cos δj sin δi �� j ⟨N⟩j sin δj �2 + �� j ⟨N⟩j cos δj �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (21) After some calculations, we obtain � j ⟨N⟩j sin(δj) = −M|R|γ S0 sin(ϕR), � j ⟨N⟩j cos(δj) = M|R|γ S0 cos(ϕR), (22) where we have made use of � i sin2(ϕR + δj) = M 2 , (23) 6 and � j sin2(ϕR − δj) cos(ϕR + δj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (24) We can easily verify numerically the validity of these expressions for several values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Finally, we obtain � ∂ϕR ∂ ⟨N⟩j �2 = 1 M 2γ2|R|2 1 S2 0 sin2(ϕR + δj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (25) The sensitivity is ⟨(∆ϕR)2⟩ = � j � ∂ϕR ∂ ⟨N⟩j �2 � ⟨N⟩j + � (∆NT)2� � = 1 M 2γ2|R|2 1 S2 0 × � 2S0 � j sin2(ϕR + δj) + 2|R|γ S0 � j sin2(ϕR + δj) cos(ϕR + δj) + � (∆NT)2� � j sin2(ϕR + δj) � = 1 M 2γ2|R|2 1 S2 0 M � S0 + ⟨(∆NT)2⟩ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (26) If we make the measurement n times, the phase sensitivity is ⟨(∆ϕR)2⟩ = 1 Mγ2|R|2 n S0 � 1 + ⟨(∆NT)2⟩ 2S0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (27) If we make use of the visibility V given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (16), we can write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (27) as ⟨(∆ϕR)2⟩ = 1 MV 2 n S0 � 1 + ⟨(∆NT)2⟩ 2S0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' (28) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Characterization of the noise We characterized the intensity and variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In front of the noise source, we placed a linear polarizer to vary its intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We also placed a light diffuser in the path to the camera that rotates at different angular frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 1, we plot experimental data for the noise variances against noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The purple circle represents an angular frequency of 0 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Similarly, the pink start, green triangle, and yellow square represent angular frequencies of 1, 2, and 3 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A theoretical black line representing ⟨(∆NT)2⟩ = ⟨NT⟩, is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' All configurations of noise employed in the experiments show super- Poissonian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0x104 Noise Intensity NT in photons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='0x105 Noise Variance ( NT)2 in photons Poissonian 0 Hz 1 Hz 2 Hz 3 Hz Figure 1: Noise variance against noise intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise properties were characterized for different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Experimental data points of the noise variances are plotted against the noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The noise variance was measured with S0 = 134 signal photons generated in a single pass through the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The detection window was set to be TD = 1 s and the detection area SD was 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='5 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A theoretical black line represents the Poissonian case, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=', � (∆NT )2� = ⟨NT ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' In all the experiments, the noise exhibit super-Poissonian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Signal intensity affected by noise In quantum holography with undetected light (QHUL), the signal intensity varies depending on the phase value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' If an additional source of noise is superimposed on the camera, the signal variance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Figure 2 shows five QHUL measurements of 12 steps for the signal intensity collected by one pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' We have also superimposed a noise in different ratios to the signal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Solid-shaded areas represent the obtained signal variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The blue area represents a ratio of r ≈ 1 : 8 and resulted in a small signal variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The orange area 8 represents a ratio of r ≈ 1 : 91 and shows an increment in the signal variance with respect to the blue one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Finally, the green area represents a ratio of r ≈ 1 : 252, and we obtained the biggest variance for the signal photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 0 2 3 phase 400 300 200 100 0 100 200 300 average centered intensity in arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' units r 1:8 r 1:91 r 1:252 Figure 2: Signal intensity in QHUL affected by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Applying a QHUL of 12 steps, we obtained the signal intensity against the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The experimental signal intensities are shown with dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' The solid-shaded areas show the signal variances for different noise intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A higher noise intensity increases the signal variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' More details in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Dayan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A 76, 043813 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Boyd, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Agarwal, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Chan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Jha, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' O’Sullivan, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 281, 3732 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Goodman, Introduction to Fourier optics, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' 5 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Brambilla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Gatti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Lugiato, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Kolobov, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' J.' metadata={'source': 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} +page_content=' A 69, 023802 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQfowEk/content/2301.02529v1.pdf'} diff --git a/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/2301.00671v1.pdf.txt b/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/2301.00671v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..51901735661b7797f4e6f93fc2fb5dcc78dfb1ce --- /dev/null +++ b/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/2301.00671v1.pdf.txt @@ -0,0 +1,1202 @@ + +1 of 20 + +Political representation bias in +DBpedia and Wikidata as a challenge +for downstream processing +Özgür Karadeniz1 *, Bettina Berendt2,3,1 *, Sercan Kıyak1, Stefan Mertens1, Leen d’Haenens1 + +1 KU Leuven, Belgium; 2 TU Berlin, Germany; 3 Weizenbaum Institute, Germany +* corresponding authors {ozgur.karadeniz|bettina.berendt}@kuleuven.be + +Abstract: Diversity Searcher is a tool originally developed to help analyse diversity in news +media texts, relying on a form of automated content analysis and thus rests on prior +assumptions and depends on certain design choices related to diversity and fairness. One +such design choice is the external knowledge source(s) used. In this article, we discuss +implications that these sources can have on the results of content analysis. We compare two +data sources that Diversity Searcher has worked with – DBpedia and Wikidata – with respect +to their ontological coverage and diversity, and describe implications for the resulting +analyses of text corpora. We describe a case study of the relative over- or under- +representation of Belgian political parties between 1990 and 2020 in the English-language +DBpedia, the Dutch-language DBpedia, and Wikidata, and highlight the many decisions +needed with regard to the design of this data analysis and the assumptions behind it, as well +as implications from the results. In particular, we came across a staggering over- +representation of the political right in the English-language DBpedia. + + + + +2 of 20 + +1 Introduction +Originally developed as a tool to help analyse diversity in news media texts, Diversity +Searcher is now ready to be used by the general public and in the context of public libraries. +The Diversity Searcher, presented in section 2, relies on a form of automated content +analysis and thus rests on prior assumptions 1 and depends on certain design choices +related to diversity and fairness. 2 Then in Section 3, we study representational biases in the +underlying data sources and what they could imply for the extent of diversity that the tool can +identify. +We analyse the two alternatives we are currently working with – DBpedia and Wikidata – +with respect to their ontological coverage and diversity, and describe implications for the +resulting analyses of text corpora. Specifically, we describe a case study that examines the +relative over- or under-representation of Belgian political parties between 1990 and 2020 +and highlights the many research-design decisions needed for this data analysis and the +assumptions behind it, as well as implications from the results. In particular, we came across +a staggering over-representation of the political right in the English-language DBpedia. +This result presents a fascinating challenge for the design of data- and AI-powered tools +such as Diversity Searcher for public libraries. Through its base functionality, but maybe +even more so through transparency regarding findings such as those in our case study, the +tool can help foster fairness and diversity and create awareness among news users that +multi-voiced news is necessary to form an accurate and fair view of the reality being reported +in a news story. +2 The Diversity Searcher as a text analysis tool +The Diversity Searcher is a semi-automated text analysis and knowledge enrichment tool +that is designed to present information to the user about the degree of diversity in news +media texts.3 The tool was developed in the interdisciplinary project DIAMOND (Diversity +and Information Media: New Tools for a Multifaceted Public Debate)4 to be used by media +professionals and media consumers, and to be integrated with iCandid,5 a research +infrastructure project offering integrated access several (social)media resources to +researchers. Over the course of the project, Diversity Searcher has been further developed +to be a tool to be used in libraries as well. +The tool analyses an uploaded text or collection of texts. It extracts actors (persons, +institutions, and geopolitical entities) and offers additional information and interaction + +1 See for example Kitchin, Rob: The Data Revolution. Big Data, Open Data, Data Infrastructures & +Their Consequences. London: Sage. 2014. +2 Berendt, Bettina; Karadeniz, Özgür; Mertens, Stefan u. a.: Fairness beyond “equal”: The Diversity +Searcher as a Tool to Detect and Enhance the Representation of Socio-political Actors in News +Media, in: Companion Proceedings of the Web Conference 2021, Ljubljana Slovenia 2021, S. 202– +212. Online: , Retrieved: 25.08.2022. +3 Ibid. +4 About DIAMOND, 07.01.2018, , Retrieved: +27.08.2022. +5 iCandid, A Snapshot of LIBIS Research Infrastructures, 03.05.2022, +, Retrieved: 25.08.2022. + + +3 of 20 + +opportunities about the individual actors and their occurrence(s) in the text(s). These range +from base statistics (such as frequencies of different actors) to a numerical evaluation of the +complex notion of “diversity”. +2.1 A quantitative measure of (actor) diversity in a text +One of our core assumptions is that diversity in media is a crucial precondition of a well- +functioning democracy and a well-informed public opinion. Diversity is considered a crucial +part of information quality, for which we draw on Stirling’s (2007)6 influential proposal of a +quantitative measure that draws on a meta-analysis of the literature on diversity from various +disciplines. Stirling identified three components, which all enhance diversity: +- Variety: many distinct entities are present, e.g. a number of people of different +backgrounds; +- Disparity: the entities are different from one another, they come from a wide range; +- Balance: the different entities are evenly distributed. This component is a simple +form of the “equality of ...” concept found in the fairness literature. + +Based on these three components Stirling defined diversity Δ as + +𝛥 = " +𝑖,𝑗(𝑖≠𝑗) +(𝑑𝑖𝑗)𝛼 ⋅ (𝑝𝑖 ⋅ 𝑝𝑗)𝛽 + +with variety being captured by the cardinality of the set of all entities i, j ∈E present in the +domain (e.g. a text, a population, ...), balance by the frequencies pi (the more uniform the +distribution, the higher this multiplicative factor), and disparity by a measure of distance or +dissimilarity d(.,.). In addition, the parameters α and β allow for a weighting of the importance +of balance or disparity. +Hence, diversity in news media means that many news stories are available (variety) and +that these express different viewpoints (disparity), with all views equally represented +(balance). As Ranaivoson points out, disparity is often a hurdle when using this formula, as it +is measured based on strong assumptions.7 The tool therefore focuses on a specific aspect +to address diversity in media content: actor diversity. +2.2 An outline of the processing in the Diversity Searcher +In this subsection, we give a brief overview of the Diversity Searcher’s processing. For +details, user interface, and a worked-out example, see (Berendt et al., 2021).8 + +6 Stirling, Andy: A general framework for analysing diversity in science, technology and society, in: +Journal of The Royal Society Interface 4 (15), 22.08.2007, S. 707–719. Online: +. +7 Ranaivoson, Heritiana: Measuring cultural diversity with the Stirling model, in: New Techniques and +Technologies for Statistics, 2013, S. 10. Online: +. +8 Berendt, Bettina; Karadeniz, Özgür; Mertens, Stefan u. a.: Fairness beyond “equal”: The Diversity +Searcher as a Tool to Detect and Enhance the Representation of Socio-political Actors in News + + +4 of 20 + +We study the diversity of socio-political actors, understood broadly in terms of three types of +actors having socio-political agency such as: persons, organisations and geo-political +entities. Our design choices were inspired by a sociological understanding9 of the extent +and the ways in which two social actors are different. +We then built a knowledge retrieval and text processing pipeline that draws on data from an +underlying knowledge source ontology. Originally, we worked with the English-language +DBpedia. The Diversity Searcher performs its analyses in two steps. The first one consists of +detecting the actors in the texts with a combination of Named Entity Recognition(NER), +Named Entity Linking (NEL), and case sensitive and insensitive text and lemma matching.10 +For NEL, we use DBpedia Spotlight, a NEL model available in multiple languages, which +links entities to DBpedia.11 NER and pattern matching tasks use the third party tool spaCy, a +freely available NLP library and language models.12 The pattern matching function was +added to the pipeline when our colleagues in media studies observed that Spotlight was +performing poorly for Flemish politicians. The Addition of this feature also paved the way for +user-defined recognition rules for named entities, and unnamed entities such as ‘refugee’, +‘criminal’, etc. +The application includes a corpus upload function. Using this function, users can upload +XML files exported from Belga Press13 and batch process them, as well as export the results +as an Excel file to be used in further analysis in Excel or SPSS. +Although detection of the actors is sufficient to address balance and variety, disparity +requires further information about the detected entities, hence the knowledge retrieval and +enrichment stage. As DBpedia contains information that is both relevant and irrelevant to our +task, the analysis also requires filtering the predicate-object pairs we obtain from DBpedia. +For this task, we developed our own ontology consisting of socio-politically relevant actors +and features. Graph data from DBpedia is thus transformed into an array of “feature_name”: +“feature_value” pairs in accordance with our ontology, which are inserted into the local +record of the resource, and later used in calculations. +The knowledge retrieval and ontology mapping processes all the resources linked to the root +resource, provided that they also belong to the three actor types. This decision is based on +the fact that properties of some types of entities that are linked to the root entity also contain +important information about the root entity. Thus, the Diversity Searcher does not stop after + +Media, in: Companion Proceedings of the Web Conference 2021, Ljubljana Slovenia 2021, S. 202– +212. Online: , Retrieved: 25.08.2022. +9 Bourdieu, Pierre: The Social Space and the Genesis of Groups. Theory and Society 14, 6. 1985. +723-744. +10 NER: The subtask of Natural Language Processing (NLP), consisting in categorizing named entities +in a text into predefined categories; NEL: The subtask of NLP consisting in linking named entities in a +text to resources on a remote knowledge base. +11 Mendes, Pablo N.; Jakob, Max; García-Silva, Andrés u. a.: DBpedia spotlight: shedding light on the +web of documents, in: Proceedings of the 7th International Conference on Semantic Systems - I- +Semantics ’11, Graz, Austria 2011, S. 1–8. Online: , +Retrieved: 25.08.2022. +12 Honnibal, Matthew; Montani, Ines: spaCy 2: Natural language understanding with Bloom +embeddings, convolutional neural networks and incremental parsing, 2017. , +Retrieved: 29.08.2022. +13 Belga Press (belgapress.be) is an online press database covering a large number of international +and Belgian news sources, where journalists and media researchers can search and export corpora. +The specific source and therefore its XML format are relevant for our target users, but the schema +could easily be made configurable. + + +5 of 20 + +retrieving, for example, the party of a politician: it will also attempt to retrieve the party’s +properties such as ideology, as different parties (variety) may have intersecting ideologies, +thus affecting disparity. Similarly, finding the country of an actor will result in a further query +into properties of the country such as EU membership or type of government. +Despite contributing to enrichment by further contextualising the actors, the integration of +information from linked entities has the disadvantage of including these resources’ errors +and biases. For example, the data quality of the entry for “US” will also affect the data about +“Hillary Clinton”, “Joe Biden” and “Donald Trump” in the local knowledge base. Similarly, any +systematic political bias related to representation of political parties of certain ideologies will +affect the tool’s representation of the root politician, even if they are detected correctly. +3 The role of the underlying ontology for bias and diversity: +Comparing DBpedia and Wikidata +Tasks like that of the Diversity Searcher critically depend on external knowledge sources. In +choosing which one(s) to use, one needs to compare different options. The question is how to +do this in a formal and automated manner and in a way that supports the goals of the +application. We opted for a method that is based on (a) a number of observations and +assumptions about the available external sources, (b) the implementation of these +assumptions into computational procedures that extract a relational representation from the +RDF structures, and (c) application-specific questions about the data obtained. The criteria +build on the comprehensive framework for determining data quality on Linked Open Data +(LOD) by Zaveri et al. on the one hand and the literature on measuring bias in language +(models) on the other hand, and they re-interpret these in the light of our questions around +actor diversity.14 15 +For example, even if a news media article is highly diverse (say, in political affiliation of the +represented actors), if the entity recognition process relies on a knowledge source that only +contains actors of one political affiliation, only these actors will be processed in the diversity +calculation and result in a minimal diversity value. So an ideal knowledge source would +exhibit high coverage, correctness and timeliness, and high diversity. + +We therefore began to question whether our initial choice of DBpedia was useful to debate +what consequences the choice of the English-language DBpedia versus (if applicable) the +language of the country or region under investigation would have (in general, local DBpedias +have better coverage of local actors), and to suspect that Wikidata, which operates with +more stringent quality controls,16 could be a better basis for the Diversity Searcher. + +14 Amrapali Zaveri, Anisa Rula, Andrea Maurino, Ricardo Pietrobon, Jens Lehmann, Sören Auer: +Quality assessment for Linked Data: A Survey. Semantic Web 7(1): 63-93 (2016) +15 Delobelle, Pieter; Tokpo, Ewoenam; Calders, Toon u. a.: Measuring Fairness with Biased Rulers: A +Comparative Study on Bias Metrics for Pre-trained Language Models, in: Proceedings of the 2022 +Conference of the North American Chapter of the Association for Computational Linguistics: Human +Language Technologies, Seattle, United States 2022, S. 1693–1706. Online: +, Retrieved: 25.08.2022. +16 Piscopo, Alessandro; Simperl, Elena: What we talk about when we talk about wikidata quality: a +literature survey, in: Proceedings of the 15th International Symposium on Open Collaboration, Skövde +Sweden 2019, S. 1–11. Online: , Retrieved: 29.08.2022. + + +6 of 20 + + +These questions present two types of challenges for tool design. The first is +pragmatic/implementation-related, and the second is conceptual. The selection of a +knowledge source may appear, in the age of LOD, to be a modular and thus easily +configurable design feature. Indeed choices, including flexible and/or user-determined +choices,17 between these knowledge sources can be implemented in a tool such as the +Diversity Searcher. However, they require preparatory studies of these datasets and +adaptations of the extraction algorithms, since predicate names and coding conventions +differ (e.g., between DBpedia and Wikidata) and because even within DBpedia, the possible +universality of predicates is not always used and different language versions exhibit +preferential uses of different predicates with related but not equal syntax and semantics. For +example, the possibility of encoding the temporal dimensions of politicians’ active terms is +used widely in the English DBpedia but not in the other sources, where instead positions +held are often specified. It is possible that such differences result from authors in one +language DBpedia-authors community copying the information-presentation style of others in +their own community. In addition, in each case there are reasons for tool builders to prefer +uniformity (the extraction of the same predicates) or to choose predicates based on other +criteria. We opted for the latter approach and chose, from semantically highly related +predicates, those with the highest coverage. +3.1 Basic statistics +We first considered the English DBpedia and Wikidata as the two external knowledge sources; +the method can easily be generalised to others. Being the two largest sources of LOD, both +DBpedia and Wikidata have the potential to provide relevant information for the entity types +that Diversity Searcher recognizes.18 19 20 Although both contain machine-readable, freely +available data based on Wikipedia, they differ in their ontologies and construction. +DBpedia is the older of the two and is defined as "a crowdsourced community effort to extract +structured content from the information created in various Wikimedia projects."21 DBpedia +extracts information from Wikipedia pages (mainly from the info-boxes) as RDFs and creates +unique (language-dependent) URIs for its resources from the pages in a semi-automatic + +17 Combinations of several knowledge sources are also possible, but require even more choices, +especially in cases when there is conflicting information in two sources. This question will therefore be +left for future work. +18 Auer, Sören; Bizer, Christian; Kobilarov, Georgi u. a.: DBpedia: A Nucleus for a Web of Open Data, +in: Aberer, Karl; Choi, Key-Sun; Noy, Natasha u. a. (Eds.): The Semantic Web, Bd. 4825, Berlin, +Heidelberg 2007 (Lecture Notes in Computer Science), S. 722–735. Online: + +19 Lehmann, Jens; Isele, Robert; Jakob, Max u. a.: DBpedia - A Large-scale, Multilingual Knowledge +Base Extracted from Wikipedia, in: Semantic Web Journal 6 (2), 2015, S. 167–195. +20 Vrandečić, Denny; Krötzsch, Markus: Wikidata: a free collaborative knowledgebase. In: +Communications of the ACM 57 (10), 2014, pp. 78–85. Online: . +21 About DBpedia, DBpedia, (n.d.). + + +7 of 20 + +way.22 This way, it forms an open language graph that is integrated well within the Semantic +Web and popular among the LOD communities.23 +Wikidata differs from DBpedia in many ways. First, they relate to Wikipedia differently. +Wikidata aims to provide structured data for all other Wikimedia projects and contains entities +that are not covered in Wikipedia. However, DBpedia is strictly limited by the information +available on Wikipedia. Second, Wikidata is edited and maintained by a large and open +community of contributors, who fact-check and enrich the data via various WikiProjects.24 +DBpedia works based on a closed community and depends more on the correctness of +information on Wikipedia. Third, the Wikidata dataset is not language-dependent and consists +of one large graph that can be displayed using any available language label or description. +Unlike the language-dependent DBpedia datasets, URIs are generated as P-numbers for +properties and Q-numbers for entities in Wikidata. Fourth, thanks to its RDF structure Wikidata +allows qualifiers and references to be added to the statements, and this extra information can +be helpful in various NLP tasks. Finally, while both offer SPARQL APIs, Wikidatahas more +restrictions and is less suitable for extracting large amounts of data.25 +DBpedia and Wikidata also differ in the amount of information they offer. The latest snapshot +of the English version of DBpedia contains more than 850 million triples describing about 5.4 +million entities.26 Wikidata, on the other hand, provides about 1.36 billion triples describing +about 97 million entities.27 28 Consequently, the average number of statements per entity is +approximately 157 in DBpedia and about 14 in Wikidata. Thus, as previous research indicated, +Wikidata describes more entities, while DBpedia has more statements about each entity.29 +While DBpedia offers various statements in many semantic languages about its entities, +Wikidata offers more resources with more structured and compact information that offers + +22 Hofer, Marvin; Hellmann, Sebastian; Dojchinovski, Milan u. a.: The New DBpedia Release Cycle: +Increasing Agility and Efficiency in Knowledge Extraction Workflows, in: Blomqvist, Eva; Groth, Paul; +Boer, Victor de u. a. (Eds.): Semantic Systems. In the Era of Knowledge Graphs, Bd. 12378, Cham +2020 (Lecture Notes in Computer Science), S. 1–18. Online: , Retrieved: 21.02.2022. +23 Abián, D.; Guerra, F.; Martínez-Romanos, J. u. a.: Wikidata and DBpedia: A Comparative Study, in: +Szymański, Julian; Velegrakis, Yannis (Eds.): Semantic Keyword-Based Search on Structured Data +Sources, Bd. 10546, Cham 2018 (Lecture Notes in Computer Science), S. 142–154. Online: +, Retrieved: 25.08.2022. +24 Wikidata:Statistics, , Wikidata, (n.d.), Retrieved: +01.03.2022. +25 The current limits of the APIs can be found in the following links: +; + +26Holze, Julia: DBpedia Snapshot 2021-12 Release Announcement, DBpedia Blog, 09.02.2022, +, Retrieved: 28.08.2022. +27Wikidata dashboards / Wikidata Datamodel Statements, Grafana, (n.d.), +, +Retrieved: 01.03.2022. +28Wikidata:Statistics, (n.d.) +29 Abián,et al.: Wikidata and DBpedia: A Comparative Study, 2018 + + +8 of 20 + +higher utility for certain specific use cases. This summarization of findings in the literature is +also reflected by our own results, which are to be presented next.30 +3.2 A brief exploration of coverage of the domain of politicians +In a first data exploration, we focused on examples of the coverage of politicians in the English +DBpedia and in Wikidata, due to their relevance for the Diversity Searcher. We formulated the +following SPARQL queries: +Query +English-language DBpedia +Wikidata +Members of the Belgian +Chamber of Representatives +31 +143 +2996 +Members of the Flemish +Parliament 32 +21 +464 +Politicians who held office in +the US Chamber of +Representatives 33 +14886 +11,160 +Table 1. Three queries to compare the coverage of politicians in the English-language +DBpedia and Wikidata. +Looking into the results of the second query, we noticed that 15 (about ¾) of the provided +Flemish politicians belong to one political party (N-VA, Nieuw-Vlaamse Alliantie, New Flemish +Alliance). Although N-VA is currently the largest party in the Flemish parliament with 35 seats + +30 Färber, Michael, Frederic Bartscherer, Carsten Menne, and Achim Rettinger. ‘Linked Data Quality +of DBpedia, Freebase, OpenCyc, Wikidata, and YAGO’. Edited by Amrapali Zaveri, Dimitris +Kontokostas, Sebastian Hellmann, Jürgen Umbrich, Amrapali Zaveri, Dimitris Kontokostas, Sebastian +Hellmann, and Jürgen Umbrich. Semantic Web 9, no. 1 (30 November 2017): 77–129. +https://doi.org/10.3233/SW-170275. pp.46-8 +31 The link to the DBpedia page: +https://dbpedia.org/page/Category:Members_of_the_Chamber_of_Representatives_(Belgium) and +the Wikidata query: https://w.wiki/4DmS . However, we should also point out that the same list of +members of the Belgian Chamber of Representatives in Dutch-Flemish DBpedia consists of 701 +resources: http://nl.dbpedia.org/page/Kamer_van_volksvertegenwoordigers .Date of Access: +01/10/2021. We will look deeper into representational problems in each of these three databases +further in the following sections. +32 The link to the DBpedia query: https://tinyurl.com/4uwbycwu and the Wikidata query: +https://w.wiki/3YG4. Date of Access: 01/10/2021. +33 The link to the DBpedia query: https://tinyurl.com/e3y2b8kh Date of access and the Wikidata query: +https://w.wiki/4DnP. Date of Access: 01/10/2021. + + +9 of 20 + +out of 124, their members are overrepresented in English DBpedia compared to other Flemish +parties.34 The data also showed some miscategorizations in DBpedia.35 +These results indicate that the English DBpedia may offer fewer entities (especially in non- +English language contexts) and that there appear to be issues in terms of bias and quality. +Our observations comply with the existing comparative literature concerning these two +information sources. 36 +3.3 A deeper look at the representation of political parties +3.3.1 Motivation: Have right-wingers taken over Wikipedia/DBpedia? And why this +shows we need to look at diversity and bias + To understand the possible representational bias in DBpedia better, we followed up with a +closer inspection of such biases and what they could mean for the Diversity Searcher’s +notion of diversity. In particular, we asked how we could capture the biases created by “over- +representation”, and how “over-presentation in a source ontology” should be defined in the +context of diversity search in the first place. Given that political analyses always need to be +understood in their historical context, in addition we asked what over-representation of (also) +historical facts with a knowledge source that has a current and ever-evolving status could +mean. + +These questions also become an investigation into DBpedia, Wikidata, etc. as an +infrastructure of cultural memory. In the case investigated here, that of representation of +politicians by party, it could be argued that normative, correct and comprehensive sources +probably exist in the form of parliament records, political party records, etc. and that the use +of DBpedia/Wikidata is primarily a pragmatic and money-saving choice; but of course the +question of representation and the needed input for the Diversity Searcher range beyond +politicians and party affiliation, such that crowdsourced sources with all their imperfections +are often the only forms of cultural memory. Thus, the investigation stands between the +consequences of a pragmatic choice of source and the study of that source in itself. +3.3.2 Method +The ensuing analysis proceeded through six stages. + +1. We concentrated on the attributes “political party affiliation” and “political alignment of a +party” as the feature on which assessments of bias and diversity are based. We limited our +analysis to “politicians from Belgium”, since we realised early on that substantial domain +knowledge is needed to understand even purely numerical phenomena and members of our +team are experts in this domain. + + +34 The party membership information is gathered manually based on the previously mentioned +DBpedia query (https://tinyurl.com/4uwbycwu). Date of Access: 01/10/2021. +35 For example, DBpedia considers the Women's Equality Party (New York) as a type of "person" and +lists it as a politician: https://dbpedia.org/page/Women%27s_Equality_Party_(New_York) +36 Abián, et al: Wikidata and DBpedia: A Comparative Study,2018 + + +10 of 20 + +2. As a null hypothesis, a knowledge source represents a political constellation in an +unbiased way if the relative number of politicians from a given party who are represented as +an entity in a knowledge source37 equals the relative number of this party in a relevant real- +life context. We chose relative numbers because it was clear from the outset that our +knowledge sources are not normative and comprehensive public records, and because we +consider “having a Wikipedia page” (etc.) as an important contributor to public visibility of a +person and their party. + +This general question is only meaningful at a given point in time, since the baseline evolves, +at the very least with elections that change political representation. + +3. The baseline is then – relatively – easy to define: the shares of the vote or the number of +seats of parties Y at times T in a given political body. We started by concentrating on the +national parliament, the Chamber of People’s Representatives (Kamer van +volksvertegenwoordigers, henceforth KVV) and used the number of seats at the beginning of +a legislature. We also looked at the regional (Flemish) parliaments (Vlaams parlement, VP), +whose election time points are not identical to those at the national level. + +These data were retrieved from Wikipedia, de vrije encyclopedie (2022a, 2022b).38 39 + +4. The representation in the ontology is more difficult to define. One may want to compare +parties’ share at T (step 3.) with what was represented in the ontology at T. We chose to +compare with what is represented in the ontology “now, but as relevant at T”. This has two +reasons. First, we wanted to study the historical evolution since 1990, and Wikipedia only +started operations at the beginning of 2001 and after that took some time to evolve, in +particular in its non-English versions (for example, the Dutch-language version was created +in 2001). Wikidata was launched in October 2012. Second, we aimed at studying these +ontologies also as infrastructures of cultural memory, which changes the question slightly: +How does a user currently see the political landscape (of the 1990s, etc.) through the lens of +an ontology? + +CSV files of actors and parties “represented now” were created using SPARQL queries.from +the ontology current at the time of writing.40 The filtering as “relevant at T” happened in the +following step 5. + +37 In the case of DBpedia, which is derived from Wikipedia, this corresponds to “a person having a +Wikipedia page”. +38 Wikipedia-bijdragers: Kamer van volksvertegenwoordigers — Wikipedia, de vrije encyclopedie, +2022. Online: +, +Retrieved: 25.08.2022. +39 Wikipedia-bijdragers: Vlaams Parlement — Wikipedia, de vrije encyclopedie, 2022. Online: +, Retrieved: +25.08.2022. +40 In line with the Diversity Searcher’s queries to online ontologies, we initially designed our queries to +acquire information for Belgian politicians as well as contextual information about their parties and +countries. Details on the queried properties are described and motivated in (Berendt et al., 2021) and +can be seen in the hyperlinked queries. Some of this information was not used in the case study’s +analysis because it was irrelevant to the question (e.g. a politician’s gender, or country information). +Some variations between queries and workarounds were necessary to accommodate differences + + +11 of 20 + + +5. Computing party representation shares. How do we then get from “politician X exists in +the ontology” to “party Y covers Z% of the representation of time T”? + +5a. We first limited the analysis to politicians who were active at T. We derived this by +creating each actor’s maximal activity period as the union of all activity periods represented +in the ontology, limited – in the absence of an explicit end date – by “today” or, if applicable, +the actor’s death date or, in a small number of cases, a well-publicised41 date of ending their +political career. We selected the politicians who were active at T = 1 January of 1990, 1996, +2000, 2005, 2011, 2015, 2020. (The exceptions from the 5-year spacing were done to +capture the effects of the general elections held in 1995 and 2010.) + +5b. We then pre-processed each actor’s list of party affiliations (which was often long and +heterogeneous). First, parties that changed their name at one or more time points were +mapped to one, identified by their current acronym. Second, small parties were grouped as +“not relevant”, “foreign party”, etc. and not further analysed due to their low shares. These +mappings were created by the domain expert in our team. + +5c. Third, this still left many politicians who belonged to different parties at different times in +their career. At first sight, this may appear to be an instance of the “valid time” problem of +temporal databases that can be handled by all state-of-the-art databases. Valid time is the +time period during which a database fact is valid in the modelled reality. In principle, a tool +that has identified an actor in a media text42 could look up what party P the actor belonged to +at the relevant time T and regard them as a representative of P in that context. However, +what is that period when it comes to assessing a text? First, which time period(s) should +count: when the article was written, what it describes, when it was read? Second, how +should the period that the text talks about be recognised and delimited? Third, is the (factual +or perceived) “identity” of an actor not shaped by their whole history? Therefore, rather than +attempting to identify and process a valid time from the database facts, the Diversity +Searcher implicitly assumes the perspective of “looking from the now” and thereby also +imitate a human user. + +But what would that imitated human user “see”? Looking from the now, the politician may be +perceived as giving visibility to only one of their parties or to all of them. If they give visibility +to all of them, they may do so equally, or one of their affiliations may “eclipse” the others. It +could even be the case that these historical changes have focussed the attention only on + +between the ontologies, their data quality, and technical restrictions on maximal number of results. +The limitations of these pragmatic choices are discussed in Section 6. +The queries can be found at https://tinyurl.com/ys2bnyy7,https://tinyurl.com/ys2bnyy7, +https://tinyurl.com/4ejp5jne, https://tinyurl.com/33fej4aw, https://tinyurl.com/2a3ph5bt (English +DBpedia); https://w.wiki/56Qc, https://w.wiki/53Lb, https://w.wiki/53ap, https://w.wiki/53gw +(Wikidata); https://tinyurl.com/2bjxp2f7, https://tinyurl.com/y5fzpu6p, https://tinyurl.com/56xkpy6m, +https://tinyurl.com/38krrb57https://tinyurl.com/38krrb57 (Dutch DBpedia). Dates of Access +(respectively): 27.05.2022, 27.05.2022, 12.04.2022, 27.05.2022, 12.04.2022, 13.04.2022, +12.04.2022, 13.04.2022, 26.04.2022, 13.04.2022, 12.04.2022, 12.04.2022. Python script used as a +workaround for the query limitations are provided in an additional document. +41 We used Wikipedia as knowledge source to determine these cases. +42 See Section 2 for a brief description of how Diversity Searcher identifies and processes actors. + + +12 of 20 + +that person and obfuscated that they also belonged to a party. Which of these apply for a +given politician and a given time point, is likely to depend heavily on the politician, the time +point, the political actions and narratives created by and around the parties43, the user, and +probably other factors. In the absence of the psychological empirical work needed to +determine which of these cases apply, we opted for delimiting the options by bounds on the +representation: +A party is made visible at time T at least by all those politicians who, over their whole career, +were only in this party44. This number defines a lower bound its visibility at T, because it +assumes that anyone who was in multiple parties does not confer any visibility on any of +them. The party is made visible at most by all those politicians who were ever in it at any +time in their career. This number defines an upper bound on the party’s visibility at T, +because it assumes that anyone who was in multiple parties confers full visibility on all of +them. All settings in which different multi-party politicians confer more or less visibility to their +various parties, lie between these bounds. + +These bounds are then compared to the baseline shares. If the lower bound is above the +baseline, the party is certainly over-represented. (Ex.: a party has 20% of the vote, but 30% +of all politicians in the ontology are pure-bred members of this party, and there may be +additional others who were in it at some point.) If the upper bound is below the baseline, it is +certainly under-represented. (Ex.: a party has 20% of the vote, but 10% of all politicians in +the ontology were in that party at all … and maybe in others too.) When the baseline lies +between lower and upper bound, no clear conclusion can be drawn about over- or under- +representation. + +Step 5 involved post-processing and querying in a MySQL relational database into which the +CSV files generated in Step 4 were imported. + +6. Studying possible over-/under-representations and their trends over time. The +analysis proceeded descriptively and via the visualisations shown in the following section. In +principle, we should clarify “higher” and “lower” in statistical terms (both in terms of statistical +significance and effect size). However, given the small number of actors and an as-yet- +absent knowledge of distributions, we limit ourselves to descriptive statistics. +3.3.3 Results and interpretation +The results are summarised in Figure 1. The figure highlights the trends in the data and the +contrast between the knowledge sources, in terms of an aligned sorting of parties from left to +right and curves lying consistently above or below the baseline over time. Details can be +seen in the higher-resolution/interactive online versions.45 Bold lines above thin lines indicate +an over-representation, bold lines below thin lines indicate an under-representation of that +party and in this sense also of the political alignment it is part of. The English-language +DBpedia has less than 20 actors up until 2005, which makes the over- and under- +representation hard to interpret. The number of politicians represented in the Dutch- + +43 such as rapprochements between right-wing and extreme-right parties that lead to the boundaries +between them becoming blurred +44 We disregard memberships in a “not relevant” minor party with negligible votes. +45 http://www.berendt.de/DIAMOND/, Retrieved: 27.12.2022. + + +13 of 20 + +language DBpedia fell sharply after the last regional and national elections in 2019, making +also this graph harder to interpret relative to the others. + +These results not only confirm our first informal observation of over-representation of right- +wing parties (especially the N-VA) in the English-language DBpedia, with a trend growing +over time. (During these years, the N-VA’s share of the popular vote increased, but the +DBpedia growth clearly exceeds the baseline growth.) Different biases seem to occur in the +Dutch-language DBpedia: although on the whole comparatively similar to the baseline, this +ontology seems to over-represent the main centrist party (CD&V). Wikidata, in contrast, +gives a rather accurate picture of party shares in the national parliament. The French- +language Walloon parties are (understandably, given the language focus) under-represented +in the Dutch-language DBpedia. Both the overrepresentation of rightist and centrist parties in +media coverage have been identified in earlier international research, such as a centrist bias +in the media coverage of the UK elections of 201746 and the right-wing overrepresentation in +social media, despite the cries of censorship in the United States.47 + +INSERT FIGURE 1 ABOUT HERE + +The seeming over-representation of the political centre in the Dutch-language DBpedia may +be an artefact of the language: The Dutch DBpedia focuses more on Flemish actors than on +the French-speaking Walloon (or Brussels or German Community) actors, and an informal +inspection of politicians’ pages in this ontology indicated that many regional and local +political actors are represented. Therefore, Figure 2 maps the shares of seats in the Flemish +parliament (which was first elected directly in 1995) as the baseline and is otherwise +analogous to Figure 1. The set of parties is a subset of that of Figure 1, since only the +Flemish parties can be elected into the Flemish parliament (while the national parliament +also contains representatives from the Walloon, Brussels, and German-Community parties). +Figure 2 suggests that the representation of the parties between extreme left and centre- +right in the Dutch-language DBpedia actually mirrors these parties’ shares in the regional +parliament rather closely; while the right and extreme right tend to be under-represented +especially in the later years, when they were highly successful especially in the Flemish +elections. + +INSERT FIGURE 2 ABOUT HERE + + +46 Deacon, David; Downey, John; Smith, David, Stanyer, James and Wring, Dominic. National News +Media Coverage of the 2017 election. Centre for Research in Communication and Culture, +Loughborough University Report 4: 5 May – 7 June 2017 Centre for Research in Communication and +Culture, Loughborough University Online: +47Scott, Mark: Despite cries of censorship, conservatives dominate social media, POLITICO, +26.10.2020, , Retrieved: 28.08.2022. + + + +14 of 20 + +Given the low shares of small-party representatives both in the baselines and in the +ontologies and given that our focus is on Flemish parties, we do not investigate the small or +the Walloon parties’ representations further. +3.3.4 Implications for the Diversity Searcher and future work +The political-science literature has hinted at the political right being skilled at using social +media before, and this appears to be an international phenomenon. As regards the “over- +representation of the centre”, this could be an artefact of the Dutch-language resource’s +focus on Dutch-language (Flemish) politicians and on majorities on the regional level, an +interpretation that is supported by the disappearance of the over-representation when +comparing with the baseline of the Flemish parliament. + +Why do such biases matter in the context of the Diversity Searcher? The tool can only +compute diversity between entities that it recognises. Imagine (a) a well-balanced text with +(say) several left-wing actors and several actors from one right-wing party, and (b) a text that +only contains actors from the right-wing and extreme-right parties. Assuming that all else is +equal between these texts, the “ground truth” would be that (a) is more diverse than (b). +However, when sourced by an ontology with a bias to the right, the tool will only (or mostly) +recognise actors from the one right-wing party in (a) and most actors from the two parties in +(b), which will lead to a higher computed diversity score for text (b) and a lower score – +maybe even zero – for text (a). This result is related to the artefact of ontological knowledge +(also observed in humans) that one perceives more distinctions and diversity in areas that +one knows well. + +The case study also illustrated the non-trivial and multiple dependencies between diversity, +fairness and biases. The feature “party” is one of the features from which the Stirling +disparity between recognised entities is computed. As a result, a text’s calculated diversity is +influenced by this feature. In general, diversity increases with the balance of the different +values of this feature (i.e., different parties). However, if these values’ shares diverge too +much from the baseline, the knowledge source is biased – and this in turn may bias the +calculation of the diversity of the text, introducing unfairness. + +Wikidata, by contrast, mirrors the representation of parties in the Belgian Parliament quite +accurately. In future work, data quality should be investigated also for other relevant +domains (such as other countries’ party systems or other socio-politically relevant topics), in +order to provide an evidence-based reason to choose one ontology over another for use in a +tool such as the Diversity Searcher. +4 Limitations, conclusions and outlook +As Leorke et al. argue, the shifts towards digital economies and smart cities resulted in a +change in the societal role of public libraries.48 The latter are now increasingly considered +and expected to function as “hubs” or “platforms” that “link people to information, services + +48 Leorke, Dale; Wyatt, Danielle; McQuire, Scott: “More than just a library”: Public libraries in the +‘smart city’, in: City, Culture and Society 15, 12.2018, S. 37–44. Online: +. + + +15 of 20 + +and to each other”.49 Moreover, public libraries also have the task of bridging digital divides +by facilitating access to digital resources and training the public in digital literacy. This +renewed societal role of the public libraries within the digital economy makes their increasing +usage of automated content analysis more problematic. Automated content analysis +software has the potential to provide insight into large digital corpora quickly, making them +valuable additions to research and critical thinking in a library setting. However, as illustrated +by the case study in this article, such tools are prone to reproducing bias in the upstream +components, even when they are designed to alleviate biases. Working with data requires +knowledge and recognition that data are not neutral and that they can be used to maintain +an unequal status quo. Thus, data are part of the problem but can also be part of the +solution if we maintain a number of principles: one is to situate the data in a historical and +social context. Library scientists can help create this awareness and a critical and informed +attitude at the user end. + +One limitation of the study was the inability to use the exact same SPARQL queries for each +database due to Wikidata, English and Dutch versions of DBpedia all having differences in +their ontologies and data quality. For some cases we found that using the same query texts +or analogous properties favours one database while bringing an empty or limited result for +another despite data being available.50 As our research and tool development focuses on +media research and library contexts, we prioritised acquiring the relevant data, even if it +meantmeans using different SPARQL queries tailored for each database. + +In future work, we plan to develop our analysis of biases in the underlying ontologies further. +This includes (a) combinations of several knowledge sources (rather than exclusive choices +between them), which requires further design decisions, especially in cases when there is +conflicting information in two sources; and (b) an investigation of Wikipedia’s edit history to +complement the “view from now” by “recreating the knowledge at time point X in the past”. +We also aim at further developing the understanding (including formalisation) of the +relationships between diversity, bias and fairness. + +In general, our findings indicate that as developers of tools, we need to monitor the potential +biases in our knowledge sources and study how they may influence the “downstream-task” +results (such as the calculation and presentation of diversity) and also user perception. Apart +from this, the authoring processes that lead to such over-representations are a highly +interesting topic in its own right. In future work, we will study both questions and also relate +them to our work on bias in large language models.51 + +It should not be forgotten that these numerical considerations can be exacerbated by the +inability of automated tools to recognise and understand context. This is another reason to +treat the numerical and categorical results as a starting point for deeper text analysis and +involve users – whether it is an individual citizen, researcher, organisation representing a +particular target group or a journalist looking for a new angle for a news story – in sense- + +49 Ibid. +50For example, the query for entities of type ‘political party’ and with country ‘Belgium’, which we used +for English DBpedia and Wikidata, returned no results for Dutch DBpedia even though the entities +exist. As a workaround, we queried the Dutch DBpedia for entities that appear as the party property of +another entity, and that have country ‘Belgium’. +51 Delobelle, Tokpo, Calders, u. a.: Measuring Fairness with Biased Rulers, 2022 + + +16 of 20 + +making by means of the interactive interface of the Diversity Searcher. This idea can and +should be extended. For future work, a two-pronged strategy is recommended: (a) identifying +and using the best-suited ontology for a given task and at the same time (b) making its +properties and shortcomings transparent to users so as to keep users aware of challenges +associated with the (and any) dataset. +Acknowledgements +We thank the Fonds Wetenschappelijk Onderzoek – Vlaanderen (FWO) for funding +DIAMOND under project code S008817N. +References +About DBpedia, DBpedia, (n.d.)., , Retrieved: 28.08.2022. + +About DIAMOND, 07.01.2018, , +Retrieved: 27.08.2022. + +Abián, D.; Guerra, F.; Martínez-Romanos, J. u. a.: Wikidata and DBpedia: A Comparative +Study, in: Szymański, Julian; Velegrakis, Yannis (Eds.): Semantic Keyword-Based Search +on Structured Data Sources, Bd. 10546, Cham 2018 (Lecture Notes in Computer Science), +S. 142–154. Online: , Retrieved: +25.08.2022. + +Amrapali Zaveri, Anisa Rula, Andrea Maurino, Ricardo Pietrobon, Jens Lehmann, Sören +Auer: Quality assessment for Linked Data: A Survey. Semantic Web 7(1): 63-93 (2016) + +Auer, Sören; Bizer, Christian; Kobilarov, Georgi u. a.: DBpedia: A Nucleus for a Web of +Open Data, in: Aberer, Karl; Choi, Key-Sun; Noy, Natasha u. a. (Eds.): The Semantic Web, +Bd. 4825, Berlin, Heidelberg 2007 (Lecture Notes in Computer Science), S. 722–735. +Online: + +Berendt, Bettina; Karadeniz, Özgür; Mertens, Stefan u. a.: Fairness beyond “equal”: The +Diversity Searcher as a Tool to Detect and Enhance the Representation of Socio-political +Actors in News Media, in: Companion Proceedings of the Web Conference 2021, Ljubljana +Slovenia 2021, S. 202–212. Online: , Retrieved: +25.08.2022. +Bourdieu, Pierre: The Social Space and the Genesis of Groups. Theory and Society 14, 6. +1985. 723-744. +Deacon, David; Downey, John; Smith, David, Stanyer, James and Wring, Dominic. National +News Media Coverage of the 2017 election. Centre for Research in Communication and +Culture, Loughborough University Report 4: 5 May – 7 June 2017 Centre for Research in +Communication and Culture, Loughborough University Online: + + + +17 of 20 + +Delobelle, Pieter; Tokpo, Ewoenam; Calders, Toon u. a.: Measuring Fairness with Biased +Rulers: A Comparative Study on Bias Metrics for Pre-trained Language Models, in: +Proceedings of the 2022 Conference of the North American Chapter of the Association for +Computational Linguistics: Human Language Technologies, Seattle, United States 2022, S. +1693–1706. Online: , Retrieved: +25.08.2022. + +Färber, Michael, Frederic Bartscherer, Carsten Menne, and Achim Rettinger. ‘Linked Data +Quality of DBpedia, Freebase, OpenCyc, Wikidata, and YAGO’. Edited by Amrapali Zaveri, +Dimitris Kontokostas, Sebastian Hellmann, Jürgen Umbrich, Amrapali Zaveri, Dimitris +Kontokostas, Sebastian Hellmann, and Jürgen Umbrich. 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Online: +, Retrieved: 29.08.2022. + +Ranaivoson, Heritiana: Measuring cultural diversity with the Stirling model, in: New +Techniques and Technologies for Statistics, 2013, S. 10. Online: +. +Scott, Mark: Despite cries of censorship, conservatives dominate social media, POLITICO, +26.10.2020, , Retrieved: 28.08.2022. +Stirling, Andy: A general framework for analysing diversity in science, technology and +society, in: Journal of The Royal Society Interface 4 (15), 22.08.2007, S. 707–719. Online: +. + +Wikipedia-bijdragers: Kamer van volksvertegenwoordigers — Wikipedia, de vrije +encyclopedie, 2022. Online: +, Retrieved: 25.08.2022. + +Wikidata dashboards / Wikidata Datamodel Statements, Grafana, (n.d.), +, Retrieved: 01.03.2022. + +Wikidata: Statistics, Wikidata, (n.d.), , +Retrieved: 01.03.2022 + +Vrandečić, Denny; Krötzsch, Markus: Wikidata: a free collaborative knowledgebase. In: +Communications of the ACM 57 (10), 2014, pp. 78–85. Online: +. + + + + +19 of 20 + + +1990 +1996 +2000 +2005 +2011 +2015 +2020 + + + + + + + + + + + + + + + + + + +6 +8 +17 +19 +83 +52 +45 + + + + + + + +343 +410 +450 +502 +515 +355 +63 + + + + + + + +110 +127 +142 +156 +167 +162 +234 + +Figure 1. Visibility of parties in the English-language DBpedia (top), the Dutch-language DBpedia (middle) and Wikidata (bottom) from 1990- +2020. Parties are ordered by alignment category (extreme left, left, center-left, center, center-right, right, extreme right, other, unknown) and +within these categories by their current acronym. Bold lines or vertical spaces between bold lines are the shares in the knowledge source, +the thin lines are the closest-in-time national parliament shares of the vote. The numbers below the graphs are the numbers of unique actors +“active at the beginning of the year” in that knowledge source.. + + + +0,3 +2005 min (of 156) +0,25 +2005 max (ot 156) +0,2 +KVV 2003 +0,150,39 +0,3 +EN +2005 min (of 19) +0.25 +2005 mas (of 19] +0.2 +0,15 +M +α.10,3 +NL +0,5 +× (ot 355) +0.2 +0,15 +0.10,3 +wd +1990 nin[of130] +0,25 +=1990 max (of 130) +0,2 +0,15 +0.1 ENNL +2996 max (of 430)0,3 +N +n(of450) +0,2 +0,15 +0,1EN0,3 +NL +(of 63) +(of 63 +0,2 +0,150,3 +wd +2020 min (of 234) +0,25 +2020 max (ot 234 +0,2 +0,15 +0,10,38 +0,3 +wd +min (of 162) +0,25 +(of 162 +0.2 +0,15 +0.10,6 +EN +0,5 +0,40,35 +EN +1996 min (of8) +.2s +=1996 max (of 8 +0,2 +KVV 1995 +0,15 +41 +0,050,3 +NL +2005 min(of 502) +0,5 +=2005 max (of 502) +0,2 +A +0,15 +α.1 +0.050,3 +EN +2000 min (of 17) +0,25 +0.2 +199 +0,15 +0,050,3 +NL +1990 nin[of343] +=1990 max (of 343) +0,2 +0,1NL +min(of 515) +max (ot 515) +0,2 +0,15 +0,1wd +min (of 167) +0,25 +2011 max (ot 167 +0.2 +0,15 +0,10,3 +wd +2000 min (of142) +0.25 +2000 max (ot 142 +0.2 +0,15 +α.1 0,3 +wd +min(of 127) +0,25 +0.2 +0,15 +α.1 0,3 +EN ++1990 min(of6) +0,25 ++1990 max (of 6) +0,2 +KVV 1987 +0,15 +0.1 +,05 +20 of 20 + + +1996 +2000 +2005 +2011 +2015 +2020 + + + + + + +410 +450 +502 +515 +355 +63 + +Figure 2. Visibility of Flemish parties in the Dutch-language DBpedia from 1996-2020. Parties are ordered by alignment category (extreme +left, left, centre-left, centre, centre-right, right, extreme right, other, unknown) and within these categories by their current acronym. Bold lines +or vertical spaces between bold lines are the shares in the knowledge source, the thin lines are the closest-in-time Flemish parliament +shares of the vote. The numbers below the graphs are the numbers of unique actors “active at the beginning of the year” in that knowledge +source.. + + + + +0,35 +NL +2005 min (of 502) +03 ++=2005 max (of 502) +0,25 +VP 2004 +0,2 +0,15 +0,1 +0,05 +0 +PVDA +Groen +Vooruit +CD&V +Open VLD +NVA +Vlaams +other/not unknown +Belang +relevant0,35 +NL ++2011 min (of 515) +0,3 ++=2011 max (of 515) +0,25 +VP 2009 +0,2 +0,15 +0.1 +0,05 +0 +PVDA +Groen +Voorut +CD&V +Open VLD +NWA +Vlaams +cther/net unknown +Belang +relevant0,4 +0,35 +NL ++2015 min (of 355) ++2015 max (of 355) +0,3 +VP 2014 +0.25 +0.2 +0,15 +0,1 +0,05 +0 +PVDA +Groen +Voorut +CD&V +Open VLD +NWA +Vlaams +other/not unknown +Belang +relevant0,35 +NL +2020 min (of 63) +0,3 ++=2020 max (of 63) +0,25 +0,2 +0,15 +0,1 +0,05 +PVDA +Groen +Vooruit +CD&V +OpenVLD +NVA +Vlaams other/not unknown +telang +relevant0,35 +NL +1996 min (of 410) +0,3 ++1996 max (of 410) +0,25 +VP 1995 +0,2 +0,15 +0,3 +0,05 +PVDA +Groen +Vooruit +CD&V +Open VLD +NVA +Belang +relevant0,35 +NL ++2000 min (of 450) +0,3 ++=2000 max (of 450) +0,25 +VP 1999 +0,2 +0,15 +0,1 +0,05 +o +PVDA +Groen +Vooruit +CD&V +Open VLD +NVA +Vlaams +Belang +other/not unknown +relevant \ No newline at end of file diff --git a/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/load_file.txt b/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ba4148d93f380c2dead5b1a0e6d34535973681e --- /dev/null +++ b/h9AyT4oBgHgl3EQfxvlP/content/tmp_files/load_file.txt @@ -0,0 +1,910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf,len=909 +page_content='1 of 20 Political representation bias in DBpedia and Wikidata as a challenge for downstream processing Özgür Karadeniz1 *, Bettina Berendt2,3,1 *, Sercan Kıyak1, Stefan Mertens1, Leen d’Haenens1 1 KU Leuven, Belgium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2 TU Berlin, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 3 Weizenbaum Institute, Germany * corresponding authors {ozgur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content='karadeniz|bettina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content='berendt}@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content='be Abstract: Diversity Searcher is a tool originally developed to help analyse diversity in news media texts, relying on a form of automated content analysis and thus rests on prior assumptions and depends on certain design choices related to diversity and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' One such design choice is the external knowledge source(s) used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' In this article, we discuss implications that these sources can have on the results of content analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' We compare two data sources that Diversity Searcher has worked with – DBpedia and Wikidata – with respect to their ontological coverage and diversity, and describe implications for the resulting analyses of text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' We describe a case study of the relative over- or under- representation of Belgian political parties between 1990 and 2020 in the English-language DBpedia, the Dutch-language DBpedia, and Wikidata, and highlight the many decisions needed with regard to the design of this data analysis and the assumptions behind it, as well as implications from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' In particular, we came across a staggering over- representation of the political right in the English-language DBpedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2 of 20 1 Introduction Originally developed as a tool to help analyse diversity in news media texts, Diversity Searcher is now ready to be used by the general public and in the context of public libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' The Diversity Searcher, presented in section 2, relies on a form of automated content analysis and thus rests on prior assumptions 1 and depends on certain design choices related to diversity and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2 Then in Section 3, we study representational biases in the underlying data sources and what they could imply for the extent of diversity that the tool can identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' We analyse the two alternatives we are currently working with – DBpedia and Wikidata – with respect to their ontological coverage and diversity, and describe implications for the resulting analyses of text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Specifically, we describe a case study that examines the relative over- or under-representation of Belgian political parties between 1990 and 2020 and highlights the many research-design decisions needed for this data analysis and the assumptions behind it, as well as implications from the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' In particular, we came across a staggering over-representation of the political right in the English-language DBpedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' This result presents a fascinating challenge for the design of data- and AI-powered tools such as Diversity Searcher for public libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Through its base functionality, but maybe even more so through transparency regarding findings such as those in our case study, the tool can help foster fairness and diversity and create awareness among news users that multi-voiced news is necessary to form an accurate and fair view of the reality being reported in a news story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2 The Diversity Searcher as a text analysis tool The Diversity Searcher is a semi-automated text analysis and knowledge enrichment tool that is designed to present information to the user about the degree of diversity in news media texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content='3 The tool was developed in the interdisciplinary project DIAMOND (Diversity and Information Media: New Tools for a Multifaceted Public Debate)4 to be used by media professionals and media consumers, and to be integrated with iCandid,5 a research infrastructure project offering integrated access several (social)media resources to researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Over the course of the project, Diversity Searcher has been further developed to be a tool to be used in libraries as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' The tool analyses an uploaded text or collection of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' It extracts actors (persons, institutions, and geopolitical entities) and offers additional information and interaction 1 See for example Kitchin, Rob: The Data Revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Big Data, Open Data, Data Infrastructures & Their Consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' London: Sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 2 Berendt, Bettina;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Karadeniz, Özgür;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Mertens, Stefan u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=': Fairness beyond “equal”: The Diversity Searcher as a Tool to Detect and Enhance the Representation of Socio-political Actors in News Media, in: Companion Proceedings of the Web Conference 2021, Ljubljana Slovenia 2021, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' 202– 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9AyT4oBgHgl3EQfxvlP/content/2301.00671v1.pdf'} +page_content=' Online: +#include + +#include + +#include +"ns3/core-module.h" +#include +"ns3/tnternet-nodule.h" +#include +"ns3/network-module.h" +#include +"ns3/applications-module.h" +#include +"ns3/mobility-nodule.h" +"ns3/mesh-module.h" +#include +#include +"ns3/mesh-helper.h" +#include +"ns3/yans-wtfi-helper.h" +using nanespace ns3; +NS_LOG_COMPONENT_DEFINE ("TestMeshscript"); +class MeshTest +A +public: +MeshTest (); +? +void Configure (int argc, char * argv); +int Run (); +private: +a +int +n_xsize; +int +n_ysize; +double +m_step; +double +m_randomstart; +double +m_totalTine; +double +m_packetInterval; +uint16_t +n_packetsize; +uint32_t +n_nifaces; +bool +n_chan; +bool +std:istring m_stack; +m_ascti; +std::string m_root; +NodeContainer nodes; +NetDeviceContainer +meshDevices; +Ipv4Interfacecontatner interfaces; +MeshHelper nesh; +private: +void CreateNodes (); +void InstallInternetstack,(); +votd InstallAppltcation (); +void Report (); +MeshTest: :MeshTest () +n_xsize (3), +n_ysize,(3) +n_step(100.0), +n_randomstart. (e.1), +n_totalTime(100.0)) +n_packetinterval,(e.1), +: +n_packetsize (1024), +nIfaces (i), + 8 apim qel++ +Ln 1, Col 1 +INSTue16:54 +aodv.cc +#include +#include + +ctostream> +#include +"ns3/core.nodule.h" +#include +"ns3/network-module.h" +#Include +winclude +"ns3/nobiltty-nodule.h" +rinclude +"ns3/aodv-module.h" +rincLude +"ns3/olsr-module.h" +tinclude +"ns3/dsdv-module.h" +#include +"ns3/dsr-nodule.h" +winclude +#include +"ns3/ocb-wtft-nac.h' +finclude +"ns3/wifi-s0211p-helper.h +incLue +"ns3/wave-mac-helper.h" +#include +"ns3/flow-monitor-module.h" +winclude +"ns3/config-store-module.h" +A +#include +"ns3/1nteger.h" +winclude +"ns3/wave-bsm-helper.h" +#nclude +"ns3/yans-wtfi-helper.h" +"ns3/wave-helper.h +winclude +ustng nanespace ns3; +a +ustng nanespace dsr; +NS_LOG_COMPONENT_DEFINE("vanet-routtng-compare"); +class:Routingstats +publie: +Routingstats (); +uint32_t GetRxBytes (); +uint32_t GetcumulativeRxBytes (); +uint32_t GetRxPkts (); +uint32_t GetCumulattveRxPkts (); +void IncRxBytes (uint32_t rxBytes); +void IncRxpkts (); +void SetRxBytes(utnt32_trxBytes); +void SetRxPkts (uint32_t rxPkts): +uint32_t GetTxBytes(); +uint32_t GetcunulativeTxBytes (); +uint32_t GetTxPkts (); +uint32 t GetcumulativeTxPkts O): ++ +Tabwidth:8 +Ln1,Col1 +INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 47 + +3. DSR +4. DSDV + + +Activiti +Tue 16:53 +dsr.cc +winclude + +#include +"ns3/core-module.h" +#include +"ns3/network-module.h" +#include +"ns3/applications-module.h" +#include +"ns3/mobility-nodule.h" +#include +"ns3/config-store-module.h" +#include +"ns3/internet-nodule.h" +#include +"ns3/dsr-nodule.h" +#include +"ns3/yans-wifi-helper.h" +using nanespace ns3; +NS_LOG_COMPONENT_DEFINE ("DsTTest"); +main (int argc, char *argv[)) +int +Users may find it convenient to turn on explicit debugging +l for selected modules; the below lines suggest how to do this +? +#ife +LogConponentEnable ("Ipv4L3Protocol", LOG_LEVEL_ALL); +a +LogConponentEnable +("UdpL4Protocol", +LOG_LEVEL_ALL); +LogConponentEnable +("udpSocketImpl", LOG, LEVEL_ALL); +LogConponentEnable +("NetDevice", LOG_LEVEL_ALL); +LogconponentEnable ("Ipv4EndPointDemux", LoG_LEVEL_ALL); +#endif +#ife +LogcomponentEnable ("Dsropttons", LoG_LEVEL_ALL); +LogconponentEnable +("DsrHelper", +", LOG,LEVEL_ALL) +LogConponentEnable +（"DsroptionHeader +("DsrRouting",LOG_LEVEL_ALL); +LogConponentEnable +LOG_LEVEL_ALL); +LogConponentEnable ("DsrFsHeader" +LOG_LEVEL_ALL) +LogConponentEnable +("DsrGraReplyTable" +,LOG_LEVEL_ALL); +LogConponentEnable +LogConponentEnable +("DsrRouteCache" +，LOGLEVEL_ALL); +LogConponentEnable +("DsrMaintainBuffer",LOG_LEVEL_ALL); +LogConponentEnable +("DsrRreqTable",LOG_LEVEL_ALL); +LogConponentEnable +C"DsrErrorBuffer +LOG_LEVEL_ALL); +LogconponentEnable ("DsrNetworkQueue", LoG_LEVEL_ALL); +#endif +NS_LOG_INFO ("creating the nodes"); +// General parameters +uint32_t nwifis= 50; +uint32_t nsinks = 10; +double TotalTime = 608.0; +double dataTime - 508.8; +uint32_t packetsize = 64; +double +ppers =1 +: +double datastart - 108.0; +// start sending data at 100s ++ ++ +Tab width: 8 +Ln 1, Col 1 +INSActivitie +Tue 16:53 +#include +#include +'ns3/core-module.h" +#include +"ns3/network-nodule.h* +#include +"ns3/applications-module.h" +#include +"ns3/nobiltty-module.h" +#include +'ns3/config-store-module.h" +#include +"ns3/internet-module.h" +#include +"ns3/dsr-module.h" +#include +"ns3/yans-wifi-helper.h" +using namespace ns3; +NS_LOG_COMPONENT_DEFINE( +("DsrTest"); +int +main (tnt argc, char *argv[]) +A +"l users may find it conventent, to turn on explict debugging. +#if8 +LogComponentEnable ("Ipv4L3Protocol", LOG_LEVEL_ALL); +a +LogConponentEnable +("UdpL4Protocol",LOG_LEVEL_ALL); +LogComponentEnable +("udpSocketInpl", +LOG_LEVEL_ALL); +LogComponentEnable +("NetDevice", +，LOG_LEVEL_ALL); +LogConponentEnable ("Ipv4EndPotntDemux", LOG_LEVEL_ALL); +#endif +#if8 +LogComponentEnable ("Dsroptlons",LoG_LEVEL_ALL); +LogComponentEnable +("DsrHelper", LOG_LEVEL_ALL): +LogComponentEnable +("DsrRouting",LoG_LEVEL_ALL); +LogComponentEnable +("DsropttonHeader +LOG_LEVEL_ALL); +LogConponentEnable +("DsrFsHeader", LOG_LEVEL_ALL); +LogComponentEnable +("DsrGraReplyTable",LOG_LEVEL_ALL); +LogConponentEnable +（"DsrSendBuffer", +LOG_LEVEL_ALL); +LogComponentEnable ("DsrRouteCache", LOG_LEVEL_ALL); +LogConponentEnable +("DsrMaintainBuffer",LOG_LEVEL_ALL); +LogComponentEnable ("DsrRreqTable",LOG_LEVEL_ALL); +LogComponentEnable +("DsrErrorBuffer", +，LOG_LEVEL_ALL); +LogConponentEnable ("DsrNetworkQueue", LoG_LEVEL_ALL); +#endif +NS_LOG_INFO ("creating the nodes"); +I/ General paraneters +uint32_t nwifis = 50; +uint32_t nsinks = 10; +double TotalTime = 600.8; +double dataTime = 5e8.0; +double ppers = 1: +utnt32_t packetsize = 64; +double datastart = 180.8; // start sending data at 180s +8m ql+ +Ln 1, Col1 +INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 48 + + +4.1 DSDV ROUTING TABLE + +5. SUMO +5.1 OSM MAP OF SRINAGAR CITY. + + +Activities + Text Editor +Tue 16:54 +dsdv-rtable.h +Openv +Save +100x +#ifndef DSDV_RTABLE_H +#define DSDV_RTABLE_H +#include +#include +#include +#include "ns3/ipv4.h" +0 +#include "ns3/ipv4-route.h" +#include +e "ns3/tiner.h" +#include +e 'ns3/net-device.h" +#include +e "ns3/output-strean-wrapper.h" +nanespace ns3 ( +namespace dsdv [ + enun RouteFlags +VALID = B, +II !< VALID +A +INVALID = 1, +1/ !< INVALID +1; +2 +class RoutingTableEntry +public: +a +RoutingTableEntry (Ptr dev = , Ipv4Address dst = Ipv4Address (), uint32_t seqNo = , +Ipv4InterfaceAddress iface = Ipv4InterfaceAddress (), uint32_t hops = §, Ipv4Address nextHop = Ipv4Address ()) +Tine lifetime = Sinulator::Now (), Time SettlingTine = Sinulator::Now (), bool changedEntries = false); +-RoutingTableEntry (); +Ipv4Address +GetDestination （)const +return m_ipv4Route->GetDestination (); +Ptr +GetRoute （) const +return m_ipv4Route; +void +SetRoute (Ptr route) +^_ipv4Route = route; +void +SetNextHop (Ipv4Address nextHop) +A_ipv4Route->SetGateway (nextHop); +Ipv4Address +GetNextHop () const +3 +return m_ipv4Route->GetGateway (); +1: +void +SetoutputDevice (Ptr device) +C/objcHeader~ Tab Width: 8- +Ln1, Col1 +INS1Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 49 + +5.2 POLY FILE CODE + + + + + + + + + + + + +Activities + Text Editor v +Tue 17:16 +osmPolyconvert.typ.xm! +Open +Save +=008 + + + + + + + + + + + + + + + + + +2 + + +a + + + + + + + + + + + + + +: +XML ~ Tab Width: 8 - +Ln 44, Col 16 +INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 50 + +CHAPTER 5 +RESULTS & COMPARISON. +1. MESH RESULTS +2. DSDV RESULT + + +Activities +Mon11:26 +mp-report-0.xml [Read-only] + + + +address +*00:00:00:80:00:01"> + +: + + +Ln 22, Col 1 +INSMon11:52 +run dsdv-routzng-protoco +./waf --run dsdv +/ns-allinone-3.29/ns-3.29/butld +a +> +:Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 51 + +3. AODV +4.DSR + + +Activities +wireshark +Mon 10:57 * +aodv-node-0-0.pcap +Eile Edit Vlew Go Capture Analyze Statistics Telephony Wireless Tools Help +Q +Apply adisplay filte ...ctrl./ +→ +*Expression... +No. +Source +Destination +ProtocolLength Info +0 +18.080000 +10.0.8.1 +10.255.255.255 +AODV +AODV +10.0.0.1 +10.255.255.255 +AODV +50.641000 +AODV +88RouteRequest,D:18.8.0.10,0:10.0.0.1Id=3Hcnt=eDSN=e0sN=3 +61.831136 +10.0.0.2 +10.255.255.255 +AODV +AODV +84 Route Reply, D: 10.@.8.2, 0: 10.8.3.2 Hcnt=8 DSN=0 Lifetime=2800 +国 +71.196000 +10.0.0.1 +10.255.255.255 +88 Route Request, +D:10.0.0.10,0: 10.0.0.1 Id=4 Hcnt=0 DSN=0 0SN=4 +81.206288 +10.0.0.2 +10.255.255.255 +AODV +91.922000 +10.0.0.1 +10.255.255.255 +88 Route Request, +88 Route Request, +D:10.8.0.10,0:10.0.8.1Id=5Hcnt=0DSN=00SN=5 +D:10.0.0.10,0:10.0.0.1 Id=4 Hcnt=1 DSN=00SN=4 +AODV +101.925288 +10.0.0.2 +10.255.255.255 +AODV +88RouteRequest,D:1e.8.0.10,0:10..8.1Id=5Hcnt=1DSN=0SN=5 +A +11 2.915806 +.255. +.255 +AODV +B4Route +122.922280 +10.0.0.2 +AODV +B4Route +133.921800 +10.0.0.1 +10 +.255.255.255 +AODV +B4Route +14 4.715999 +10.0.0.1 +10.255.255.255 +AODV +? +15 4.723288 +10.0.0.2 +10.255.255.255 +AODV +88RouteRequest,D:18.8.0.10,0:10.8.8.1Id=6Hcnt=1DsN=00SN=6 +Frame 1:88 bytes on wire (704 bits),88 bytes captured (7e4 bits) +IEEE 802.11 Data, Flags:........ +a +Logical-Link Control +Internet Protocol Version4,Src:1e.8.8.1,Dst:18.255.255.255 +User Datagram Protocol,Src Port:654,Dst Port:654 +Ad hoc on-demand Distance Vector Routing Protocol,Route Request, Dest IP: 1o.o.8.1o, orig IP:1o.o.o.1 +圆 +08 +0000 +ff +ff +80 +808001 +01 +00 +0800 +0020 +00 +Ba 00 +00 01 +0030 +eafft +02 8e 82 8e +28 08 80 +8128000 +00000001 +0a90g008 +00000088800001 +(. +8040 +00 00 00 01 +00000801 +8050 +: +07 +aodv-node-0-0.pcap +Profile: Default0 +Retry=8, MoreData=0 Duration/ID=ous, DA=ff:ff:ff:ff:ff:ff, +A=00:00:00:00:00:29 +h:12)ns3::WifiMacTrailer +101DsssRate11Mbps/NodeList/17/DeviceList/e/Sns3::WifiNetDevice/Phy/State/Rx0k +Retry=o,MoreData=o Duration/ID=ous,DA=ff:ff:ff:ff:ff:ff +A=00:00:00:08:00:29 +DS3: +SCP Default ECN Not-ECT ttl e id protocol 48 offset (bytes) e flags +ns3::WifiMacTrailer() +101DsssRate11ME +,MoreData=o Duration/ID=ous,DA=ff:ff:ff:ff:ffff, +A=00:00:00:00:00:29.BSSID=00:00:00:00:00:29, +SCP Default ECN Not-ECT ttl id protocol 48 offset (bytes) flags +none] length:48 10.1.1. +>10.1.1.255)ns3::DsrRouttng-eader +nextHeader: +length: 12) ns3: :WifiMacTrailer () +101DsssRate11Mbp +y=0, MoreData=8 Duration/ID=ous, DA=ff:ff:ff:ff:ff:ff +A=00:00:00:00:00:29,BSSID=00:00:00:00:00:29,FraqNunber=0 +DSCP Default ECN Not-ECT ttl e id 8 protocol 48 offset (bytes) flags +nonellength:4810.1.2 +10.1.1.255)ns3::DsrRoutingHeader +nextHeader: +17 messageType +ns3::WifiMacTrailer +101DsssRate11Mbp +Retry=0, MoreData=8 Duration/ID=ous, DA=ff:ff:ff:ff:ff:ff, +A=00:00:00:00:00:29 +FraqNunber=, SegNunber=) ns3::LlcSnapHeader +SCP Default ECN Not-ECT ttl 8 id 8 protocol 48 offset (bytes) e flags +a +nextHeader +ns3::WifiMacTrailer +5A=00:00:00:00:00:2 +ault ECN Not-ECT ttl id protocol 48 offset (bytes) flags +ns3::WifiMacTrailerC +101 DsssRate11Mbps/NodeList/1/DeviceList/e/Sns3::WifiNetDevice/Phy/State/Rxok ns3::Wi +Retry=0,MoreData=o Duration/ID=ous,DA=ff:ff:ff:ff:ff:ff. +A=00:00:00:00: +ult ECN Not-ECT ttl id protocol 48 offset (bytes) flags +nonelength:40 10.1.1.41>18.1.1.255)ns3::DsrRoutingHeader +ns3::WifiMacTrailer () +101DsssRate11Mbps/NodeList/3/DeviceList/e/Sns3::WifiNetDevice/Phy/State/Rx0k +Retry=o,MoreData= Duration/ID=ous,DA=ff:ff:ff:ff:ff:ff +国 +A=00:00:00:00:00:29.BSSID=00:00:00:00:00:29 +DSCP Default ECN Not-ECT ttl e id e protocol 48 offset (bytes) e flags +nonellength:40 +>10.1.1.255)ns3::bsrRouttng-eader +tHeade +h:12)ns3::WifiMacTrailer() +101DsssRate11Mbp +,MoreData=8Duration/ID=ous,DA=ff:ff:ff:ff:ff:f +Default ECN Not-ECT ttl 8 id 8 protocol 48 offset (bytes) flags +nonelength:40 +ns3::WifiMacTrailerC +101DsssRate11Mbp +Retry=0, MoreData= Duration/ID=ous, DA=ff:ff:ff:ff:ff:ff. +A=00:00:00:00:00:29 +Bs5tD=08:00:80:80:00:29.EragNunber=8. +ns3::-LcSnaD +SCP Default ECN Not-ECT ttl e id e protocol 48 offset (bytes) flags +nonel length: +ns3::WifiMacTrailerC +101DsssRate11ME +NodeList/4b/DeviceList/e/Sns3::witiNetDevlce/phv/State/Rx0kns3:: +, Retry=0, MoreData=8 Duration/I0=ous, DA=ff:ff:ff:ff:ff:ff, +oneengt +h:12)ns3::WifiMacTrailer（ +/DeviceList/o/Sns3::wutiNetDevice/phy/State/Rxokns +Retry=8,MoreData=o Duration/ID=ous,DA=ff:ff:ff:ff:ff:ff +A=00:00:0 +DS3: +SCP Default ECN Not-ECT ttl e id protocol 48 offset (bytes) e flags +none +h:12)ns3::WifiMacTrailer () +o,MoreData=8 Duration/ID=ous,DA=ff:ff:ff:ff:ff:ff +A=00:00:00:08:00: +Default ECN Not-ECT ttl id protocol 48 offset (bytes) flags +noneLength:4 +:12)ns3::WifiMacTrailer() +101DsssRate11ME +,MoreData=8Duration/ID=ous,DA=ff:ff:ff:ff:ff:f +SA=00:0000:00:00:29 +Default ECN Not-ECT ttl 8 id 8 protocol 48 offset (bytes) flags +nonelength:40 +10.1.1.255 +)ns3::DsrRoutingHeader +nextHeader: +17messageType: +ns3::WifiMacTrailer +101DsssRate11MbpsNodeList/23/DeviceList/e/Sns3::WifiNetDevice/Phy/State/Rx0kns3::WifiMacHeao +Retry=0, MoreData=8 Duration/ID=ous, DA=ff:ff:ff:ff:ff:ff, + (tos exe DscP Default ECN Not-ECT ttl 8 id protocol 48 offset (bytes) e flags +nonelength:4010.1.1.41>10.1.1.255)ns3::DsrRoutingHeader +CnextHeader: +sourceid: +255length:12)ns3::WifiMacTrailer +t 181.93 /NodeList/48/DeviceList/e/Sns3:WifiNetDevice/Phv/State/Tx DsssRate11Mbps ns3::WifiMacHeader (DATA ToDS=o,FronDS=, MoreFraq=§, Retry=8. MoreData= Duration/ID=8us. DA=ff:ff:ff:ff:ffff. +Plain Text Tab Width: 8 +Ln 7, Col 560 +INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 52 + +COMPARISON +1. SIMULATION DATA FOR DSDV. +2. SIMULATION DATA FOR DSR. +Table – 4 + + +Simulation Data For DSR +Simulation +Packet Sent +Packet +PDR +Packet +Recieved +Forwarding +Time +25 +12208 +976 +1250.82 +3233 +50 +24416 +1916 +1274.32 +6497 +75 +36622 +2835 +1291.78 +9783 +100 +48830 +3765 +1296.95 +13059 +125 +61036 +4693 +1300.58 +16336 +150 +73224 +5618 +1303.74 +19621 +175 +85450 +6550 +1304.58 +22898Simulation Data For DSDv +Simulation +Packet Sent +Packet +PDR +Packet +Recieved +Forwarding +Time +25 +12208 +125 +7145 +872 +50 +24416 +397 +6150.13 +1779 +75 +36622 +768日 +4768.49 +5585 +100 +48830 +1536 +3179.04 +9004 +125 +61063 +2501 +2440.46 +12241 +150 +73244 +3485 +2101.69 +15439 +175 +85450 +4520 +1890.49 +18606Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 53 + +3.SIMULATION DATA FOR AODV. +4. SIMULATION (AODV & DSDV) vs PDR(Packet Dessimation Rate). + + + +Simulation(AODV & DSDV) vs PDR +Surulation,TinelsPOF +20000 +1S.c0 +LR.0 +7.080 +16,000 +15,0006 +14,000 +E.000 +12,0000 +11,0000 +0,0000 +RCo3 +.0000 +2.000 +5.0000 +2,000 +30.000X +30.00 +40,000 +50,0000 +61,0000 +70,0000 +0,0000 +.0000 +L00,0000 +110,0000 +120,0000 +131,000 +150.0004AODv Simulation Data +PDR +Simulation Time +Packet Sent +Packet +Packet +Recieved +Forwarding +25 +12208 +713 +1712.70 +3506 +50 +24416 +1431 +1706.22 +7007 +75 +36622 +2156 +1698.61 +10501 +100 +48830 +2889 +1690.20 +13990 +125 +61063 +3618 +1687.01 +17473 +150 +73244 +4363 +1678.75 +20948 +175 +85450 +5106 +1672.85 +24427Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 54 + +5. AODV vs DSDV THROUGHPUT +6. AODV vs DSDV vs DSR THROUGHPUT. + + + +Average Throughput +10 +120 +140 +160Simulation Time vs Throughput +rrent_tie,inetrcexs +ragps +0.0 +dodrt-15o-throwgput, +28,000 +25,80 +24,00 +22,800 +20.00 +13,000 +5,00 +134,000 +12,000 +8,000 +6,000 +4,000 +2.000 +Trugot +20004000000000.00000.00010Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 55 + + +The above grahps and results depict that initaially with lesser number of nodes all the +protocols perform same with a slight edge to DSR protocol on the routes of Srinagar city +but later as the number of nodes increase AODV shows the greater output as compared to +others +Average Throughput for lesser number of nodes: DSR > AODV > DSDV. +Average Throughput for increased number of nodes: AODV > DSDV > DSR. +Since the number of nodes is large so second scenario fits the best for routing in Srinagar +city. + +PDR is higher for AODV is good for lesser number of nodes but it falls down as the +nodes increase and DSR takes up. +PDR for lesser number of nodes: AODV > DSR > DSDV +PDR for increased number of nodes: DSR > AODV > DSDV. + +End to end delay of DSR is least and that of DSDV is highest for lesser number of nodes +but when the number of nodes is increased the trend changes e2e delay gets decreased +for DSDV and AODV but increases for DSR. +E2e delay for lesser number of nodes: DSR < AODV < DSDV. +E2e delay for increased number of nodes: DSDV < AODV < DSR. + +Normalized Routing Load is initially zero but increases gradually with time giving less +values for DSDV than AODV and DSR. +Initial NRL is equal to zero for all the three protocols. +NRL after some time: DSDV < AODV < DSR. + +So from above comparison we can depict that AODV is having highest Throughput and +is average in other cases of comparison for the routes of srinagar, So based on above +comparison I would preffer AODV over DSDV and DSR. + +Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 56 + +CHAPTER 6 +CONCLUSION AND FUTURE WORK +A) Average Throughput Throughput indicate rate of communication per unit time. +Throughput in this experiment evaluated for AODV, DSDV and DSR for all these three +mobility models. The throughput (bytes per simulation time ) versus increasing number +of nodes of protocols by using column mobility of nodes in given environment. In this +AODV perform better than other protocols, but in lesser number of nodes performance of +all protocols is almost same. Throughput of protocols with respect increasing nodes +shows that throughput of AODV is better than DSDV, and DSR perform least. +Throughput by using random mobility model is much better than other two mobility +models. In this case all three protocols perform better but AODV is much better. +B) Packet delivery ratio +It is the ratio of data packets delivered to the destination to those generated by the source. +It is calculated by dividing the number of packet received by the destination through the +number of packet originated by the source. The packet delivery ratio of AODV is good +for lesser number of nodes but when nodes increase then PDR fall down. DSR is having +increased PDR than AODV and DSDV for greater number of packets. +C) End to end delay +It is the amount of time taken by packet to reach from one node to other. End to end delay +versus increasing number of nodes by using column mobility model. At lesser number of +nodes the e2e delay of DSR is at its least value but DSDV at its peak value, after +increasing number of nodes e2e of DSR start increasing but oppositely DSDV and +AODV start decreasing. DSR have higher e2e delay but DSDV have least. DSR is on its +lowest value and DSDV have its largest point. Generally, With the increasing number of +nodes the delay of these protocols gradually decrease. + + +Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 57 + +D) Normalized routing load +It is the metadata and network routing information sent by a source node to destination +node, which uses a portion the available bandwidth of a protocol. AODV have a higher +NRL but DSDV is on bottom. Initially value of NRL is zero, but with number of nodes +it’s gradually starts increasing. Value of NLR in DSDV is very lesser. In case of group +mobility NRL is high when routing protocol is DSR and less when DSDV. If random +way point mobility model is used it shows a less values for DSDV but high value for +DSR and AODV. +Srinagar city needs implementation of these protocols and also intelligent traffic system +with modification of some roads with better planning. Some junctions are normal with +current flow of traffic while others need some modification as well some extra roads are +needed. +FUTURE WORK +1. Protocols need modification and some new protocols are needed in general. +2. Stable and more actualised simulators are needed. +3. Implementation of these protocols in our traffic system is required. +4. Better Planning is necessity. +5. Upgraded roads and intelligent vehicles. + + + + + + + +Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 58 + +References +[1] +IEEE 1609. IEEE 1609 Family of Standards for for Wireless Access in Ve- +hicular Environments (WAVE), available from IEEE standards. + +[2] +IEEE 802.11. Information Technology - Telecommunications and Informa- +tion Exchange between Systems - Local and Metropolitan Area Networks - +Speci c Requirements - Part 11: Wireless (LAN) Medium Access Control +(MAC) and Physical Layer (PHY) Speci cations. ANSI/IEEE Std. 802.11, +ISO/IEC 8802-11, 1999. + +[3] +IEEE 802.11p. IEEE Draft Standard for Information Technology - +Telecommunications and information exchange between systems - Local and +metropolitan area networks - Speci c requirements - Part 11: Wireless LAN +Medium Access Control (MAC) and Physical Layer (PHY) speci cations +Amendment 6: Wireless Access in Vehicular Environments. 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In Future Com- +puter and Communication (ICFCC), 2010 2nd International Conference on, +volume 1, pages V1{183 {V1{187, May 2010. + + +Analysis and Design of VANET Protocols for Srinagar City | 2019 + + +Department of Information Technology, NIT Srinagar +Page 62 + +[39] +O. Tonguz, N. Wisitpongphan, F. Bai, P. Mudalige, and V. Sadekar. Broad- +casting in vanet. In 2007 Mobile Networking for Vehicular Environments, +pages 7 {12, May 2007. + +[40] +Y. Toor, P. Muhlethaler, and A. Laouiti. Vehicle ad hoc networks: ap- +plications and related technical issues. IEEE Comm. Surveys & Tutorials, +10(3):74 {88, Sep. 2008. + + + + + + + + + + + + + + + + + + diff --git a/j9E1T4oBgHgl3EQfgQRf/content/tmp_files/load_file.txt b/j9E1T4oBgHgl3EQfgQRf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b08984902ef8e1787a527b55223ba3baea69cce7 --- /dev/null +++ b/j9E1T4oBgHgl3EQfgQRf/content/tmp_files/load_file.txt @@ -0,0 +1,2029 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf,len=2028 +page_content='Project Report On Analysis and Design of VANET Protocols for Srinagar City Submitted in partial fulfilment of the requirements for the award of the degree of BACHELOR OF TECHONOLOGY IN INFORMATION TECHNOLOGY FURQAN YAQUB KHAN IT/03/15 Under the supervision of DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Shabir Ahmad Sofi Department of Information Technology National Institute of Technology Srinagar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' J&K June 2019 FRO TUTE OF SRINAGAR KASHMIR DEPARTMENT OF INFORMATION TECHNOLOGY NATIONAL INSTITUTE OF TECHNOLGY SRINAGAR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='J&K JUNE 2019 CERTIFICATE This is to certify that the project titled Analysis and Design of VANET Protocols for Srinagar City has been completed by Furqan Yaqub Khan (IT/03/15) under my supervision in partial fulfilment of the requirements for the award of the degree of Bachelor of Technology in Information Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It is also certified that the project has not been submitted or produced for the award of any other degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Shabir Ahmad Sofi Supervisor Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' of Information Technology NIT, Srinagar RO TUTE OF SRINAGAR KASHMIR STUDENTS DECLARATION We, hereby declare that the work, which is being presented in the project entitled Analysis and Design of VANET Protocols for Srinagar City in partial fulfilment of the requirements for the award of the degree of Bachelor of Technology in Information Technology in the session 2019, is an authentic record of our own work carried out under the supervision of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Shabir Ahmad Sofi, Department of Information Technology, National Institute of Technology, Srinagar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The matter embodied in this project has not been submitted by us for the award of any other degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Dated:13 06 2019 Name : Furqan Yaqub Khan Signature: furkaan RO TUTE OF SRINAGAR KASHMIR ACKNOWLEDGEMENT I would like to express my special thanks of gratitude to my guide Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Shabir Ahmad Sofi as well as our Head of Department (Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Arooj Nissar who gave me the golden opportunity to do this wonderful project on the topic Analysis and Design of VANET Protocols for Srinagar City, which also helped me in doing a lot of Research and i came to know about so many new things, I am really thankful to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Secondly I would also like to thank my parents and friends who helped me a lot in finalizing this project within the limited time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Furqan Yaqub Khan IT/03/15 8th Semester B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Tech, Information Technology NIT Srinagar I Abstract Vehicular ad-hoc network (VANET) is subclass of mobile ad-hoc network which is vehicle to vehicle and vehicle to infrastructure communication environment, where nodes involve themselves as servers and/or clients to exchange and share information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET have some unique characteristics like high dynamic topology, frequent disconnections, restricted topology etc, so it need special class of routing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To simulate the VANET scenarios we require two types of simulators, traffic simulator for generating traffic and network simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this project I created a sample scenario of VANET for AODV, DSDV, DSR routing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' I have used SUMO for generating traffic mobility files and NS-3 for testing performance of routing protocols on the mobility files created using Traffic simulator SUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' II Contents List of Contents……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. III List of Tables ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' IV List of Figures ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Protocols ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Literature Survey 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' SUMO: contributors and participants……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 and VANET protocols……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Theory 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Unicast, Multicast and Broadcast……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Proactive, Reactive and Hybrid protocols……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Categorization of next hop selection……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Types of Routing protocols……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Methodology 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 Simulation Modeling Methodology……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Code Section……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Result and Comparison 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Result……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Comparison……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Conclusion and Future work……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 56 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' References……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 58 III LIST OF FIGURES S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='No Topic Page No 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MESH ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AODV ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSR……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…… 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSDV……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSDV ROUTING TABLE……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…… 48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' OSM MAP OF SRINAGAR CITY……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…… 48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' POLY FILE CODE……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…… 49 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' RESULTS & COMPARISON……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 50 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MESH RESULT……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSDV RESULT……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AODV RESULT……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 51 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSR RESULT……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 51 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Simulation ( AODV & DSDV) ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 53 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Simulation Time vs Throughput……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 54 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Average Throughput……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='. 54 IV LIST OF TABLES S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='No Topic Page No 1) SUMO ( Contributors and Participants)……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 6 2) Simulation parameter Setup……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='… 45 3) Simulation data for DSDV……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 52 4) Simulation data for DSR……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 52 5) Simulation data for AODV……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='………' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='…… 53 V Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 1 CHAPTER 1 INTRODUCTION Today, Vehicular Ad-hoc Network (VANET) is one of the most emerging areas for the improvement of Intelligent Transportation System (ITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET is a special form of MANET, where Mobile Ad-hoc Network (MANETs) are self-configuring network of mobile nodes connected by wireless links, while, VANET are distributed and self- assembling communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A technology that uses moving vehicles as nodes to create a mobile network is termed as VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Here, node movement is restricted by factors like road course, encompassing traffic and traffic regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The primary goal of VANET is to provide road safety and other value added services such as email, audio/video sharing etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET From the last decade, mobile communication techniques have transformed the automotive industry by providing anytime anywhere communication between different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This ease of communication allows exchange of valuable information between devices just on the go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The seamless exchange of information on real time bases has turned out to become a new paradigm in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Correspondingly, the advances in the information technology and communication have easily supported the idea of communication between mobile devices [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Among these advancements, the concept of Vehicular Ad-hoc NETworks (VANET) came into limelight which has opened new possibilities to avail the use of safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET refers to a network created in an ad-hoc manner where different moving vehicles and other connecting devices come in contact over a wireless medium and exchange useful information to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A small network is created at the same moment with the vehicles and other devices behaving as nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Whatever information the nodes possess is transferred to all other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Similarly all the nodes after transferring their set of data receive the data being transmitted by other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' After accumulating all of such data, nodes then work to generate useful information out of the data and then again transmit the information to other devices [2][4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The communication between devices expands in such as way where Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 2 nodes are free to join and leave the network i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' it is an open network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The new vehicles being launched in the market are now coming with equipped on board sensors which make it easy for the vehicle to easily join and merge in the network and leverage the benefits of VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET is a variation of MANET (Mobile Ad-hoc NETwork).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MANET comprises of nodes which communicate without central network and where nodes are equipped with networking capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET on the other side has emerged as a challenging and more liable class or variation of MANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The freedom of nodes to enter or leave the network in VANET calls for different routing protocols than MANET .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This inter vehicle communication leads to passing and receiving of information so as to increase traffic efficiency, detect road conditions, decrease collisions, detect emergency situations and overall increase the efficiency of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET transfers the information to distant devices as well with the help of multi hops [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET can be characterized by following factors: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Dynamic topology- The speed and direction of vehicles changes constantly thereby resulting in high dynamic topology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Intermittent connectivity- Connectivity between devices changes very frequently like connection between two devices exchanging information can disconnect anytime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The reason behind frequent disconnection is high dynamic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mobility Patters: A large section of vehicles follow a certain patterns to move which is generally a function of traffic signals, speed limits, highways, streets, road conditions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' These patterns when observed help in the creation of routing protocols for VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Unlimited power and storage: It is assumed that the nodes in VANET are capable of possessing an unlimited amount of power as well as storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore the nodes are free to exchange the data without the foundations of power consumption or storage wastage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 3 On board sensors: VANET assumes that the nodes are seldom equipped with on board sensors which are capable of transmission of information to other devices or nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 NS3 a tool for simulating the real world network on one computer by writing scripts in C++ or Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Normally if we want to perform experiments, to see how our network works using various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' We don’t have required number of computers and routers for making different topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Even if we have these resources it is very expensive to build such a network for experiment purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' So to overcome these drawbacks we used NS3, which is a discrete event network simulator for Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 helps to create various virtual nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=', computers in real life) and with the help of various Helper classes it allows us to install devices, internet stacks, application, etc to our nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Using NS3 we can create Point To Point, Wireless, CSMA, etc connections between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Point To Point connection is same as a LAN connected between two computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Wireless connection is same as WiFi connection between various computers and routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' CSMA connection is same as bus topology between computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' After building connections we try to install NIC to every node to enable network connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When network cards are enabled in the devices, we add different parameters in the channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=', real world path used to send data) which are data-rate, packet size, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Now we use Application to generate traffic and send the packets using these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' PROTOCOLS Wireless network can be classified into infrastructure based and infrastructure less network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In the case of infrastructure based networks, Access point are used for communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' They act as routers for the nodes within their communication range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Whereas, in infrastructure less networks, also known as, ad hoc networks, nodes act as routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A Mobile ad hoc network (MANET) is a type of ad hoc network in which nodes can change locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 4 The routing protocols in MANET are broadly classified into three categories, namely, proactive protocols, reactive protocols, hybrid protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Proactive protocols, also known as table-driven protocols, maintain routing information in the routing table of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The proactive routing protocols are Destination-Sequenced Distance-Vector (DSDV) routing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The reactive protocols are Ad-hoc On- demand Distance Vector (AODV), Dynamic Source Routing (DSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Ad-hoc On-demand Distance Vector (AODV) AODV is a combination of on-demand and distance vector i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' hop-to-hop routing methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When a node needs to know a route to a specific destination it creates a ROUTE REQUEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Next the route request is forwarded by intermediate nodes which also create a reverse route for itself for destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When the request reaches a node with route to destination it creates again a REPLY which contains the number of hops that are require to reach the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' All nodes that participate in forwarding this reply to the source node create a forward route to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This route created from each node from source to destination is a hop-by-hop state and not the entire route as in source routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Destination Sequenced Distance Vector (DSDV): DSDV is a hop-by-hop distance vector routing protocol requiring each node to periodically broadcast routing updates based on the idea of classical Bellman-Ford Routing algorithm [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node maintains a routing table listing the “next hop” for each reachable destination, number of hops to reach destination and the sequence number assigned by destination node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The sequence number is used to distinguish stale routes from new ones and thus avoid loop formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The stations periodically transmit their routing tables to their immediate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A station also transmits its routing table if a significant change has occurred in its table from the last update sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' So, the update is both time-driven and event-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The routing table updates can be sent in two ways: a “full dump” or an “incremental”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Dynamic Source Routing(DSR): DSR is a simple and efficient protocol designed specifically for use in multiple wireless adhoc networks of mobile nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It allows nodes to dynamically discover a source route across multiple network hops to any destination in adhoc network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each data packet sent then carries in its header the complete ordered list of modes through which packet must pass, allowing packet routing to be a trivially loop free and avoiding the need for up-to-date routing information in the intermediate nodes through which the packet is forwarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' With the inclusion of this source route in the hearder of each data packet, other nodes forwarding or overhearing any of the packets may easily cache this route information for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 6 CHAPTER 2 LITERATURE SURVEY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' SUMO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Contributors and Participants Org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Name Topics / Contribution Christian Rössel Initial microsimulation core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' initial detectors implementation Peter Wagner Models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' organisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' spiritual lead Daniel Krajzewicz Everything Julia Ringel Traffic Light & WAUT Algorithms Eric Nicolay Everything Michael Behrisch Everything Yun-Pang Wang User Assignment Danilot Teta Boyom Vehicular Communication Model (removed from the source) Sascha Krieg Lena Kalleske Laura Bieker Tests,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Python scripts Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NIT Srinagar Page 7 Jakob Erdmann network import,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NETEDIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Andreas Gaubatz Maik Drozdzynski Uni Lübeck Axel Wegener TraCI initiator Thimor Bohn TraCI Friedemann Wesner TraCI Felix Brack Tino Morenz Christoph Sommer TraCI merge with Veins, Subscription Interface, Misc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' David Eckhoff TraCI, deterministic simulation behavior Falko Dressler TraCI Tobias Mayer Traffic model abstraction, IDM model port Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 8 HU Berlin Matthias Heppner Unittests Piotr Woznica ACTIVITYGEN Walter Bamberger Development of ACTIVITYGEN as a base for the evaluation of trust scenarios in VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The work is part of the project Fidens: Trust between cooperative systems featuring trusted probabilistic knowledge processing in vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' IIT Bombay, India Ashutosh Bajpai randomDepart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='py, a python script to generate the real traffic pattern by exponential Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Enrico Gueli TraCI4J Leontios Papaleontiou Sumo Traffic Modeller Karol Stosiek Documentation, network building Table 1 POLITECNICO DI TORINOUniwersytet WroclawskiAnalysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 and VANET Protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Chia-Chen Hung et al (2008), had analyzed and demonstrated ,the Intelligent Transportation System (ITS), a worldwide initiative program that utilized novel information and communication technology for transport infrastructure and vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Among extensive ITS components, efficient communication system was the most important role that connects numerous vehicles with roadside infrastructure and management center in the ITS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper a new Heterogeneous Vehicular Network (HVN) architecture and a mobility pattern aware routing(MPARP) for HVN was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' HVN integrates Wireless Metropolitan Area Network (WMAN) with VANET technology and reserves advantages of better coverage in WMAN and high data rate in VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Vehicles in HVN can communicate with each other and access Internet ubiquitously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Nisha Devi and Adiline T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=" Macriga (2010), had presented about today's communication industry that concentrates more on the live information transfer without altering the existing infrastructure and hence required a single convergence platform of all networks’ access." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The next generation systems support multimode, multi-access and reconfigurable devices to support inter-working of heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The network selection is user-centric and based on multiple QOS (Quality of Service) parameters like bandwidth, cost, security level, call drop probability etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=', to select appropriate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The proposed algorithm used a distance function to generate an ordered list of various access technologies called networks in a particular region according to multiple user preferences and level of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Further level of customization was done with user preference in terms of giving priority to few parameters and was implemented by weighted distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Benslimane Abderrahim et al (2011), had presented coupling the high data rates of IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11p-based VANETs and the wide coverage area of 3GPP networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=', UMTS), that envisions a VANET-UMTS integrated network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this architecture, vehicles are dynamically clustered according to different related metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 10 From these clusters, a minimum number of vehicles, equipped with IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11p and UTRAN interfaces, are selected as vehicular gateways to link VANET to UMTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Issues pertaining to gateway selection, gateway advertisement and discovery, service migration between gateways (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=', when serving gateways lose their optimality) are all addressed and an adaptive mobile gateway management mechanism was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Simulations were carried out using NS2 to evaluate the performance of the envisioned architecture incorporating the proposed mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mohamad Yusof Darus and Abu Bakar Kamalrulnizam (2011) had studied some of the Congestion control algorithms in Vehicular Networks (VANETs) The study further exposed the weaknesses and advantages of some of these congestion control algorithms which could assist researchers to tackle the inherent problems of congestions in VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This paper also concluded with a planned future research for disseminating uni-priority of event-driven safety messages while solving congestion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Kalyani B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Amit (2012), had analyzed, a heterogeneous network framework for providing seamless connectivity and data transfer between various network technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This framework provides QoS seamless heterogeneous network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The heterogeneous network of wireless communication was expected to integrate potentially a large number of heterogeneous wireless technologies which could be considered a huge step forward towards a universal seamless access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' One of the main challenges for seamless mobility was the availability of reliable horizontal (intra system) and vertical (inter system) handoff schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Singhal Manav and Shukla Anupam (2012), had proposed the implementation of Location based services through Google Web Services and Walk Score Transit APIs on Android Phones to give multiple services to the user based on their Location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Location based Services offer many advantages to the mobile users to retrieve the information about their current location and process that data to get more useful information near to their location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' With the help of A-GPS in phones and through Web Services using GPRS, Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 11 Location based Services can be implemented on Android based smart phones to provide these value-added services: advising clients of current traffic conditions, providing routing information, helping them find nearby hotels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Sharma Manish and Gurpadam Singh (2012), had discussed various ad-hoc routing protocols, Reactive, Proactive & Hybrid, taking in to consideration parameters like speed, altitude, mobility etc in real VANET scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The AODV and DYMO (Reactive), OLSR (Proactive) and ZRP (hybrid) protocols were compared for IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11(MAC) and IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11(DCF) standard using Qualnet as a Simulation tool, Since IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11, covered both physical and data link layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Hence performance of the protocols in these layers helped, to make a right selection of Protocol for high speed mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Varying parameters of VANET showed that in the real traffic scenarios proactive protocol performs more efficiently for IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 (MAC) and IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11(DCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Muhammad Rizwan Arshad et al (2012), had examined, WiMAX and WiFi on Vehicular Ad-hoc Networks (VANET) which were used to evaluate the best service provider technology for VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In VANET the nodes are moving very fast and change their network infrastructure rapidly, which have very short time to communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Both WiMAX and WiFi is to be used as per their features in the long distances areas and then their practice in real model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The focus of this research was to reduce the delay time of message passing, authentication and to find the best suitable and qualitative service from WiMAX and WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Gadkari .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Y Mushtak and Sambre .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='B Nitin (2012), had made an attempt for identifying major issues and challenges associated with different VANET protocols, security and simulation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Security and privacy are indispensable in vehicular communications for successful acceptance and deployment of such a technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The vehicular safety application should be thoroughly tested before it is deployed in a real world to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Simulator tool has been preferred over outdoor experiment because it simple, easy and Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 12 cheap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The increasing popularity and attention in VANETs has prompted researchers to develop accurate and realistic simulation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper, we make a survey of several publicly available mobility generators, network simulators, and VANET simulators was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Patel Nikhil and Parmar Kiran (2012), had presented several optimizations methods for the execution of vertical handoff decision algorithms, with the goal of maximizing the quality of service experienced by each user and a method to select the handover target network .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Future wireless networks must be able to coordinate services within a diverse- network environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' One of the challenging problems for coordination is vertical handoff, which is the decision for a mobile node to handoff between different types of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' While traditional handoff is based on received signal strength comparisons, vertical handoff must evaluate additional factors, such as monetary cost, security, power consumption, network conditions, and user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper, the method proposed is a combination of weight distribution and cost factor calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Weights of various network parameters are generated based on user preferences and the power level of mobile terminal, and cost factors of candidate networks are calculated using a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The network with the Highest Qi and lowest cost is selected as the handover target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This method was able to maximize the user’s satisfaction level by choosing the one access network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Vetrivelan et al (2012), had suggested a Multi-Constraint Realtime Vehicular (MCRV) mobility framework that is equipped with important criterion like collision avoidance between vehicles and traffic reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Also certain vehicles such as ambulance, fire service vans, police patrols needed to be given a high priority in our envisioned network architecture, as their requirements are crucial during emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Hence, enabling QoS for differentiating the services according to vehicular priorities and providing group communications, alongside vehicular collision avoidance, was implemented using NS3 and SUMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The vehicle-to-vehicle (V2V) and vehicle-to- infrastructure (V2I) communications were done using WAVE and WiMAX/UMTS Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 13 heterogeneous networks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The horizontal and vertical handovers were chosen at appropriate rite decision for their communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Bhat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='S Vijaylaxmi and Shah .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='N Pragnesh (2012), had studied, the problem of multihop routing in vehicular ad hoc networks (VANET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11p and other vehicular network standards advocate vehicles to issue periodic broadcast messages at regular intervals called beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The physical rate adaptation in 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 was deeply investigated, though still open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Since 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 used the random access Distributed Coordination Function (DCF) mechanism to access the medium, collisions could occur when two or more stations wanted to transmit data simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper the authors had proposed a rate adaptation algorithm that behaved like Auto Rate Fallback (ARF), but made use of the RTS/CTS handshake, when necessary, to decide whether the physical transmission rate should be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The performance of rate adaptation algorithm, was then compared with other well known algorithms, AODV and DSDV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Yuvraj Singh(2012), had suggested that, the radio propagation was essential for emerging technologies with appropriate design, deployment and management strategies for any wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It is heavily site specific and can vary significantly depending on terrain, frequency of operation, velocity of mobile terminal, interface sources and other dynamic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Large scale path loss modeling plays a fundamental role in designing both fixed and mobile radio systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Predicting the radio coverage area of a system is not done in a standard manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, before setting up a system, one has to choose a proper method depending on the channel’s BTS antenna height gain to show good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mor Annu (2013), had discussed about, Vehicular Ad Hoc Network (VANET) which is a sub class of mobile ad hoc networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET is most advanced technology for intelligent transportation system that provides wireless communication among vehicles and vehicle to road side equipments, according to IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11p standard for end to end communication between vehicles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' One of the most important routing protocols used in ad hoc networks was AODV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This protocol is connectivity based reactive protocol that Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 14 searches routes only when they are needed because bandwidth is limited and topology frequently changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It always exchanges control packets between neighbor nodes for routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this article author presented cross layer technique that found channel security at link layer to AODV routing protocol to improve the communication in vehicles for safety purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To reduce the packet delay in AODV , the routing protocol (AODV_BD),was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It reduced the packet delay in AODV and made routes more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Burde Ashwini and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Pingat (2013) , had focussed on the use of VANET technology for efficient traffic management and route planning while vehicle heads from source to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET technology was used as a medium to generate updated information for the vehicle when it headed from source to its destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Performance of SA, PSO and CA algorithm about traffic alerts were also compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Sandhu K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Gurveen et al (2013), had tried to discuss two latest wireless technologies: Wi-Fi and WiMAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Option way out to the trouble of accessing information in remote areas where wired network are inaccessible was offered by Wireless Networking Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Wireless Networking had changed the way people communicate and share information by eliminating the boundaries of distance and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Komala and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indumathi (2013), had discuss about, A Heterogeneous Network abbreviated as HetNet , which is a mix of macrocells, picocells,femtocells, remote radio heads and relays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A HetNet is a network consisting of various wireless access technologies and functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The Next Generation mobility technique called the 4G deals with the multi-access heterogeneous wireless networks which provide seamless connectivity of multimedia services at a higher data rate to the end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' There are various research challenges like self organization, backhauling, handover and interference for the 4G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The various handover mechanisms with respect to intra-domain and inter-domain are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' An experimental testbed for the seamless mobility of heterogeneous wireless techniques such as mobility from IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 to IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='16e Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 15 was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The performance results of a seamless handoff with minimum packet loss and delay proved the efficiency in the mobility of the HetNets using the Network Simulator (NS-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Ghonge M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mangesh and Gupta G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Suraj (2013), had described WLAN, WPAN and WiMAX technologies that were introduced and comparatively studied in terms of peak data rate, bandwidth, multiple access techniques, mobility, coverage, standardization, and market penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Detailed technical comparative analysis between WLAN, WPAN, WiMAX wireless networks that provide an alternative solution to the problem of information access in remote inaccessible areas where wired networks are not cost effective had been looked into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Their work had proved that the WiMAX standard goal was not to replace Wi-Fi in its applications, but rather to supplement it in order to form a wireless network web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Humayun Kabir (2013) had presented the aspects related to Vehicular Ad Hoc Network (VANET) , which is a kind of special wireless ad hoc network, that has the characteristics of high node mobility and fast topology changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' VANET has become an active area of research, standardization, and development because it has tremendous potential to improve vehicle and road safety, traffic efficiency and convenience as well as comfort to both drivers and passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Vehicular networks will not only provide safety and lifesaving applications, but they will become a powerful communication tool for their users and help researchers and developers to understand and distinguish the main features surrounding VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Jain Sapan Kumar and Badhe Vivek (2013), had presented about, how a Wireless Sensor Network varies from Heterogeneous Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In the wireless sensor network , hundreds of sensor node are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If energy, range and hardware capabilities are different in various nodes in the network then , these types of network are mainly known as heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Energy in the sensors is a scarce resource .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It must be managed Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 16 in an efficient manner to expand the life of network and a secure multi-hop reactive protocol for heterogeneous wireless sensor network with clustering was designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Gaur Saurabh Kumar et al (2013), had analyzed the critical factors in deciding the networking framework over which the future vehicular applications would be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A reactive research effort is needed for making VANETs a reality in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A vehicular Ad hoc network (VANETs) can be used as an alert system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' By this we get the alert about the traffic jam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It helps to create balance in traffic load to reduce travelling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This system is also useful to broadcast emergency signal to the driver of the vehicle behind the accident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It also helps to send message to ambulance and traffic police in the case of traffic emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Jaiswal Siddhant and Dr D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Adane (2013), had provided a routing algorithm which works on a hybrid scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' it will have both static and dynamic infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The approach used was Cluster based routing which will help in transmitting packets even in a network with low vehicle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Tejpreet Singh et al (2013), had discussed about VANETs ,that are highly dynamic in nature due to mobility of nodes and this dynamic nature caused topological change in the network, which may affect the communication and security of whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='There are various attacks which may effect the network, but wormhole attack is one the harmful attack which may affect the communication in VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This is so because wormhole may lead to attacks like Denial of service attack, data tampering, masquerading etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper performance of different routing protocols were analysed on the basis of metrics like throughput, end-to-end delay and jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Performance of routing protocols were analysed in two cases first , without wormhole attack and second is with wormhole attack and it has been checked how much performance of routing protocols AODV, OLSR and ZRP was degraded with wormhole attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Vishal Kumar et al (2013), had primarily categorized the various possible applications of vehicular network, along with its features, and implementations in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 17 applications of VANETs are of the classes :1) Safety oriented, 2) Commercial oriented 3) Convenience oriented and 4) Productive Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' E Abinaya and R Sekar (2014), had proposed an idea to optimize signal control at traffic intersections which used vehicular ad hoc networks (VANETs) to collect and aggregate real-time speed and position information on individual vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' An online algorithm, referred to as the oldest job first (OJF) algorithm was used to minimize the delay across the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The results were compared with vehicle-actuated methods ,Webster’s method ,and pre-timed signal control methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mathew Ann Bittu, Joseph Sumy (2014), had discussed about ,Heterogeneous network, which is an important component of cellular networks to meet the increasing mobile data demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Due to uneven traffic distribution, some nodes suffer from heavy load, and their adjacent nodes may carry only light load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This load imbalance among nodes restricts the network from fully utilizing its capacity and providing better services to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' So there is a need for load balancing mechanism to be present in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this paper a load balancing scheme was proposed that moves the load of heavy nodes to lightly loaded or idle nodes by finding next shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This may lead to efficient utilization of nodes and nodes can allocate resource efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Jaswal Kamini et al (2014) , had studied the selection of a network simulator for evaluating research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' According to the previous researches on the performance of OPNET, it had been proved that OPNET is comparatively more reliable, easy to understand and implement network simulation tool than its other counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The main focus of this paper was to study the Wimax performance when implemented with OPNET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' OPNET is based on a mechanism called discrete event system which means that the system behaviour can simulate by modeling the events in the system in the order of scenarios the user has set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Hierarchical structure is used to organize the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 18 Anwer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Shahid and Guy Chris (2014), had surveyed some of the key vehicular wireless access technology standards such as 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11p, P1609 protocols, Cellular System, CALM, MBWA, WiMAX, Microwave, Bluetooth and ZigBee which served as a base for supporting both Safety and Non Safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The survey also analysed and compared the wireless standards using various parameters such as bandwidth, ease of use, upfront cost, maintenance, accessibility, signal coverage, signal interference and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, the work discussed some of the issues associated with the interoperability among those protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Mohammed Shafeeq Ahmed (2014), had discussed and illustrated, security solution, various vulnerabilities and possible attacks to WiMAX network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='16, security has been considered as the main issue during the design of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, security mechanism of the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='16 (WiMAX) still remains a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' WiMAX is relatively a new technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' not deployed widely to justify the evidence of threats, risk and vulnerability in real situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This paper addressed, the security aspects of the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='16 Standard and pointed out the security vulnerabilities, threats and risks associated with both layers in WiMAX physical and MAC Layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The threats apply to both layers of WiMAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' At PHY layers, jamming can be considered a major threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' At MAC layer, critical threats include eavesdropping of management messages, masquerading, management message modification or DoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Pooja Rani et al (2014), had surveyed different outdoor and indoor propagation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In wireless communication, path loss was caused by different obstacles between the transmitter and receiver that absorb power due to which signal strength is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To calculate the path loss between the transmitter and receiver, different path loss models are used like okumara, hata, cost 231 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' These path loss models may give different results in urban, suburban and rural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' These models depend on various parameters like mobile-station antenna height, transmitter-receiver distance, base - station antenna height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 19 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Karthikeyan and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Subramani (2014), had surveyed about QoS based agent routing algorithms in MANET, WSN and VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' One of the most challenging tasks in Ad-hoc Network(MANET &VANET) is Quality of Service (QoS) which is determined by numerous parameters such as bandwidth and delay constraints, varying channel conditions, power limitations, node mobility, dynamic topology, packet delivery ratio, end-to-end delay and connection duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' With the increasing demand for real time applications in the Wireless Senor Network (WSN), real time critical events anticipate an efficient quality-of-service (QoS) based routing for data delivery from the network infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Designing such QoS based agent routing protocol to meet the reliability and delay guarantee of critical events while preserving the energy efficiency was a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Vishalakshi Prabhu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Nagaraja (2014), had surveyed on the Worldwide mobile operators, industry experts, and researchers that have diverse visions of potential 4th generation (4G) features and its implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This paper had given a survey and classification of the important QoS approaches proposed for 4G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Classification was based on the work done in each protocol layer and Cross Layer Design (CLD) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, this paper presented outcomes of survey which included significant observations, limitations and idea of further research in improving QoS in 4G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Anuradha Singh and Mintu Singh (2015), have discussed the overview of Vehicular Ad-hoc Network (VANET), which is a most critical class of mobile ad-hoc network (MANET) that enables intelligent communication among vehicles and also between vehicle and roadside infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It is a promising approach for the Intelligent Transport System (ITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' There are many challenges to be addressed when employing VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It has a very high dynamic topology and constrained mobility which makes the traditional MANET protocols unsuitable for VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The aim of this review paper was to give an overview of the vehicular ad hoc networks, its standards, applications, security issues and the existing VANET routing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 20 Kshirsagar Suresh Nikhil and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Sutar (2015), have thrown light on accident prevention and traffic signal control for ambulance, police van, and normal vehicles too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To overcome this they have implemented a highway model, intersection model that manages vehicle mobility and shows the actual communication between vehicle to vehicle (V2V) and vehicle to infrastructure (V2I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='The security of VANET technology is one of the most critical issues because their information transmission is propagated in open access environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Over a period of years, VANET has received increased attention as the potential technology to enhance active and preventative safety on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Sadek M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Noha et al (2015), have discussed about Intelligent Transportation Systems (ITS) that have been receiving significant interest from various stakeholders worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' ITS promise major enhancements to the efficiency, safety, convenience and sustainability of transportation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To satisfy the diverse vehicular application requirements, this paper had proposed, an integration of IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11-based VANET and LTE cellular network using mobile vehicular gateways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 g is used for V2V communications and LTE for V2I communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A burst communication technique is applied to prevent packet losses in the critical uplink ITS traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A performance simulation-based study was conducted to validate the feasibility of the proposed system in an urban vehicular environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The system performance was evaluated in terms of data loss, data rate, delay and jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The results indicated that the proposed Multi-RAT system offers acceptable performance that meets the requirements of the different vehicular applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Chib Randeep Singh et al (2015), have suggested that Radio wave propagation models are extremely important in radio network planning, design as well as in interference planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Radio propagation is essential for emerging technologies with appropriate design, deployment and management strategies for any wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It is heavily site specific and can vary significantly depending on terrain, frequency of operation, velocity of mobile terminal, interface sources and other dynamic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Accurate characterization Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 21 of radio channel through key parameters and a mathematical model is important for predicting signal coverage, achievable data rates, BER and Antenna gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Path loss models for macro cells such as Okumura, Hata and COST 231 models were analyzed and compared with their parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 22 CHAPTER 3 THEORY A routing protocol defines the way the dissemination of a message in a network is handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It defines the creation of a route from the source of the message its destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In Ad-Hoc networks, every node is responsible to deal with the routing and do not rely on specified devices like network with infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This part will firstly identify several ways to classify routing algorithms, secondly, it will present the routing protocols available in VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, a table will summarize this chapter by classifying the algorithms according to their different routing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In the next section, I would introduce important routing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Unicast, Multicast and Broadcast Those three types of communication define the number of point concerned by a transmitted message in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Unicast describes a communication between two points, the sender and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Multicast defines a communication between a sender and several receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, broadcast describes a communication where the sender’s message is sent to all the other nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In VANETs, the three types of communication are used, depending on the types of application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Environment and entertainment application will use more often unicast and multicast because the message does not concern every vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' On the other Hand, safety application will mainly use broadcast communication to reach all nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Unicast and multicast transmissions need the establishment of a communication between the sender and the receiver(s) before the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The absence of routing devices (such as gateway) in ad-hoc network implies that the communication will consist of a succession of hops from the sender to the receiver(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Where each node in the path will forward the message until it reaches its destination(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The way the algorithm defines this path is called route discovery and is the first phase for most of unicast and multicast protocol in VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Most of the time it uses broadcasting methods during this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, even if this Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 23 thesis focusses more on safety application and on broadcasting algorithms, we will also go through some unicast and multicast protocols that use broadcasting techniques in their first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, the results obtained for a new broadcasting algorithm could also impact the efficiency of unicast and multicast algorithm by improving the route discovery phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 Proactive, Reactive and Hybrid protocols Routing protocols can be classified under different criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' From the way it handles routes in the network to the way it discovers those routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this section, we identify three behaviors defining when a route is established and maintained in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those three types are proactive protocols, reactive protocols and hybrid protocols and will be detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Proactive Protocols Proactive protocols use a route discovery phase before sending any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, routes are calculated and maintained up to date continuously by transmitting periodic routing information on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each time a node wants to transmit a packet, the packet is sent only if a route to the destination is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Otherwise the packet will wait in queue until a route has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those types of protocols are difficult to maintain in highly dynamic and scalable network such as VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, they require a significant amount of routing information to be transmitted, increasing significantly the bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Accordingly, Proactive protocols are not the most suitable for VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Examples of proactive protocols such as Destination Sequenced Demand Vector (DSDV) is detailed later in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 Reactive Protocols For reactive protocols, the route discovery phase is initiated only when a packet needs to be sent over network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If a route is found it will be maintained using maintenance route packets sent periodically until the destination is not reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Reactive protocols present lower overhead than proactive protocols, but the end to delay is more important due to the route discovery phase started every time a packet needs to be send.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 24 reactive protocols present also some inconvenient in VANETs where the transmission require a low end to end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Examples of reactive protocols such as Ad-hoc On Demand Distance Vector (AODV), Distance Vector Routing (DSR) are detailed later in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 Hybrid Protocols Hybrid protocols are designed to compensate the overhead of proactive protocols and the end to end delay of reactive protocols by combining properties of both types of protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In most of hybrid protocols in VANETs, each node of the network broadcast its routing information (using beacon messages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Thus, nodes create and maintain a table of neighbors based on those beacons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The route discovery is then initialized when a packet needs to be sent and use the neighbors’ tables to find destination faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Hybrid protocols have been made to handle dynamic networks such as Mobile Ad-Hoc Networks (MANETs) and then modified to fit the high speed and scalability of VANETs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Examples of Hybrid protocols such as Zone Routing Protocol (ZRP) and Temporarily Ordered Routing Algorithm (TORA) are given in more details later in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 Categorization of next hop selection In routings protocols in MANET and VANET, the route discovery phase uses different methods to select the next node in the route (called next hop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed the selection of the next hop is made on several criteria depending on the method used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' We distingue height types of next hop selection which are further node, best quality link, most demanding node, probability base, backbone node, stochastic method, counter based, distance to mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those method are not necessary independent and are explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Distance-based The distance-base technique consists of selecting the next hop based on the distance between the current node and the closest to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This technique can either rely on node geographical location especially in VANET but can also rely on the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' An important amount of algorithm uses this technique during the route Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 25 discovery phase named greedy forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In greedy forwarding, each node selection the next hop the closest geographically to the destination (This technique is called furthest node selection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In broadcasting routing protocols where the destination position is not known, each node selects the node the furthest away from itself in its transmitting range considering that close nodes will already have received the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The furthest node can also consist of a of each node decide by itself whether to forward the message by comparing its distance to the precedent hop and comparing it to a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Examples of Routings protocols using greedy forwarding and further nodes are contained in geographical routing detailed in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 Best Quality Link Selecting the next hop based on the quality of the link means considering real word channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This selection may rely on distance with next hop, the received power from beacon packets or other channel criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of protocols can increase the end-to-end delay because of the information gathering phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, it shows a better reliability than furthest node selection that ignores channel condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 Most Demanding node The most demanding node method prioritizes certain nodes of the networks according to their locations in the graph or their time to react to previous messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The idea is to transmit the message to the nodes the most concerned by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of method is mainly used in warning message propagation in VANET to provide security relative information to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' These type of selection ensure the delivery of important packets to demanding nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, it increases calculation considerably to identify the demanding area and therefore the end to end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 Probability based forwarding Each node will forward the packet depending on a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This is used to decrease the number of packet rebroadcasting a packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The assigned probability is either defined or dynamically change depending on the network density or node location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 26 method is mainly used for some broadcasting protocols where all nodes of the network are concerned by the message so the objective is to decrease the number of rebroadcasting nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 Backbone node Backbone node selection consider the existence of infrastructure in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those can be physical, using road side units for examples like in some trajectory based routing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Certain can also be dynamically created by forming groups of nodes where some of them will be entitled to the routing (like in cluster based routing algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those infrastructures will be used to either calculate the route to destination or to select next forwarding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The goals of this selection is to decrease the overhead by allowing only backbone nodes to transmit However, the node selection requires calculation that are needed to be kept as simple as possible to maintain a reliable end to end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='6 Stochastic method In stochastic method, the next hop is selected randomly among the neighbors available in transmitter based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In receiver based algorithm, each node selects randomly either to forward or not the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of selection do not ensure the packet delivery and is therefore not reliable in route request or broadcast routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='7 Counter based The counter based method consists on counting the number of time a message is received by a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When a node receives a message, it will set up a time to wait before forwarding the message and if it receives again the message before this time is over, it will cancel the forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This defined time is called a back off timer and can be calculated over some parameters such as node location for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This solution can be most of the time used with another selection technique sur as distance-based for example in the form of a back off timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The Counter-based method is mainly used in geographic broadcast routing protocol to propagate message to all node and decreasing overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 27 However, it can also be used to create route to destination among the first node to forward to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8 Distance to mean The Distance to mean selection can be compared to the distance-based technique and is mostly a receiver-based method, where each node decides by itself to forward the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, here the decision is made on the distance between the node and the spatial mean of precedent forwarders of the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This value is then compared to a threshold value calculated over several parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This selection method shows better results than distance based ones such as further node in terms of reachability, bandwidth consumption and end to end delay [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, the algorithm proposed on this paper will use this type of next hop selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Algorithms based on this technic will be detailed in more depth in the next sections of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 Types of routing protocols Now that three types of communication have been distinguee, the routing protocols based on those communication can be separated into six categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' We identify topology based, location based, cluster based, Geocast based, trajectory based and geographic based routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' All those categories are sorted by the way the transmitted message is conveyed to its destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Topology Based Routing Topology based routing consist on the establishment and the maintenance of routing table for each node of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It means that every node in the network knows the path to reach other node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of protocol is not fitted for VANET because of the high mobility and scalability of vehicular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Maintaining such table is hard when the topology of a network is constantly changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, certain protocol adapts such protocols to make it more efficient in VANET environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 AODV Ad-hoc On-Demand Vector (AODV) routing is a neighborhood awareness protocol because each node uses Beacon hello messages to keep track of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AODV is also a reactive routing protocol because the route generation is started only when a node wants to send a packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, a Route Request (RREQ) is sent to neighbors and propagated between net-hop neighbors to find a path to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, a Route Reply (RRep) is sent back to the source using the reverse route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AODV control if those routes contain no loop and find the shortest path among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node then stores next hop to destination in a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It can also create new routes or modify existent by handling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AODV is one of the most used protocol in wireless networks because of its viability, however, it is not fit to handle high mobility and scalability among VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, a lot of protocol used in VANET are adapted from AODV with modification to fit more VANET the needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Finally, AODV provide Unicast and Multicast by establishing routes to destination which is node needed when you send warning messages in VANET which only needs Broadcasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 MAV-AODV Multicast with Ant Colony Optimization of AODV (MAV-AODV) is based on AODV and is a Bio inspired algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Based on Ant colony, MAV-AODV use nodes’ mobility information to build multicast tree and sustain its lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A neighbor table is maintained using periodic beacons to obtain mobility information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then the Request Response phase of AODV is improved using node’s position and a Best quality link next hop selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 OLSR Optimized Link State Routing (OLSR) Protocol is a proactive protocol because it creates and maintain a routing table based on topology information regularly exchanged by the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then certain nodes are classified as Multi-Point Relay (MPR) by their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Information that they broadcast periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those nodes are then used to form the route from a node to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Like AODV, maintaining a routing table in VANET networks is not efficient because of the ephemeral state of the network but also in terms Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 29 of bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, OLSR also provide Unicast and Multicast which are not needed for warning message propagation in VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 DSDV Destination Sequenced Distance Vector (DSDV) uses routing table scheme where the path is calculated based on the Bellman-Ford algorithm which is, in graph theory, an algorithm to find the shortest path form a single vertex to other vertices in a weighted digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The main used of this algorithm is to avoid the routing loop problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 DSR Like AODV, Dynamic Source Routing (DSR) starts route discovery operation when a node wants to send a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' But unlike AODV, the route is kept in full in the table and maintained for a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Likewise, the RREQs are made using flooding, not by using a maintained table of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Furthermore, every node is responsible to wait for reception confirmation from next hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Until then, it will keep sending the packet until it reaches a defined maximum threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 TORA TORA is alike DSR but in addition to the route discovery phase, this protocol provides a route erase phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Thanks to the first phase, every node constructs and maintains a route table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Nodes are then able to detect network partition, in that case, they will trigger the erase phase by sending a clean packet that will delete the invalid route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='6 FSR FSR uses flooding broadcasting to propagate packets in the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, with the latest location information contains in those messages, each node builds and maintains a Topology Table which allows node to build routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='7 ZRP In ZRP, every node’s neighborhood is delimited by a defined zone depending on the transmission range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' For nodes inside this area, routes are discovered reactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 30 to transmit to nodes outside of the transmission zone, a route request is emitted to other zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' ZRP is a hybrid protocol, between topology based and cluster based because nodes are grouped in zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' But in Cluster based routing algorithm a cluster head is designed to deal with all routing inside and outside of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8 MAR-DYMO Mobility-aware Ant colony optimization Routing (MAR-DYMO) is an integration in Mobile Ad-Hoc Networks (MANET) of Ant Colony Optimization (ACO) [20] by combining it to the routing protocol Dynamic MANET On-demand (DYMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DYMO is the successor of AODV protocol and is based on the same principle of multi-hop propagation between neighbors until it reaches destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' ACO works with several principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' One of them is the pheromones which are used to grade route to increase reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In MAR-DYMO, more pheromones are added on RReq route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then if a RRep crossed the same route, more pheromones are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Route is then selected according to pheromones density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Furthermore, pheromone evaporates with time and are added by transmitted packet to maintain and modify route if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MAR-DYMO also uses a Kinetic Graph framework to make prediction about node’s neighbors trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It uses aperiodic HELLO message sending compared to DYMO and reduce Bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='9 QoSBeeVANET Quality of Service Bee Swarm routing protocol (QoSBeeVANET) is a topology based protocol designed for unicast routing inspired over bee swarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In this protocol, the first phase which is route request (RReq) is implemented using stochastic broadcasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Which means, every node of the network is given a random or determined probability to forward a message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of broadcasting helps reducing the number of bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' As soon as the destination is reached a RRep is sent back to the source and the route is stored in a routing table with the following information: next hop, number of hop before destination, hop count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The algorithm maintains routes by sending periodically packets to neighbors and if it detects a missing node or a loss of Quality of Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 31 Service (such a too high bandwidth consumption or end to end delay), it will warn all other nodes concerned on the degraded path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' All nodes on error are removed and a new route discovery phase is started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' After all, this algorithm can easily flood the network because of the number of packets send, especially if an error occurs (due to node missing or QoS not respected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='10 HyBR Hybrid Bee Swarm Routing (HyBR) [23] has been designed to overcome drawbacks faced by QoSBeeVANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Still designed to propose unicast and multicast routing, HyBR use two types of routing depending on the density of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Topology based routing when the density is high and geography-based routing when the density is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The topology based routing RReq and RRep is executed like in QoSBeeVANET using stochastic flooding for RReq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The RRep is then routed back to destination throughout discovered path and stored in a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The geography-based routing is based on shortest node algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A broadcasting flooding is executed to determine all routes to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then the route is selected hop by hop with the node the closest in hop distance to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Results obtained by this algorithm are close to those obtained with AODV and GPSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 Datataxis Datataxis is a topology based unicast routing protocol inspired by the behavior of Bio- System: Escherichia coli bacteria (an active component of in the natural immune system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Datataxis is designed to collect metadata (such as location, time stamp, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=') in urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then those data were proposed to be diffused using the protocol MobEyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Datataxis estimate firstly the meta-data density by road segment and then send multi- agents systems allowed to move from nodes to nodes to collect those data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This protocol has been proposed essentially for distributed surveillance and monitoring, for police car for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, we will not detail it in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='12 MURU Multi-hop routing for Urban VANET is a topology based routing protocol designed for unicast and multicast communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The protocol is based on AODV but instead of using hop count to find optimal path to destination, the Expected Disconnection Degree (EDD) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' EDD is calculated over the Packet Error Rate (PER) of link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' PER being function of hop distance, EDD being the probability that a link break then depends mainly on hop distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' EDD depends also independently to predicted speed, movement trajectory and vehicle location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Knowing that in the paper, the vehicle’s mobility is approximated to a first order Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The Markov chain define a stochastic process in which the conditional probability distribution of future state depends only on the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='Besides this improvement, the first phase of the protocol, the route request broadcasting is constrained by vehicle movement trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This protocol shows good result but the number of information required to calculate EDD can be difficult to obtain in real life scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Furthermore, as other topology protocol, maintaining path topology decrease scalability of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 Position Based Routing The lack of scalability and robustness of topology-based protocol has lead research to find other type of routing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, the creation and maintaining of routing table in highly scalable networks may not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Position-based routing protocols which use network location of nodes to decide how to route messages are a new area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Although, Geocast routing can be defined as a position based routing protocol because it defines area based on geographical coordinates where nodes are concerned about the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 GPSR Greedy Perimeter Stateless Protocol (GPSR) [26] uses periodic beacon messages to build neighbors table on each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The next-hop selection is distance based and uses GPS node’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' GPSR also integrates recovery strategies based on perimeter routing to Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 33 eliminate redundant routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This protocol is the most used in VANET to run simulation because it presents good reachability and end-to-end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 AMAR Adaptive Movement Routing (AMAR) like GPSR, uses a greedy forwarding technique for next hop selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' But in addition to the distance, AMAR also use neighbors’ position, direction and speed to select the next hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 GYTAR Like precedent protocol, Greedy Traffic Aware Routing (GYTAR) bases its next hop selection on greedy forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' GYTAR also use periodic beacon to maintain a table of neighbors containing position, velocity and direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Secondly, it defines junction based of nodes density close to the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' GYTAR uses then the table and the junction density to select next hop between its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 DREAM Distance Routing Effect Algorithm Mobility (DREAM) acquires each node’s position using local services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It then calculates the possible destination area position and use directional flooding to reach it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The directional flooding consists of restricting the flooding graph to nodes in the area that leads to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 LABAR Location Area Based Ad-hoc Routing (LABAR) [27] uses a backbone next hop selection using V2I communication (with road side unit) to create an infrastructure in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LABAR then routes message from mobiles nodes using fixed backbone nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To determine the route among fix nodes it uses directional routing such as AODV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='6 ROVER Robust Vehicular Routing (ROVER) [28] represents an example of Geocast routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Geocast routing consider that only certain area is concerned by the message sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 34 area are called Zone Of Relevance (ZOR) and are defined by their GPS location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each packet is then affected to one ZOR and will be forwarded to each node in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node discovers in which ZOR it belongs using local services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='7 pPSO (AODV) The parallel Particle Swarm Optimization (pPSO) for VANET is inspired by swarm and is a parametrization of the protocol AODV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It consists of calculating the best position and speed for the vehicles to occupy in order to make the protocol AODV reduce its packet overhead, end-to-end delay and delivery ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8 GSR Geographic Source Routing (GSR) uses a Reactive Location Service (RLS) and a digital map to handle routing in urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' With the location of the destination acquired with RLS and fixed node in the network (RSU at intersections), GSR use Dijkstra to calculate the shortest path between junction (fixed nodes) and greedy forwarding to disseminate the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='9 CAR Connectivity-Aware Routing (CAR) uses neighbor recognition using beacon messages sent with a time interval depending on the number of neighbors detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The route request phase use anchor points selected over best quality link method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then the packet is forwarded using a greedy forwarding method among those anchor points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 Cluster Based Routing Cluster based routing consists of dividing the network into smaller groups of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Among each group a cluster head will be selected, basically a node will handle every communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The ones inside the cluster but also the one outside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The main objective of cluster based routing is to handle high scalability of VANET by handling smaller connected infrastructure networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Although, this type of protocol often uses GPS information to delimit and organize its cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 CBLR The Cluster Based Location Protocol (CBLR) builds its cluster using bacon hello messages as an initialization phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This operation is also used to define the cluster head that will construct and maintain a table containing information over nodes in the cluster and others cluster heads of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Dissemination inside the cluster is effected using a greedy forwarding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Communication outside of the cluster is handled firstly by finding the location of the destination using other cluster head information 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 CBDRP Cluster Based Directional Routing Protocol (CBDRP) builds its cluster on nodes velocity vector (speed and direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The routing is then effected like in the CBLR protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 EDCBRP Euclidian Distance Cluster Routing Protocol (EDCBRP) bases the network division in cluster on Euclidian distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Nodes with a Euclidian distance with each other under a defined threshold form a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The distance is calculated using GPS information of nodes and acquired by hello message beacon periodically send.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Topology tables are maintained inside clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' For communication with other cluster, a route-request route response is sent in order to build the route to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 TACR Trust Dependent Ant Colony Routing (TACR) is a Bio inspired routing algorithm for VANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In TACR, cluster are builds on position and speed of moving nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The cluster head is selected on lowest node speed with priority to RSU because of their fixed position and infrastructure network available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The communication between cluster is achieve using the Ant Colony Optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It consists of a route request sent to every other cluster that check if the destination is in its member table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If yes, the cluster head do not forward the route request but instead, send a route response backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Inside cluster communication are maintained with maintaining a member table using Beacon messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 Trajectory Based Routing Trajectory based routing are developed mainly for urban environment with Road Side unit (RSU) positioned on roads intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Those routing protocols propose to use a fixed infrastructure composed by the RSUs and disseminate message to moving node using trajectory calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This type of algorithm modifies the V2I and I2V communication by adding several information transmitted to and by fixed RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Thereby, traffic statistics such as density, average speed, average direction or digital map of the area can be transmitted over V2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 AMR The Adaptive Message Routing (AMR) algorithm has for main objective to reduce the end-to-end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' To achieve its goal, it builds route using a genetic algorithm based on the probability of connectivity and the hop count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' AMR uses backbone next hop selection, using fixed RSU to convey messages over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The algorithm calculates the intersection between the source and the destination and the infrastructure network build on RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 IGRP The Intersection-based Geographical Routing Protocol (IGRP) is mostly used to send packet from vehicles to the internet using a genetic optimization algorithm over intersection routing protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This algorithm uses a backbone next hop selection technique among RSUs until it reaches an internet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 TBD The Trajectory Based Data (TBD) algorithm uses vehicular density, mobility patterns, average speeds and digital map transmitted over V2I communication to evaluate its best next-hop to reach the closest RSU with the lower end-to-end delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, it shares this delay with close nodes for them to build their own path to RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Packets are sent over RSU network that will disseminate it to destination using classical infrastructure networking routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 TSF Trajectory-based Statistical Forwarding (TSF) calculate end-to-end delay in the opposite way of TBD, meaning from the fixed node to the moving vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then based on the minimal delay between nodes and RSU, the route between source and destination is calculated with destination trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' An optimal target point is identified between the destination node’s trajectory and an RSU on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, the packet will be forwarded over the RSU network to reach the optimal target point at the same moment as the destination node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 TMS Trajectory-based Multi-Anycast (TMS) assumes the existence of a Traffic Control Center (TCC) containing information of all nodes in the network (position and velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Based on those information, every time a packet is sent, the TCC identify a rendezvous point between destination node’s trajectory and a forwarding tree build on moving nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='6 STDFS Shared-Trajectory-based Data Forwarding Scheme (STDFS) uses RSU to propagate nodes’ trajectory over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' With those information, every node can calculate a rendezvous point with the destination and disseminate the packet over V2V communication to the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 Geographic and Broadcast Routing Broadcast routing protocol are mainly used to transmit warning information or other data that concern every vehicle on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, this type of routing is also used as the route request phase for some multi-cast or unicast protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' For the route discovery phase for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The main goal of such routing is to reduce bandwidth consumption by skipping the route discovery phase and therefore, decreasing the number of routing packet sent on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Hybrid-DTN Hybrid geographic and Delay Tolerant Networks (Hybrid DTN) uses a greedy forwarding with a furthest node next hop selection as a route discovery phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, if the greedy forwarding fail, it uses the perimeter forwarding (or right-hand rule) method to reach the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This method consists for a node, of covering a counterclockwise circle around itself and forwarding the packet to its first neighbor found in this circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='2 SRB Secure Ring Broadcasting (SRB) is based on the best quality link selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, it classify nodes based on receiving power to estimate the distance between the receiver node and the last broadcasting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then only those at the preferred distance from the last broadcasting node can forward the packet several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This algorithm is based on a flooding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, depending on the estimated distance between graph level, nodes can forward one or several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='3 PBSM Parameter less Broadcast in Static to highly Mobile (PBSM) only checks if nodes in the neighborhood has received the packet and then retransmit the packet only to those who did not receive the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4 EAEP Edge-Aware Epidemic Protocol (EAEP) uses probabilistic techniques to decide whether to forward the packet or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A time probability is calculated by each node to decide when to forward the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5 DV-CAST Distributed Vehicular broadcast (DVCAST) uses periodic beacon to maintain a table of neighbors for each node to know local connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then, depending on the connectivity of each node, action of forwarding is decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DV-CAST sort the node in two categories, the well-connected ones and the sparely connected ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The next forwarding node in the Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 39 well-connected one is selected over the distance with the sender node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, a back off timer is calculated inversely proportional to the distance with the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The node with the smaller back off time (the furthest node) will then rebroadcast the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' For the second category, the sparely connected ones, if a node has a neighbor in the opposite direction road it will rebroadcast immediately, if not, then it will keep the packet until it finds another node in the opposite direction road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='6 TRADE TRAck DEtection (TRADE) protocol gives nodes a table of neighbors maintained using periodic beaconing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' A node sorts its neighbors in three categories depending on their position and velocity: same road ahead, same road behind and different road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then it uses the furthest node selection technique as next hop selection in the two first categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' For the third one, the node just rebroadcast to every node in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='7 MAC The Media Access Control (MAC) protocol also uses periodic beaconing to maintain a table of neighbors on each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Consequently, every node calculates its relative direction to the sender and the ones between the sender and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then if the node’s relative direction corresponds to the packet direction, the node will check in its neighbor table if it is the more dedicated to forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This selection is based on the furthest node criteria using the segment define by the distance and the relative direction calculated before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8 REAR Reception Estimation Alarm Routing (REAR) protocol is based on probabilistic next hop selection among neighbors in the direction of the message propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' REAR makes node maintain a neighbor table and each warning message sent contains sender position and a list of sender’s neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node in the direction of the message’s direction will then calculate its reception probability and wait a back off time inversely proportional to this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The first node to reach its back off time will then rebroadcast the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='9 GPCR Greedy Perimeter Coordinator Routing (GPCR) defines junction as a link between one node reached by a message and another one out of the message’s range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It uses 2-hop neighbors table to locate a junction between to nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, by sending periodic beacon containing its position and its neighbors, each node can construct such a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Therefore, if a node has two neighbors that does not have each other on their respective table, that means this node is a junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Every node calculates the number of junction it represents and the those that have the bigger coefficient are called coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Coordinators informs other nodes of their new role and form the backbone of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Consequently, the next forwarding hop selection during the broadcasting will prefer a backbone node as the next forwarding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If no coordinator is found, the furthest node method is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='10 TDR Three-Dimensional scenario oriented Routing (TDR) protocol proposes an improvement of GPCR protocol and its upgrade GyTAR to fit 3-Dimensional environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The main difference comes from the size of hello beacons which contain 3 coordinates instead of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 Multicast Routing for Message dissemination protocol This protocol also uses beaconing to maintain neighbor’s awareness table and the next hop selection is based on the most demanding node criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='12 OAPB Optimistic Adaptive Probabilistic Broadcast (OAPB) constructs and maintains for each node a 2-hop neighbors table using periodic beacon sending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then each node calculates a probability of rebroadcasting based on that information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' it create a back off timer, function of the probability of rebroadcasting and a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The node with the smallest back off timer will be selected as the next hop for the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='13 UMB Urban Multi-Hop Broadcast (UMB) protocol considers that every message should be transmitted to RSU disposed to every intersection and that should behave as repeater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If there is any RSU available, the algorithm will function on a reactive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When message needs to be sent it will send a directional request to relay (RTS) containing its position and direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The nodes in the broadcast area defined by the packet direction emit a signal of a duration proportional to the distance with the sender and the number of jamming signal in its transmission range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The node with the longest jamming (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' the furthest away from the sender) signal will send the Clear To Broadcast answer (CTB) to the sender and will be designed as next hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This method can be classified as furthest node selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' When the warning message is sent, elected relay will send back an ACK message to the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If it fails to do so or if several CTB are received, UMB will start the recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='14 SB Smart Broadcast (SB) is based on UMB but replaces the jamming signal of the furthest node selection method by a classic back off timer inversely proportional to the distance with the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This way the next relay is not the one that wait the most of time like in UMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' SB also handle CTB collision better with a random selection between the two possible routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='15 STAR Intersection based routing protocol that uses RSU placed at red light intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Packet are forwarded on every red light connected and to every first car on green light that are the closest to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='16 VanetDFCN VANET Delayed Flooding with Cumulative Neighborhood (VANETDFCN) is an improvement of the protocol DFCN for MANET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' It transmit more information about neighbors (such as the position) to improve the forwarding node selection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 42 next hop selection is distance based, if the distance between the receiver and the sender is over a Distance To Live, the node will not forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The packet must also have been received only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Based on those criteria, a coefficient is calculated which represent the number of TCP chunk a packet can be divided in and transmitted on the liaison created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='17 xChangeMobile xChangeMobile is an improvement of VanetDFCN with the addition of a Threshold on the minimal number of chunk that can be transmitted over a liaison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Only node that can provide a liaison with a coefficient higher than the Threshold will be considered as forwarding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='18 MOCell MOCell routing protocol is based on a genetic algorithm to improve xChangeMobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The goal is to reduce bandwidth consumption and the number of lost chunks in the TCP process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='19 RBLSM Reliable Broadcasting of Life Safety Messages (RBLSM) is also a reactive protocol that sends RTS and waits for CTB when a packet needs to be sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, RBLSM uses the nearest node next hop selection instead of the furthest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='20 LW-RBMD Light Weight Reliable Broadcast Message Delivery (LW-RBMD) protocol uses the furthest node technique using only the header of warning messages to transmit sender’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Nonetheless, the sending node will wait for a rebroadcasted message and take it into an acknowledgment (ACK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If the ACK is not received, the sender will resend the packet after a timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This protocol is designed to limit the network overhead and still maintaining high reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='21 MHVB Multi-Hop Vector Broadcasting (MHVB) is a broadcasting routing algorithm using the furthest node selection technique with a classical inversely proportional back off timer set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, MHVB integrates a congestion detection algorithm, which consist of detecting when the number of neighbors is above a certain Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' In that case, the Broadcasting interval is increased to lower the bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='22 DTM Distance-To-Mean (DTM) algorithm uses the distance to mean as next hop selection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node maintains a table of neighbors using beaconing and defines a distance threshold based on the number of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This distance represents the minimum the distance to mean value must exceed so that the node is considered as a possible forwarding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Firstly, introduced in 2011, it presents better results than distance based greedy forwarding in terms of reachability and of network overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This paper being based on this paper, DTM will be explained in more details in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='23 DADCQ Distribution-Adaptive Distance with Channel Quality (DADCQ) protocol uses both distance, and best quality link as next hop selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Here, warning messages also propagate their neighbor table in their header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This way, each node has access to two-hop neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='24 CSBD CSBD is a MAC and network cross layer with density-adaptive contention window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This means that the MAC layer is directly influenced by the information obtained on the network layer using geographical routing and distance-to-mean heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='25 SLAB Statistical Location Assisted Broadcast (SLAB) enhances DADCQ using the distance-to- mean next-hop selection method instead of the distance-based used in DADCQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Also, SLAB uses machine learning techniques to improve the Threshold function definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='26 FLB The Fuzzy Logic Based (FLB) protocol is based on the DTM algorithm but uses fuzzy- based techniques to calculate the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, each node is classified according to several factors defined on its information such as velocity, position and number of neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' the Coverage factor, the Mobility Factor and the Connectivity Factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' This improvement in the broadcasting node selection presents better results than DTM in terms of number of rebroadcast per covered nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='27 BEFLAB Bandwidth Efficient Fuzzy Logic-Assisted Broadcast (BEFLAB) for VANET, like FLB presents a Fuzzy-logic based receiver based rebroadcasting node selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Each node is here classified according to two factors, the Mobility Factor and the Coverage factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then a table of Fuzzy rules decide whether a node will rebroadcast or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Unlike FLB, BEFLAB does not use the distance-to-mean heuristic method but only the fuzzy technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='28 IHAB Intelligent Hybrid Adaptive Broadcast (IHAB) for VANET is based on FLB and BEFLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Indeed, IHAB first calculate the node potential transmit density (PTD) using two-hop neighbor’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Then IHAB chooses between FLB and BEFLAB which protocol to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' If the network is said dense (PTD superior to a threshold), IHAB will chose to use BEFLAB, if the network is sparse (PTD inferior to threshold), IHAB will chose to use FLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Analysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 45 CHAPTER 4 METHODOLOGY Simulations have been carried out on NS3 to compare and analyze routing algorithms, such as the DSDV, AODV, and DSR, based on various performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' However, performance comparison and analysis between the two classical MANET routing protocol types, proactive and reactive, have rarely been done using NS3 in the Linux Ubuntu operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' The procedures and simulation results presented in this project report will help VANET researchers and designers tune their systems to meet particular requirements in a more efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS3 Simulation Modeling Methodology To establish NS3 simulations, several classes such as core-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h and network- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h need to be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' These classes plus their detailed descriptions can be found in NS3 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Moreover, NS3 employs C++ and Python languages, and several simulation steps need to be followed to start any NS3 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 Simulation Parameter Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Table-2 Simulation Parameter Setup Description Parameters Channel/Wireless channel Channet Type Propagation Radio-Propagation Model Phy/Wireless Phy Network Interface Type Mac/802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='11 MAC Type Queue/Droptail/Priqueue Interface Queue Type LL Link Layer Type Antenna/OmniAntenna Antenna Model 50 Max Packet 100 Number Of Mobile Nodes AODV,DSR,DSDV Routing Protocol 867 Topographical Dimention 561 Topographical Dimention 25s,50s,75s,100s,125s,150s,175s,200s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Time Of SimulationAnalysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 46 CODE SECTION CODE SNIPPETS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='MESH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='AODV Activiti Tue 16:53 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='cc #include #include #include #include "ns3/core-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/tnternet-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/network-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/applications-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/mobility-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" "ns3/mesh-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include #include "ns3/mesh-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/yans-wtfi-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" using nanespace ns3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS_LOG_COMPONENT_DEFINE ("TestMeshscript");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' class MeshTest A public: MeshTest ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void Configure (int argc, char * argv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' int Run ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' private: a int n_xsize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' int n_ysize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double m_step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double m_randomstart;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double m_totalTine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double m_packetInterval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint16_t n_packetsize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t n_nifaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' bool n_chan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' bool std:istring m_stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' m_ascti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' std::string m_root;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NodeContainer nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NetDeviceContainer meshDevices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Ipv4Interfacecontatner interfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MeshHelper nesh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' private: void CreateNodes ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void InstallInternetstack,();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' votd InstallAppltcation ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void Report ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' MeshTest: :MeshTest () n_xsize (3), n_ysize,(3) n_step(100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='0), n_randomstart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1), n_totalTime(100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='0)) n_packetinterval,(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1), : n_packetsize (1024), nIfaces (i), 8 apim qel++ Ln 1, Col 1 INSTue16:54 aodv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='cc #include #include ctostream> #include "ns3/core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/network-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #Include winclude "ns3/nobiltty-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" rinclude "ns3/aodv-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" rincLude "ns3/olsr-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" tinclude "ns3/dsdv-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/dsr-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" winclude #include "ns3/ocb-wtft-nac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h\' finclude "ns3/wifi-s0211p-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h incLue "ns3/wave-mac-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/flow-monitor-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" winclude "ns3/config-store-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" A #include "ns3/1nteger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" winclude "ns3/wave-bsm-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #nclude "ns3/yans-wtfi-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" "ns3/wave-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h winclude ustng nanespace ns3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' a ustng nanespace dsr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS_LOG_COMPONENT_DEFINE("vanet-routtng-compare");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' class:Routingstats publie: Routingstats ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetRxBytes ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetcumulativeRxBytes ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetRxPkts ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetCumulattveRxPkts ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void IncRxBytes (uint32_t rxBytes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void IncRxpkts ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void SetRxBytes(utnt32_trxBytes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' void SetRxPkts (uint32_t rxPkts): uint32_t GetTxBytes();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetcunulativeTxBytes ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t GetTxPkts ();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32 t GetcumulativeTxPkts O): + Tabwidth:8 Ln1,Col1 INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' DSDV Activiti Tue 16:53 dsr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='cc winclude #include "ns3/core-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/network-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/applications-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/mobility-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/config-store-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/internet-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/dsr-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/yans-wifi-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" using nanespace ns3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS_LOG_COMPONENT_DEFINE ("DsTTest");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' main (int argc, char *argv[)) int Users may find it convenient to turn on explicit debugging l for selected modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' the below lines suggest how to do this ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #ife LogConponentEnable ("Ipv4L3Protocol", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' a LogConponentEnable ("UdpL4Protocol", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("udpSocketImpl", LOG, LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("NetDevice", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogconponentEnable ("Ipv4EndPointDemux", LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #endif #ife LogcomponentEnable ("Dsropttons", LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogconponentEnable ("DsrHelper", ", LOG,LEVEL_ALL) LogConponentEnable （"DsroptionHeader ("DsrRouting",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrFsHeader" LOG_LEVEL_ALL) LogConponentEnable ("DsrGraReplyTable" ,LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable LogConponentEnable ("DsrRouteCache" ，LOGLEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrMaintainBuffer",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrRreqTable",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable C"DsrErrorBuffer LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogconponentEnable ("DsrNetworkQueue", LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #endif NS_LOG_INFO ("creating the nodes");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' // General parameters uint32_t nwifis= 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t nsinks = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double TotalTime = 608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double dataTime - 508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t packetsize = 64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double ppers =1 : double datastart - 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=" // start sending data at 100s + ++ Tab width: 8 Ln 1, Col 1 INSActivitie Tue 16:53 #include #include 'ns3/core-module." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/network-nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h* #include "ns3/applications-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/nobiltty-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include \'ns3/config-store-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/internet-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/dsr-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" #include "ns3/yans-wifi-helper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h" using namespace ns3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' NS_LOG_COMPONENT_DEFINE( ("DsrTest");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' int main (tnt argc, char *argv[]) A "l users may find it conventent, to turn on explict debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #if8 LogComponentEnable ("Ipv4L3Protocol", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' a LogConponentEnable ("UdpL4Protocol",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("udpSocketInpl", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("NetDevice", ，LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("Ipv4EndPotntDemux", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #endif #if8 LogComponentEnable ("Dsroptlons",LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsrHelper", LOG_LEVEL_ALL): LogComponentEnable ("DsrRouting",LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsropttonHeader LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrFsHeader", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsrGraReplyTable",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable （"DsrSendBuffer", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsrRouteCache", LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrMaintainBuffer",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsrRreqTable",LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogComponentEnable ("DsrErrorBuffer", ，LOG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' LogConponentEnable ("DsrNetworkQueue", LoG_LEVEL_ALL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' #endif NS_LOG_INFO ("creating the nodes");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' I/ General paraneters uint32_t nwifis = 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' uint32_t nsinks = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double TotalTime = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double dataTime = 5e8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double ppers = 1: utnt32_t packetsize = 64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' double datastart = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' // start sending data at 180s 8m ql+ Ln 1, Col1 INSAnalysis and Design of VANET Protocols for Srinagar City | 2019 Department of Information Technology, NIT Srinagar Page 48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 DSDV ROUTING TABLE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' SUMO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='1 OSM MAP OF SRINAGAR CITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content=' Activities Text Editor Tue 16:54 dsdv-rtable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E1T4oBgHgl3EQfgQRf/content/2301.03227v1.pdf'} +page_content='h Openv Save 100x #ifndef DSDV_RTABLE_H #define DSDV_RTABLE_H #include #include #include 2 a 0 is a parameter weighting +the influence of both terms. In our experimental setting, we consider degradation +models y = Ax∗ + ν ∈ Rm for some groundtruth signal x∗ ∈ Rn, a linear +operator A ∈ Rn×m and a white Gaussian noise ν ∈ Rn. We thus deal with +convex data-fidelity terms of the form f(x) = 1 +2||Ax − y||2, with Lf = ||AT A||. +1 +arXiv:2301.13731v1 [stat.ML] 31 Jan 2023 + +Our analysis can nevertheless apply to a broader class of convex or nonconvex +functions f. +To find an adequate solution of the ill-posed problem of recovering x∗ from y, +the choice of the regularization φ is crucial. Convex [19] and nonconvex [26] +handcrafted functions are now largely outperformed by learning approaches [18, +27], that may not even be associated to a closed-form regularization function φ. +1.1 +Proximal algorithms +Estimating a local or global optima of problem (1) is classically done us- +ing proximal splitting algorithms such as Proximal Gradient Descent (PGD) +or Douglas-Rashford Splitting (DRS). Given an adequate stepsize τ > 0, these +methods alternate between explicit gradient descent, Id −τ∇h for smooth func- +tions h, and/or implicit gradient steps using the proximal operator Proxτh(x) ∈ +arg minz +1 +2τ ||z −x||2 +h(z) , for proper lower semi-continuous functions h. Proxi- +mal algorithms are originally designed for convex functions, but under appropriate +assumptions, PGD [2] and DRS [23] algorithms converge to a stationary point +of problem (1) associated to nonconvex functions f and φ. +1.2 +Plug-and-Play algorithms +Plug-and-Play (PnP) [25] and Regularization by Denoising (RED) [18] methods +consist in splitting algorithms in which the descent step on the regularization +function is performed by an off-the-shelf image denoiser. They are respectively +built from proximal splitting schemes by replacing the proximal operator (PnP) +or the gradient operator (RED) of the regularization φ by an image denoiser. +When used with a deep denoiser (i.e parameterized by a neural network) these +approaches produce impressive results for various image restoration tasks [27]. +Theoretical convergence of PnP and RED algorithms with deep denoisers +has recently been addressed by a variety of studies [20, 21, 17, 9]. Most of these +works require specific constraints on the deep denoiser, such as nonexpansivity. +However, imposing nonexpansivity of a denoiser can severely degrade its denoising +performance. +Another line of works tries to address convergence by making PnP and RED +algorithms be exact proximal algorithms. The idea is to replace the denoiser of +RED algorithms by a gradient descent operator and the one of PnP algorithm +by a proximal operator. +Theoretical convergence then follows from known +convergence results of proximal splitting algorithms. The authors of [7, 10] thus +propose to plug an explicit gradient-step denoiser of the form D = Id −∇g, for +a tractable and potentially nonconvex potential g parameterized by a neural +network. As shown in [10], such a constrained parametrization does not harm +denoising performance. The gradient-step denoiser guarantees convergence of +RED methods without sacrificing performance, but it does not cover convergence +of PnP algorithms. An extension to PnP has been addressed in [11]: following [8], +when g is trained with contractive gradient, the gradient-step denoiser can be +written as a proximal operator D = Id −∇g = Proxφ of a nonconvex potential φ. +A PnP scheme with this proximal denoiser becomes again a genuine proximal +splitting algorithm associated to an explicit functional. +Following existing +convergence results of the PGD and DRS algorithms in the nonconvex setting, +[11] proves convergence of PnP-PGD and PnP-DRS with proximal denoiser. +2 + +The main limitation of this approach is that the proximal denoiser D = Proxφ +does not give tractability of Proxτφ for τ ̸= 1. Therefore, to be a provable +converging proximal splitting algorithm, the stepsize of the overall PnP algorithm +has to be fixed to τ = 1. For instance, for the PGD algorithm with stepsize +τ = 1: +xk+1 ∈ Proxφ(Id −λ∇f)(xk) +(2) +the convergence of xk to a stationary point of (1) is only ensured for regularization +parameters λ satisfying Lfλ < 1 [2]. This is an issue for low noise levels ν, for +which relevant solutions are obtained with a dominant data-fidelity term in (1). +Our objective is to design a convergent PnP algorithm with a proximal de- +noiser, and with minimal restriction on the regularization parameter λ. Contrary +to previous work on PnP convergence [20, 21, 22, 7, 10, 9], we not only wish to +adapt the denoiser but also the original optimization scheme of interest. We +study a new proximal algorithm able to deal with a proximal operator that can +only be computed for a predefined and fixed stepsize τ = 1. +1.3 +Contributions and outline +In this paper, we propose a relaxation of the Proximal Gradient Descent algorithm, +called αPGD, such that when used with a proximal denoiser, the corresponding +PnP scheme Prox-PnP-αPGD can converge for any regularization parameter λ. +In section 2, extending the result from [11], we show how building a denoiser +D that corresponds to the proximal operator of a M-weakly convex function φ. +Then we introduce a relaxation of the denoiser that allows to control M, the +constant of weak convexity of φ. +In section 3, we give new results on the convergence of Prox-PnP-PGD +[11] with regularization constraint λ(Lf + M) < 2. In particular, using the +convergence of PGD for nonconvex f and weakly convex φ given in Theorem 1, +Corollary 1 improves previous PnP convergence results [11] for M < Lf. +In section 4, we present αPGD, a relaxed version of the PGD algorithm1 +reminiscent of the accelerated PGD scheme from [24]. Its convergence is shown +Theorem 2 for a smooth convex function f and a weakly convex one φ. Corol- +lary 2 then illustrates how the relaxation parameter α can be tuned to make +the proposed PnP-αPGD algorithm convergent for regularization parameters λ +satisfying λLfM < 1. Having a multiplication, instead of an addition, between +the constants Lf and M opens new perspectives. In particular, by plugging a +relaxed denoiser with controllable weak convexity constant, Corollary 3 demon- +strates that, for all regularization parameter λ, we can always decrease M such +that λLfM < 1 i.e. such that the αPGD algorithm converges. +In section 5, we provide experiments for both image deblurring and image +super-resolution applications. We demonstrate the effectiveness of our PnP- +αPGD algorithm, which closes the performance gap between Prox-PnP-PGD +and the state-of-the-art plug-and-play algorithms. +1There are two different notions of relaxation in this paper. One is for the relaxation of the +proximal denoiser and the other for the relaxation of the optimization scheme. +3 + +2 +Relaxed Proximal Denoiser +This section introduces the denoiser used in our PnP algorithm. We first redefine +the Gradient Step denoiser and show in Proposition 1 how it can be constrained +to be a proximal denoiser; and finally introduced the relaxed proximal denoiser. +2.1 +Gradient Step Denoiser +In this paper, we make use of the Gradient Step Denoiser introduced in [10, 7]. +It writes as a gradient step over a differentiable potential gσ parametrized by +a neural network. +Dσ = Id −∇gσ. +(3) +This denoiser can then be trained to denoise white Gaussian noise νσ of various +standard deviations σ by minimizing the ℓ2 denoising loss E[||Dσ(x+νσ)−x)||2]. +When parametrized using a DRUNet architecture [27], it was shown in [10] that +the Gradient Step Denoiser (3), despite being constrained to be a conservative +vector field (as in [18]), achieves state-of-the-art denoising performance. +2.2 +Proximal Denoiser +We propose here an new version of the result of [11] on the characterization +of the gradient-step denoiser as a proximal operator of some potential φ. In +particular, we state a new result regarding the weak convexity of the φ function. +The proof of this result, given in Appendix A relies on the results from [8]. +Proposition 1 (Proximal denoisers) Let gσ : Rn → R a C2 function with +∇gσ Lgσ-Lipschitz with Lgσ < 1. Then, for Dσ := Id −∇gσ, there exists a po- +tential φσ : Rn → R ∪ {+∞} such that Proxφσ is one-to-one and +Dσ = Proxφσ +(4) +Moreover, φσ is +Lgσ +Lgσ +1-weakly convex and it can be written φσ = ˆφσ + K on +Im(Dσ) (which is open) for some constant K ∈ R, with ˆφσ : X → R ∪ {+∞} +defined by +ˆφσ(x) := +� +gσ(Dσ +−1(x))) − 1 +2 +����Dσ +−1(x) − x +����2 +if x ∈ Im(Dσ), ++∞ otherwise. +(5) +Additionally ˆφσ verifies ∀x ∈ Rn, ˆφσ(x) ≥ gσ(x). +To get a proximal denoiser from the denoiser (3), the gradient of the learned +potential gσ must be contractive. In [11] the Lipschitz constant of ∇gσ is softly +constrained to satisfy Lgσ < 1, by penalizing the spectral norm +����∇2gσ(x + νσ) +���� +S +in the denoiser training loss. +2.3 +Relaxed Denoiser +Once trained, the Gradient Step Denoiser Dσ = Id −∇gσ can be relaxed as +in [10] with a parameter γ ∈ [0, 1] +Dγ +σ = γDσ + (1 − γ) Id = Id −γ∇gσ. +(6) +4 + +Applying Proposition 1 with gγ +σ = γgσ which has a γLgσ-Lipschitz gradient, we +get that if γLg < 1, there exists a +γLgσ +γLgσ +1-weakly convex φγ +σ such that +Dγ +σ = Proxφγ +σ, +(7) +satisfying φ0 +σ = 0 and φ1 +σ = φσ. Hence, one can control the weak convexity of +the regularization function by relaxing the proximal denoising operator Dγ +σ. +3 +Plug-and-Play Proximal Gradient Descent (PnP- +PGD) +In this section, we give convergence results for the Prox-PnP-PGD algorithm, +xk+1 = Dσ ◦ (Id −λf)(xk) = Proxφσ ◦(Id −λf)(xk), +which is the PnP version of PGD, with plugged Proximal Denoiser (4). The +authors of [11] proposed a suboptimal convergence results as the semiconvexity of +φσ was not exploited. Doing so, we improve the condition on the regularization +parameter λ for convergence. +We present properties of smooth functions and weakly convex functions +in Section 3.1. Then we show in Section 3.2 the convergence of PGD in the +smooth/weakly convex setting. We finally apply this result to PnP in Section 3.3. +3.1 +Useful inequalities +We present two results relative to weakly convex functions and smooth ones. +We make use of the subdifferential of a proper, nonconvex function φ defined as +∂φ(x)= +� +v ∈ Rn, ∃(xk), φ(xk)→φ(x), vk →v, limz→xk +φ(z)−φ(xk)−⟨vk,z−xk⟩ +||z−xk|| +≥0 ∀k +� +. +Proposition 2 (Weakly convex functions, proof in Appendix B) For φ +proper lsc and M-weakly convex with M > 0, +(i) ∀x, y and t ∈ [0, 1], +φ(tx + (1 − t)y) ≤ tφ(x) + (1 − t)φ(y) + M +2 t(1 − t) ||x − y||2 ; +(8) +(ii) ∀x, y, we have ∀z ∈ ∂φ(y), +φ(x) ≥ φ(y) + ⟨z, x − y⟩ − M +2 ||x − y||2 ; +(9) +(iii) Three-points inequality. For z+ ∈ Proxφ(z), we have, ∀x +φ(x) + 1 +2 ||x − z||2 ≥ φ(z+) + 1 +2 +����z+ − z +����2 + 1 − M +2 +����x − z+����2 . +(10) +Lemma 1 (Descent Lemma for smooth functions) For f proper differen- +tiable and with a Lf-Lipschitz gradient, we have ∀x, y +f(x) ≤ f(y) + ⟨∇f(y), x − y⟩ + Lf +2 ||x − y||2 . +(11) +5 + +3.2 +Proximal Gradient Descent with a weakly convex func- +tion +We consider the following minimization problem for a smooth nonconvex func- +tion f and a weakly convex function φ that are both bounded from below: +min +x F(x) := λf(x) + φ(x). +(12) +We now show under which conditions the classical Proximal Gradient Descent +xk+1 ∈ Proxτφ ◦(Id −τλ∇f)(xk) +(13) +converges to a stationary point of (12). We first show convergence of function +values, and then convergence of the iterates, if F verifies the Kurdyka-Lojasiewicz +(KL) property [2]. Large classes of functions, in particular all the proper, closed, +semi-algebraic functions [1] satisfy this property, which is, in practice, the case +of all the functions considered in this analysis. +Theorem 1 (Convergence of PGD algorithm (13)) Assume f and φ proper +lsc, bounded from below with f differentiable with Lf-Lipschitz gradient, and φ +M-weakly convex. Then for τ < 2/(λLf + M), the iterates (13) verify +(i) (F(xk)) monotonically decreases and converges. +(ii) ||xk+1 − xk|| converges to 0 at rate mink≤K ||xk+1 − xk|| = O(1/ +√ +K) +(iii) All cluster points of the sequence xk are stationary points of F. +(iv) If the sequence xk is bounded and if F verifies the KL property at the +cluster points of xk, then xk converges, with finite length, to a stationary +point of F. +The proof follows standard arguments of the convergence analysis of the +PGD in the nonconvex setting [4, 2, 16]. We only demonstrate here the first +point of the theorem, the rest of the proof is detailed in Appendix C. Proof.[i] +Relation (13) leads to +xk−xk+1 +τ +− λ∇f(xk) ∈ ∂φ(xk+1), by definition of the +proximal operator. As φ is M-weakly convex, Proposition 2 (ii) leads to +φ(xk)≥φ(xk+1)+||xk − xk+1||2 +τ ++λ⟨∇f(xk), xk+1− xk⟩ − M +2 ||xk − xk+1||2. +(14) +The descent Lemma 1 gives for f: +f(xk+1) ≤ f(xk) + ⟨∇f(xk), xk+1 − xk⟩ + Lf +2 ||xk − xk+1||2 . +(15) +Combining both inequalities, for Fλ,σ = λf + φσ, we obtain +F(xk) ≥ F(xk+1) + +�1 +τ − M + λLf +2 +� +||xk − xk+1||2 . +(16) +Therefore, if τ < 2/(M + λLf), (F(xk)) is monotically deacreasing. As F is +assumed lower-bounded, (F(xk)) converges. +□ +6 + +3.3 +Prox-PnP Proximal Gradient Descent (Prox-PnP-PGD) +Equipped with the convergence of PGD, we can now study the convergence of +Prox-PnP-PGD, the PnP-PGD algorithm with plugged Proximal Denoiser (4): +xk+1 = Dσ(Id −λf)(xk) = Proxφσ(Id −λf)(xk). +(17) +This algorithm targets stationary points of the functional Fλ,σ defined as: +Fλ,σ := λf + φσ. +(18) +The following result, obtained from Theorem 1, improves [11] using the fact +that the potential φσ is not any nonconvex function but a weakly convex one. +Corollary 1 (Convergence of Prox-PnP-PGD (17)) Let gσ : Rn → R ∪ +{+∞} of class C2, coercive, with Lgσ-Lipschitz gradient, Lgσ < 1, and Dσ := Id −∇gσ. +Let φσ be defined from gσ as in Proposition 1. Let f : Rn → R ∪ {+∞} differen- +tiable with Lf-Lipschitz gradient. Assume f and Dσ respectively KL and semi- +algebraic, and f and gσ bounded from below. Then, for λLf < (Lgσ +2)/(Lgσ +1), +the iterates xk given by the iterative scheme (17) verify the convergence properties +(i)-(iv) of Theorem 1 for F = Fλ,σ. +The proof of this result is given in Appendix D. It is a direct application +of Theorem 1 using τ = 1 and the fact that φσ defined in Proposition 1 is +M = Lgσ/(Lgσ + 1)-weakly convex. +By exploiting the weak convexity of φσ, the convergence condition λLf < 1 +of [11] is here replaced by λLf < Lgσ +2 +Lgσ +1. Even if the bound is improved, we are +still limited to regularization parameters satisfying λLf < 2. In the next section, +we propose a modification of the PGD algorithm to relax this constraint. +4 +PnP Relaxed Proximal Gradient Descent (PnP- +αPGD) +In this section, we study the convergence of a relaxed PGD algorithm applied +to problems (1) involving a smooth convex function f and a weakly convex +function φ. Our objective is to design a convergent algorithm in which the +proximal operator is only computable for τ = 1 and the data-fidelity term +constraint is less restrictive than the bound τ < 2/(M + λLf) of Theorem 1. +4.1 +αPGD algorithm +We present our main result which concerns, for weakly convex functions φ, the +convergence of the following α-relaxed PGD algorithm, defined for 0 < α < 1 as +� +� +� +� +� +qk+1 = (1 − α)yk + αxk +xk+1 = Proxτφ(xk − τλ∇f(qk+1)) +yk+1 = (1 − α)yk + αxk+1. +(19a) +(19b) +(19c) +Algorithm (19) with α = 1 exactly corresponds to the PGD algorithm (13). This +scheme is reminiscent of [24] (taking α = θk and τ = +1 +θkLf in Algorithm 1 of [24]), +which generalizes Nesterov-like accelerated proximal gradient methods [4, 15]. +7 + +As shown in [12], there is a strong connection between the proposed algorithm (19) +and the Primal-Dual algorithm [5] with Bregman proximal operator [6]. In the +convex setting, one can show that ergodic convergence is obtained with τλLf > 2 +and small values α. Convergence of a close algorithm is also shown in [14] for a +M-semi convex φ and a c > M-strongly convex f. However, φσ is here nonconvex +while f is only convex, so that a new convergence result needs to be derived. +Theorem 2 (Convergence of αPGD (19)) Assume f and φ proper lsc, bounded +from below, f convex differentiable with Lf-Lipschitz gradient and φ M-weakly +convex. Then2 for α ∈ (0, 1) and τ < min +� +1 +αλLf , α +M +� +, the updates (19) verify +(i) F(yk) + α +2τ +� +1 − 1 +α +�2 ||yk − yk−1||2 monotonically decreases and converges. +(ii) ||yk+1 − yk|| converges to 0 at rate mink≤K ||yk+1 − yk|| = O(1/ +√ +K) +(iii) All cluster points of the sequence yk are stationary points of F. +The proof, given in Appendix E, relies on Lemma 1 and Proposition 2. It follows +the general strategy of the proofs in [24], and also requires the convexity of f. +With this theorem, αPGD is shown to verify convergence of the iterates and +of the norm of the residual to 0. Note that we do not have here the analog of +Theorem 1(iv) on the iterates convergence using the KL hypothesis. Indeed, +as we detail in Appendix E.0.1, the nonconvex convergence analysis with KL +functions from [2] or [16] do not extend to our case. +When α = 1, Algorithms (19) and (13) are equivalent, but we get a slightly +worst bound in Theorem 2 than in Theorem 1 (τ < min +� +1 +λLf , 1 +M +� +≤ +2 +λLf +M ). +Nevertheless, when used with α < 1, we next show that the relaxed algorithm is +more relevant in the perspective of PnP with proximal denoiser. +4.2 +Prox-PnP-αPGD algorithm +We can now study the Prox-PnP-αPGD algorithm obtained by taking the +proximal denoiser (4) in the αPGD algorithm (19): +� +� +� +� +� +qk+1 = (1 − α)yk + αxk +xk+1 = Dσ(xk − λ∇f(qk+1)) +yk+1 = (1 − α)yk + αxk+1 +(20a) +(20b) +(20c) +This scheme targets the minimization of the functional Fλ,σ given in (18). +Corollary 2 (Convergence of Prox-PnP-αPGD (20)) Let gσ : Rn → R ∪ +{+∞} of class C2, coercive, with Lg < 1-Lipschitz gradient and Dσ := Id −∇gσ. +Let φσ be the M = Lgσ/(Lgσ + 1)-weakly convex function defined from gσ as +in Proposition 1. Let f be proper, convex and differentiable with Lf-Lipschitz +gradient. Assume f and gσ bounded from below. Then, if λLfM < 1 and for +any α ∈ [0, 1] +M < α < 1/(λLf) +(21) +the iterates xk given by the iterative scheme (20) verify the convergence properties +(i)-(iii) of Theorem 2 for F = Fλ,σ defined in (18). +2As shown in the proof, a better bound can be found, but with little numerical gain. +8 + +This PnP corollary is obtained by taking τ = 1 in Theorem 2 and using the +M = (Lgσ)/(Lgσ + 1)-weakly convex potential φσ defined in Proposition 1. +The existence of α ∈ [0, 1] satisfying relation (21) is ensured as soon as +λLfM < 1. As a consequence, when M gets small (i.e φσ gets "more convex") +λLf can get arbitrarily large. This is a major advance compared to Prox-PnP- +PGD that was limited (Corollary 1) to λLf < 2 even for convex φ (M = 0). To +further exploit this property, we now consider the relaxed denoiser Dγ +σ (6) that +is associated to a function φγ +σ with a tunable weak convexity constant M γ. +Corollary 3 (Convergence of Prox-PnP-αPGD with relaxed denoiser) +Let F γ +λ,σ := λf +φγ +σ, with the M γ = +γLg +γLg+1-weakly convex potential φγ +σ introduced +in (7) and Lg < 1. Then, for M γ < α < 1/(λLf), the iterates xk given by +the Prox-PnP-αPGD (20) with γ-relaxed denoiser Dγ +σ defined in (6) verify the +convergence properties (i)-(iii) of Theorem 2 for F = F γ +λ,σ. +Therefore, using the γ-relaxed denoiser Dγ +σ = γDσ + (1 − γ) Id, the overall +convergence condition on λ is now λ < +1 +Lf +γLg +γLg+1. +Provided γ gets small, λ can be arbitrarily large. Small γ means small +amount of regularization brought by denoising at each step of the PnP algorithm. +Moreover, for small γ, the targeted regularization function φγ +σ gets close to a +convex function and it has already been observed that deep convex regularization +can be sub-optimal compared to more flexible nonconvex ones [7]. Depending on +the inverse problem, and on the necessary amount of regularization, the choice +of the couple (γ, λ) will be of paramount importance for efficient restoration. +5 +Experiments +The efficiency of the proposed Prox-PnP-αPGD algorithm (20) is now demon- +strated on deblurring and super-resolution. For both applications, we consider +a degraded observation y = Ax∗ + ν ∈ Rm of a clean image x∗ ∈ Rn that +is estimated by solving problem (1) with f(x) = +1 +2 ||Ax − y||2. Its gradient +∇f = AT (Ax−y) is thus Lipschitz with constant Lf = +����AT A +���� +S. We use for eval- +uation and comparison the 68 images from the CBSD68 dataset, center-cropped +to n = 256 × 256 and Gaussian noise with 3 noise levels ν ∈ {0.01, 0.03, 0.05}. +For deblurring, the degradation operator A = H is a convolution performed +with circular boundary conditions. As in [28, 10, 17, 27], we consider the 8 +real-world camera shake kernels of [13], the 9 × 9 uniform kernel and the 25 × 25 +Gaussian kernel with standard deviation 1.6. +For single image super-resolution (SR), the low-resolution image y ∈ Rm is +obtained from the high-resolution one x ∈ Rn via y = SHx+ν where H ∈ Rn×n +is the convolution with anti-aliasing kernel. The matrix S is the standard s-fold +downsampling matrix of size m × n and n = s2 × m. As in [27], we evaluate SR +performance on 4 isotropic Gaussian blur kernels with standard deviations 0.7, +1.2, 1.6 and 2.0; and consider downsampled images at scale s = 2 and s = 3. +The proximal denoiser Dσ defined in Proposition 1 is trained following [11] +with Lg < 1. For both Prox-PnP-PGD and Prox-PnP-αPGD algorithm, we +use the γ-relaxed version of the denoiser (6). All the hypotheses on f and gσ +from Corollaries 1 and 3 are thus verified and convergence of Prox-PnP-PGD +and Prox-PnP-αPGD are theoretically guaranteed provided the corresponding +9 + +conditions on λ are satisfied. Hyper-parameters γ ∈ [0, 1], λ and σ are optimized +via grid-search. In practice, we found that the same choice of parameters γ and +σ are optimal for both PGD and αPGD, with values depending on the amount +of noise ν in the input image. We thus choose λ ∈ [0, λlim] where for Prox- +PnP-PGD λPGD +lim += +1 +Lf +γ+2 +γ+1 and for Prox-PnP-αPGD λαPGD +lim += +1 +Lf +γ+1 +γ +≥ λPGD +lim . +For both ν = 0.01 and ν = 0.03, λ is set to its maximal allowed value λlim. +Prox-PnP-αPGD is expected to outperform Prox-PnP-PGD at these noise levels. +Finally, for Prox-PnP-αPGD, α is set to its maximum possible value 1/(λLf). +We numerically compare in Table 1 the presented methods Prox-PnP-PGD +(that improves [11]) and Prox-PnP-PGD against three state-of-the-art deep PnP +approaches: IRCNN [28], DPIR [27], and GS-PnP [10]. Among them, only +GS-PnP has convergence guarantees. Both IRCNN and DPIR use PnP-HQS, the +PnP version of the Half-Quadratic Splitting algorithm, with well-chosen varying +stepsizes. GS-PnP uses the gradient-step denoiser (3) in PnP-HQS. +As expected, by allowing larger values for λ, we observe that Prox-PnP-αPGD +outperforms Prox-PnP-PGD in PSNR at low noise level ν ∈ {0.01, 0.03}. The +performance gap is significant for deblurring and super-resolution with scale 2 +and ν = 0.01, in which case only a low amount of regularization is necessary, that +is to say a large λ value. In these conditions, Prox-PnP-αPGD almost closes the +PSNR gap between Prox-PnP-PGD and the state-of-the-art PnP methods DPIR +and GS-PnP that do not have any restriction on the choice of λ. We assume +that the remaining difference of performance is due to the γ-relaxation of the +denoising operation that affects the regularizer φγ. We qualitatively verify in +Figure 1 (deblurring) and Figure 2 (super-resolution), on "starfish" and "leaves" +images, the performance gain of Prox-PnP-αPGD over Prox-PnP-PGD. We +also plot the evolution of Fλ,σ(xk) and ||xk+1 − xk||2 to empirically validate the +convergence of both algorithms. +Deblurring +Super-resolution +scale s = 2 +scale s = 3 +Noise level ν +0.01 +0.03 +0.05 +0.01 +0.03 +0.05 +0.01 +0.03 +0.05 +IRCNN [28] +31.42 +28.01 +26.40 +26.97 +25.86 +25.45 +25.60 +24.72 +24.38 +DPIR [27] +31.93 28.30 +26.82 +27.79 +26.58 +25.83 +26.05 +25.27 +24.66 +GS-PnP [10] +31.70 +28.28 +26.86 +27.88 +26.81 26.01 +25.97 +25.35 24.74 +Prox-PnP-PGD +30.91 +27.97 +26.66 +27.68 +26.57 +25.81 +25.94 +25.20 +24.62 +Prox-PnP-αPGD +31.55 +28.03 +26.66 +27.92 +26.61 +25.80 +26.03 +25.26 +24.61 +Table 1: PSNR (dB) results on CBSD68 for deblurring (left) and super-resolution +(right). PSNR are averaged over 10 blur kernels for deblurring (left) and 4 blur +kernels along various scales s for super-resolution (right). +6 +Conclusion +In this paper, we propose a new convergent plug-and-play algorithm built from +a relaxed version of the Proximal Gradient Descent (PGD) algorithm. When +used with a proximal denoiser, while the original PnP-PGD imposes restrictive +conditions on the parameters of the problem, the proposed algorithms converge, +with minor conditions, towards stationary points of a weakly convex functional. +We illustrate numerically the convergence and the efficiency of the method. +10 + +(a) Clean +(b) Observed +(c) Prox-PnP-PGD +(33.32dB) +(d) Prox-PnP-αPGD +(33.62dB) +(e) Fλ,σ(xk) +Prox-PnP-PGD +(f) Fλ,σ(xk) +Prox-PnP-αPGD +(g) ||xi+1 − xi||2 +Figure 1: Deblurring of “starfish" blurred with the shown kernel and noise +ν = 0.01 . +(a) Clean +(b) Observed +(c) Prox-PnP-PGD +(28.19dB) +(d) Prox-PnP-αPGD +(28.59dB) +(e) Fλ,σ(xk) +Prox-PnP-PGD +(f) Fλ,σ(xk) +Prox-PnP-αPGD +(g) ||xi+1 − xi||2 +Figure 2: SR of "leaves" downscaled with the shown kernel, scale 2 and ν = 0.01. +Acknowledgements +This work was funded by the French ministry of research through a CDSN +grant of ENS Paris-Saclay. 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SIAM J. on Opt. 30(1), +149–181 (2020) +[24] Tseng, P.: On accelerated proximal gradient methods for convex-concave +optimization. submitted to SIAM Journal on Optimization 2(3) (2008) +[25] Venkatakrishnan, S.V., Bouman, C.A., Wohlberg, B.: Plug-and-play priors +for model based reconstruction. In: IEEE GlobalSIP. pp. 945–948 (2013) +[26] Wen, F., Chu, L., Liu, P., Qiu, R.C.: A survey on nonconvex regularization- +based sparse and low-rank recovery in signal processing, statistics, and +machine learning. IEEE Access 6, 69883–69906 (2018) +[27] Zhang, K., Li, Y., Zuo, W., Zhang, L., Van Gool, L., Timofte, R.: Plug- +and-play image restoration with deep denoiser prior. IEEE Trans. on Im. +Proces. (2021) +[28] Zhang, K., Zuo, W., Gu, S., Zhang, L.: Learning deep CNN denoiser prior +for image restoration. In: IEEE CVPR. pp. 3929–3938 (2017) +13 + +A +Proof of Proposition 1 (Characterization of the +Proximal Denoiser) +Proof. This result is an extension of the Proposition 3.1 from [11]. Under the +same assumptions as ours, it is shown, using the characterization of the Proximity +operator from [8], that for ˆφσ : Rn → R defined as +ˆφσ(x) := +� +gσ(Dσ +−1(x))) − 1 +2 +����Dσ +−1(x) − x +����2 +if x ∈ Im(Dσ), ++∞ otherwise +(22) +we have Dσ = Prox ˆφσ and ∀x ∈ Rn, ˆφσ(x) ≥ gσ(x). +We would like to show that ˆφσ is semiconvex. However, as dom(ˆφσ) = Im(Dσ) +is non necessarily convex, we can not obtain this result. We thus propose another +choice of function φσ such that Dσ = Proxφσ. As we suppose that ∇gσ = Id −Dσ +is L Lipschitz, Dσ is L + 1 Lipschitz. By [8, Proposition 2], we have ∀x ∈ X, +Dσ(x) ∈ Proxφσ(x) +(23) +for some proper lsc fonction φσ : X → Rn such that x → φσ(x)+ +� +1 − +1 +L+1 +� +||x||2 +2 +is convex, i.e. such that φσ is 1 − +1 +L+1 = +L +L+1 weakly convex. For all y ∈ Rn, +the function x → φσ(x) + 1 +2 ||x − y||2 is then strongly convex and the proximal +operator is single valued. Then ∀x ∈ X, +Dσ(x) = Proxφσ(x) = Prox ˆφσ(x) +(24) +By [8, Corollary 5], for any C ⊂ Im(Dσ) polygonally connected, there is K ∈ +R such that φσ = ˆφσ + K on C. +As X is convex, it is connected and, as +Dσ is continuous, Im(Dσ) is connected and open, thus polygonally connected. +Therefore, ∃K ∈ R such that φσ = ˆφσ + K on Im(Dσ). +□ +B +Proof of Proposition 2 (Three points inequality +for weakly convex functions) +Proof. (i) and (ii) follow from the fact that φ + M +2 ||x||2 is convex. We now prove +(iii). Optimality conditions of the proximal operator z+ ∈ Proxφ(z) gives +z − z+ ∈ ∂φ(z+). +(25) +Hence, by (ii), we have ∀x, +φ(x) ≥ φ(z+) + ⟨z − z+, x − z+⟩ − M +2 +����x − z+����2 , +(26) +and therefore, +φ(x) + 1 +2 ||x − z||2 ≥ φ(z+) + 1 +2 ||x − z||2 + ⟨z − z+, x − z+⟩ − M +2 +����x − z+����2 += φ(z+) + 1 +2 +����x − z+����2 + 1 +2 +����z − z+����2 − M +2 +����x − z+����2 += φ(z+) + 1 +2 +����z − z+����2 + 1 − M +2 +����x − z+����2 . +□ +14 + +C +Proof of Theorem 1 (PGD for nonconvex and +weakly convex optimization) +We here demonstrate the last three points of Theorem 1. Proof. +(i) We have demonstrated Section 3.2 that (F(xk)) is monotically deacreasing +and converges. We call F ∗ its limit. +(ii) Summing (16) over k = 0, 1, ..., m gives +m +� +k=0 +||xk+1 − xk||2 ≤ +1 +1 +τ − M+Lf +2 +(F(x0) − F(xm+1)) +≤ +1 +1 +τ − M+Lf +2 +(F(x0) − F ∗) . +(27) +Therefore, limk→∞ ||xk+1 − xk|| = 0 with the convergence rate +γk = min +0≤i≤k ||xi+1 − xi||2 ≤ 1 +k +F(x0) − F ∗ +1 +τ − M+Lf +2 +(28) +(iii) Suppose that a subsequence (xki)i is converging towards y. Let’s show +that x is a critical point of F. We had +xk+1 − xk +τ +− λ∇f(xk+1) ∈ ∂φ(xk+1). +(29) +From the continuity of ∇f, we have ∇f(xki) → ∇f(y). As ||xk+1 − xk|| → +0, we get +xki − xki−1 +τ +− λ∇f(xki) → −λ∇f(y). +(30) +We recall that the subdifferential of a proper, nonconvex function φ is +defined as the limiting subdifferential +∂φ(x) := +� +v ∈ Rn, ∃xk, φ(xk) → φ(x), vk → v, +lim +z→xk +φ(z) − φ(xk) − ⟨vk, z − xk⟩ +||z − xk|| +≥ 0 ∀k +� +. +(31) +which verifies +{v ∈ Rn, ∃xk, φ(xk) → φ(x), vk → v, vk ∈ ∂φ(xk)} ⊆ ∂φ(x). +(32) +Therefore, if we can show that φ(xki) → φ(y), we get −λ∇f(y) ∈ ∂φ(y) +i.e. y is a critical point of F. Using the fact that φ is lsc, +lim inf +i→∞ φ(xki) ≥ φ(y). +(33) +15 + +On the other hand, by weak convexity of φ, Proposition 2 (ii) gives, for +zki = +xki−1−xki +τ +− λ∇f(xki) ∈ ∂φ(xki), +φ(xki) ≤ φ(y) − ⟨zki, xki − y⟩ + M +2 ||xki − y||2 +≤ φ(y) + ||zki|| ||xki − y|| + M +2 ||xki − y||2 +≤ φ(y) + +�||xki−1 − xki|| +τ ++ λ ||∇f(xki)||+ M +2 ||xki − y|| +� +||xki − y|| . +(34) +Therefore, +lim sup +i→∞ +φ(xki) ≤ φ(y), +(35) +and +lim +i→∞ φ(xki) = φ(y). +(36) +(iv) We wish to apply Theorem 2.9 from [2]. We need to satisfy H1, H2, and H3 +specified in [2, Section 2.3]. Condition H1 corresponds to the sufficient +decrease condition shown in (i). For condition H3, we use that, as (xk) +is bounded, there exists a subsequence (xki) converging towards y. Then +F(xki) → F(y) has been shown in (iii). Finally, for condition H2, from (14), +we had that +zk+1 + λ∇f(xk+1) = xk+1 − xk +τ +(37) +where zk+1 ∈ ∂φ(xk+1). Therefore, +||zk+1 + λ∇f(xk+1)|| = ||xk+1 − xk|| +τ +(38) +which gives condition H2. +□ +D +Proof of Corollary 1 (Prox-PnP-PGD conver- +gence) +Proof. The convergence results is obtained applying Theorem 1 for φ = φσ defined +in Proposition 1 and τ = 1. The condition τλLf + M < 2 from Theorem 1 +becomes, with τ = 1 and the fact that φσ is M = Lgσ/(Lgσ + 1)-weakly convex +λLf < 2 − M = Lgσ + 2 +Lgσ + 1 +(39) +Point (iv) requires Fλ,σ to verify the Kurdyka-Lojasiewicz (KL) property at the +cluster points of xk, and the iterates xk to be bounded. We now show these two +points. +16 + +• As f is supposed KL, we only need to show that φσ is KL on the open +Im(Dσ). Indeed, cluster points of xk are fixed points of the PGD operator +Dσ ◦(Id −τ∇f) and thus belong to Im(Dσ). We supposed Dσ to be a semi- +algebraic mapping. As mentioned in [2], by the Tarski–Seidenberg theorem, +the composition and inverse of semi-algebraic mappings are semi-algebraic +mappings. Moreover, semi-algebraic scalar functions are KL. Therefore, +by the definition of ˆφσ equation (5), we get that φσ is semi-algebraic and +thus KL on Im(Dσ). +Let us precise here that, in practice, Dσ will be defined, as in [11], as the +gradient of a neural network. It is easy to check, using the chain rule +and the fact that semi-algebraicity is stable by product, sum, inverse and +composition that Dσ is a then a semi-algebraic mapping. +• Using Proposition 1 with the notations of the same Proposition, we have +that ∀x ∈ Rn, gσ(x) ≤ ˆφσ(x) Thus, the coercivity of gσ implies the +coercivity of ˆφσ. Moreover ∀x ∈ Im(Dσ), φσ = ˆφσ + K. Therefore, as +xk ∈ Im(Dσ), we have along the sequence that Fλ,σ = λf(xk) + φσ(xk) = +λf(xk) + ˆφσ + K. Therefore, by coercivity of ˆφσ (and lower boundedness +of f) and the fact that Fλ,σ(xk) monotonically decreases, the sequence xk +remains necessarily bounded. +□ +E +Proof of Theorem 2 (αPGD for convex and +weakly convex optimization) +Proof. (i) and (ii) : We can write (19b) as +xk+1 ∈ arg min +y∈Rn +φ(y) + λ⟨∇f(qk+1), y − xk⟩ + 1 +2τ ||y − xk||2 +∈ arg min +y∈Rn +φ(y) + λf(qk+1) + λ⟨∇f(qk+1), y − qk+1⟩ + 1 +2τ ||y − xk||2 +∈ arg min +y∈Rn +Φ(y) + 1 +2τ ||y − xk||2 +(40) +with Φ(y) := φ(y)+λf(qk+1)+λ⟨∇f(qk+1), y −qk+1⟩. As φ is M-weakly convex, +so does Φ. The three-points inequality of Proposition 2 (iii) applied to Φ thus +gives ∀y ∈ Rn, +Φ(y) + 1 +2τ ||y − xk||2 ≥ Φ(xk+1) + 1 +2τ ||xk+1 − xk||2 + +� 1 +2τ − M +2 +� +||xk+1 − y||2 +(41) +that is to say, +φ(y) + λf(qk+1) + λ⟨∇f(qk+1), y − qk+1⟩ + 1 +2τ ||y − xk||2 ≥ +φ(xk+1) + λf(qk+1) + λ⟨∇f(qk+1), xk+1 − qk+1⟩ ++ 1 +2τ ||xk+1 − xk||2 + +� 1 +2τ − M +2 +� +||xk+1 − y||2 +(42) +17 + +Using relation (19c), and the descent Lemma 1 as well as the convexity on f, +f(qk+1) + ⟨∇f(qk+1), xk+1 − qk+1⟩ +=f(qk+1) + +� +∇f(qk+1), 1 +αyk+1 + +� +1 − 1 +α +� +yk − qk+1 +� += 1 +α +� +f(qk+1) + ⟨∇f(qk+1), yk+1 − qk+1⟩ +� ++ +� +1 − 1 +α +� � +f(qk+1) + ⟨∇f(qk+1), yk − qk+1⟩ +� +≥ 1 +α +� +f(yk+1) − Lf +2 ||yk+1 − qk+1||2 +� ++ +� +1 − 1 +α +� � +f(qk+1) + ⟨∇f(qk+1), yk − qk+1⟩ +� +≥ 1 +α +� +f(yk+1) − Lf +2 ||yk+1 − qk+1||2 +� ++ +� +1 − 1 +α +� +f(yk). +(43) +Since yk+1 −qk+1 = α(xk+1 −xk) (from relations (19a) and (19c)), by combining +relations (42) and (43), we now have for all y ∈ Rn, +φ(y) + λ +� 1 +α − 1 +� +f(yk) + 1 +2τ ||y − xk||2 + λf(qk+1) + λ⟨∇f(qk+1), y − qk+1⟩ ≥ +φ(xk+1)+ λ +αf(yk+1)+ +� 1 +2τ − αλLf +2 +� +||xk+1 − xk||2+ +� 1 +2τ − M +2 +� +||xk+1 − y||2 . +(44) +Using again the convexity of f we get for all y ∈ Rn, +φ(y) + λf(y) + λ +� 1 +α − 1 +� +f(yk) + 1 +2τ ||y − xk||2 ≥ +φ(xk+1)+ λ +αf(yk+1) + +� 1 +2τ − αλLf +2 +� +||xk+1 − xk||2+ +� 1 +2τ − M +2 +� +||xk+1 − y||2 . +(45) +Now, the weak convexity of φ with relation (19c) gives +φ(xk+1) ≥ 1 +αφ(yk+1) + +� +1 − 1 +α +� +φ(yk) − M +2 (1 − α) ||yk − xk+1||2 . +(46) +Combining (45) and (46), and using F = λf + φ leads to +∀y ∈ Rn +� 1 +α − 1 +� +(F(yk) − F(y)) + 1 +2τ ||y − xk||2 ≥ +1 +α(F(yk+1) − F(y)) + +� 1 +2τ − αλLf +2 +� +||xk+1 − xk||2 ++ +� 1 +2τ − M +2 +� +||xk+1 − y||2 − M +2 (1 − α) ||yk − xk+1||2 . +(47) +18 + +For y = yk, we get +1 +α(F(yk) − F(yk+1)) ≥ − 1 +2τ ||yk − xk||2 + +� 1 +2τ − αλLf +2 +� +||xk+1 − xk||2 ++ +� 1 +2τ − M(2 − α) +2 +� +||xk+1 − yk||2 . +(48) +For constant α ∈ (0, 1), using that +yk − xk+1 = 1 +α(yk − yk+1) +yk − xk = +� +1 − 1 +α +� +(yk − yk−1) +(49) +(50) +we get +F(yk) − F(yk+1) ≥ − α +2τ +� +1 − 1 +α +�2 +||yk − yk−1||2 ++ α +� 1 +2τ − αλLf +2 +� +||xk+1 − xk||2 ++ 1 +α +� 1 +2τ − M(2 − α) +2 +� +||yk+1 − yk||2 . +(51) +With the assumption ταλLf < 1, the second term of the right-hand side is +non-negative and therefore, +F(yk) − F(yk+1) ≥ − α +2τ +� +1 − 1 +α +�2 +||yk − yk−1||2 ++ 1 +α +� 1 +2τ − M(2 − α) +2 +� +||yk+1 − yk||2 . += −δ ||yk − yk−1||2 + δ ||yk+1 − yk||2 ++ (γ − δ) ||yk+1 − yk||2 +(52) +with +δ = α +2τ +� +1 − 1 +α +�2 +γ = 1 +α +� 1 +2τ − M(2 − α) +2 +� +(53) +(54) +We now make use of the following lemma. +Lemma 2 ([3]) Let (an)n∈N and (bn)n∈N be two real sequences such that bn ≥ 0 +∀n ∈ N, (an) is bounded from below and an+1 + bn ≤ an ∀n ∈ N. Then (an)n∈N +is a monotonically non-increasing and convergent sequence and � +n∈N bn < +∞ +To apply Lemma 2 we look for a stepsize satisfying +γ − δ > 0 +i.e. +τ < α +M . +(55) +Therefore, hypothesis τ < min +� +1 +αλLf , α +M +� +gives that (F(yk) + δ ||yk − yk−1||2) +is a non-increasing and convergent sequence and that � +k ||yk − yk+1||2 < +∞. +19 + +Note that a slightly more precise bound can be found keeping the second term +in (51). For sake of completeness, we develop it here. Keeping the assumption +ταλLf < 1 in (51). We can use that +xk+1 − xk = 1 +α(yk+1 − yk) + +� +1 − 1 +α +� +(yk − yk−1). +(56) +Then, by convexity of the squared ℓ2 norm, for 0 < α < 1, we have +||yk+1 − yk||2 ≤ α ||xk+1 − xk||2 + (1 − α) ||yk − yk−1||2 +(57) +and +||xk+1 − xk||2 ≥ 1 +α ||yk+1 − yk||2 + +� +1 − 1 +α +� +||yk − yk−1||2 . +(58) +which gives finally +F(yk) − F(yk+1) ≥ +� +α +� +1 − 1 +α +� � 1 +2τ − αLf +2 +� +− α +2τ +� +1 − 1 +α +�2� +||yk − yk−1||2 ++ +� 1 +2τ − αλLf +2 ++ 1 +α( 1 +2τ − M(2 − α) +2 +) +� +||yk+1 − yk||2 . += −1 − α +2ατ +� +1 − α2τλLf +� +||yk − yk−1||2 ++ +1 +2ατ +� +1 + α − α2τλLf − τM(2 − α) +� +||yk+1 − yk||2 . += −δ ||yk − yk−1||2 + δ ||yk+1 − yk||2 + (γ − δ) ||yk+1 − yk||2 +(59) +with +δ = 1 − α +2ατ +� +1 − α2τλLf +� +γ = +1 +2ατ +� +1 − α2τλLf + α − τM(2 − α) +� +(60) +(61) +The condition on the stepsize becomes +γ − δ > 0 +⇔ τ < +2α +α3Lf + (2 − α)M +(62) +And the overall condition is +τ < min +� 1 +αLf +, +2α +α3Lf + (2 − α)M +� +(63) +(iii) The proof of this result is an extension of the proof proposed in Ap- +pendix C in the context of the classical PGD. Suppose that a subsequence (yki) +is converging towards y. Let’s show that y is a critical point of F. From (19b), +we have +xk+1 − xk +τ +− λ∇f(qk+1) ∈ ∂φ(xk+1). +(64) +20 + +First we show that xki+1 − xki → 0. We have ∀k > 1, +||xk+1 − xk|| = +���� +���� +1 +αyk+1 + (1 − 1 +α)yk − 1 +αyk − (1 − 1 +α)yk−1 +���� +���� +≤ 1 +α ||yk+1 − yk|| + ( 1 +α − 1) ||yk − yk−1|| +→ 0. +(65) +From (19a), we also get ||qk+1 − qk|| → 0. Now, let’s show that xki → y and +qki → y. First using (19c), we have +||xki − y|| ≤ ||xki+1 − y|| + ||xki+1 − xki|| +≤ 1 +α ||yki+1 − y|| + ( 1 +α − 1) ||yki − y|| + ||xki+1 − xki|| +→ 0. +(66) +Second, from (19a), we get in the same way qki → y. From the continuity of ∇f, +we get ∇f(qki) → ∇f(y) and therefore +xki − xki−1 +τ +− λ∇f(qki) → −λ∇f(y). +(67) +As explained in the proof Appendix C (iii), if we can also show that φ(xki) → φ(y), +we get from the subdifferential characterization (32) that −λ∇f(y) ∈ ∂φ(y) i.e. +y is a critical point of F. +Using the fact that φ is lsc and xki → y. +lim inf +i→∞ φ(xki) ≥ φ(y). +(68) +On the other hand, with Equation (45) for k + 1 = ki, taking i → +∞, +||y − xki+1|| → 0, ||y − xki|| → 0, f(yki) → f(y), f(yki+1) → f(y) and we +get +lim sup +i→∞ +φ(xki) ≤ φ(y), +(69) +and therefore +lim +i→∞ φ(xki) = φ(y). +(70) +E.0.1 +On the convergence of the iterates with the KL hypothesis +In order to prove a result similar to Theorem 1 (iv) on the convergence of +the iterates with the KL hypothesis, we can not directly apply Theorem 2.9 +from [2] on F as the objective function F(xk) by itself does not decrease along +the sequence but F(xk) + δ ||xk+1 − xk||2 does (where δ = +α +2τ +� +1 − 1 +α +�2). +Our situation is more similar to the variant of this result presented in [16, +Theorem 3.7]. +Indeed, denoting F : Rn × Rn → R defined as F(x, y) = +F(x) + δ ||x − y||2 and considering ∀k ≥ 1, the sequence zk = (yk, yk−1) with yk +following our algorithm, we can easily show that zk verifies the conditions H1 +and H3 specified in [16, Section 3.2]. However, condition H2 does not extend to +our algorithm. We plan as future work to derive a new version of [16, Section 3.2] +that fits to our case of interest. +□ +21 + diff --git a/lNFST4oBgHgl3EQfJDg3/content/tmp_files/load_file.txt b/lNFST4oBgHgl3EQfJDg3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9dcb782829e63ad37c83c990ca7c86695f4f8ae7 --- /dev/null +++ b/lNFST4oBgHgl3EQfJDg3/content/tmp_files/load_file.txt @@ -0,0 +1,650 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf,len=649 +page_content='A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser Samuel Hurault1, Antonin Chambolle2, Arthur Leclaire1, and Nicolas Papadakis1 1Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France 2CEREMADE, CNRS, Université Paris-Dauphine, PSL, Palaiseau 91128, France Abstract This paper presents a new convergent Plug-and-Play (PnP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' When plugged with a relaxed proximal denoiser, we show that the proposed PnP-αPGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1 Introduction We focus on the convergence of Plug-and-Play methods associated to the class of inverse problems: ˆx ∈ arg min x λf(x) + φ(x), (1) where f is a Lf-Lipschitz gradient function acting as a data-fidelity term with respect to a degraded observation y, φ is a M-weakly convex regularization function (such that φ + M 2 ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='||2 is convex);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' and λ > 0 is a parameter weighting the influence of both terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In our experimental setting, we consider degradation models y = Ax∗ + ν ∈ Rm for some groundtruth signal x∗ ∈ Rn, a linear operator A ∈ Rn×m and a white Gaussian noise ν ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We thus deal with convex data-fidelity terms of the form f(x) = 1 2||Ax − y||2, with Lf = ||AT A||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='13731v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='ML] 31 Jan 2023 Our analysis can nevertheless apply to a broader class of convex or nonconvex functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' To find an adequate solution of the ill-posed problem of recovering x∗ from y, the choice of the regularization φ is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Convex [19] and nonconvex [26] handcrafted functions are now largely outperformed by learning approaches [18, 27], that may not even be associated to a closed-form regularization function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 Proximal algorithms Estimating a local or global optima of problem (1) is classically done us- ing proximal splitting algorithms such as Proximal Gradient Descent (PGD) or Douglas-Rashford Splitting (DRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Given an adequate stepsize τ > 0, these methods alternate between explicit gradient descent, Id −τ∇h for smooth func- tions h, and/or implicit gradient steps using the proximal operator Proxτh(x) ∈ arg minz 1 2τ ||z −x||2 +h(z) , for proper lower semi-continuous functions h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proxi- mal algorithms are originally designed for convex functions, but under appropriate assumptions, PGD [2] and DRS [23] algorithms converge to a stationary point of problem (1) associated to nonconvex functions f and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 Plug-and-Play algorithms Plug-and-Play (PnP) [25] and Regularization by Denoising (RED) [18] methods consist in splitting algorithms in which the descent step on the regularization function is performed by an off-the-shelf image denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' They are respectively built from proximal splitting schemes by replacing the proximal operator (PnP) or the gradient operator (RED) of the regularization φ by an image denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' When used with a deep denoiser (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e parameterized by a neural network) these approaches produce impressive results for various image restoration tasks [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Theoretical convergence of PnP and RED algorithms with deep denoisers has recently been addressed by a variety of studies [20, 21, 17, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Most of these works require specific constraints on the deep denoiser, such as nonexpansivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' However, imposing nonexpansivity of a denoiser can severely degrade its denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Another line of works tries to address convergence by making PnP and RED algorithms be exact proximal algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The idea is to replace the denoiser of RED algorithms by a gradient descent operator and the one of PnP algorithm by a proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Theoretical convergence then follows from known convergence results of proximal splitting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The authors of [7, 10] thus propose to plug an explicit gradient-step denoiser of the form D = Id −∇g, for a tractable and potentially nonconvex potential g parameterized by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As shown in [10], such a constrained parametrization does not harm denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The gradient-step denoiser guarantees convergence of RED methods without sacrificing performance, but it does not cover convergence of PnP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' An extension to PnP has been addressed in [11]: following [8], when g is trained with contractive gradient, the gradient-step denoiser can be written as a proximal operator D = Id −∇g = Proxφ of a nonconvex potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' A PnP scheme with this proximal denoiser becomes again a genuine proximal splitting algorithm associated to an explicit functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Following existing convergence results of the PGD and DRS algorithms in the nonconvex setting, [11] proves convergence of PnP-PGD and PnP-DRS with proximal denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 2 The main limitation of this approach is that the proximal denoiser D = Proxφ does not give tractability of Proxτφ for τ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, to be a provable converging proximal splitting algorithm, the stepsize of the overall PnP algorithm has to be fixed to τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For instance, for the PGD algorithm with stepsize τ = 1: xk+1 ∈ Proxφ(Id −λ∇f)(xk) (2) the convergence of xk to a stationary point of (1) is only ensured for regularization parameters λ satisfying Lfλ < 1 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' This is an issue for low noise levels ν, for which relevant solutions are obtained with a dominant data-fidelity term in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Our objective is to design a convergent PnP algorithm with a proximal de- noiser, and with minimal restriction on the regularization parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Contrary to previous work on PnP convergence [20, 21, 22, 7, 10, 9], we not only wish to adapt the denoiser but also the original optimization scheme of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We study a new proximal algorithm able to deal with a proximal operator that can only be computed for a predefined and fixed stepsize τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='3 Contributions and outline In this paper, we propose a relaxation of the Proximal Gradient Descent algorithm, called αPGD, such that when used with a proximal denoiser, the corresponding PnP scheme Prox-PnP-αPGD can converge for any regularization parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In section 2, extending the result from [11], we show how building a denoiser D that corresponds to the proximal operator of a M-weakly convex function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then we introduce a relaxation of the denoiser that allows to control M, the constant of weak convexity of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In section 3, we give new results on the convergence of Prox-PnP-PGD [11] with regularization constraint λ(Lf + M) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In particular, using the convergence of PGD for nonconvex f and weakly convex φ given in Theorem 1, Corollary 1 improves previous PnP convergence results [11] for M < Lf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In section 4, we present αPGD, a relaxed version of the PGD algorithm1 reminiscent of the accelerated PGD scheme from [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Its convergence is shown Theorem 2 for a smooth convex function f and a weakly convex one φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Corol- lary 2 then illustrates how the relaxation parameter α can be tuned to make the proposed PnP-αPGD algorithm convergent for regularization parameters λ satisfying λLfM < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Having a multiplication, instead of an addition, between the constants Lf and M opens new perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In particular, by plugging a relaxed denoiser with controllable weak convexity constant, Corollary 3 demon- strates that, for all regularization parameter λ, we can always decrease M such that λLfM < 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' such that the αPGD algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In section 5, we provide experiments for both image deblurring and image super-resolution applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We demonstrate the effectiveness of our PnP- αPGD algorithm, which closes the performance gap between Prox-PnP-PGD and the state-of-the-art plug-and-play algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1There are two different notions of relaxation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' One is for the relaxation of the proximal denoiser and the other for the relaxation of the optimization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 3 2 Relaxed Proximal Denoiser This section introduces the denoiser used in our PnP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We first redefine the Gradient Step denoiser and show in Proposition 1 how it can be constrained to be a proximal denoiser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' and finally introduced the relaxed proximal denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 Gradient Step Denoiser In this paper, we make use of the Gradient Step Denoiser introduced in [10, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' It writes as a gradient step over a differentiable potential gσ parametrized by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Dσ = Id −∇gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (3) This denoiser can then be trained to denoise white Gaussian noise νσ of various standard deviations σ by minimizing the ℓ2 denoising loss E[||Dσ(x+νσ)−x)||2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' When parametrized using a DRUNet architecture [27], it was shown in [10] that the Gradient Step Denoiser (3), despite being constrained to be a conservative vector field (as in [18]), achieves state-of-the-art denoising performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 Proximal Denoiser We propose here an new version of the result of [11] on the characterization of the gradient-step denoiser as a proximal operator of some potential φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In particular, we state a new result regarding the weak convexity of the φ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The proof of this result, given in Appendix A relies on the results from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proposition 1 (Proximal denoisers) Let gσ : Rn → R a C2 function with ∇gσ Lgσ-Lipschitz with Lgσ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then, for Dσ := Id −∇gσ, there exists a po- tential φσ : Rn → R ∪ {+∞} such that Proxφσ is one-to-one and Dσ = Proxφσ (4) Moreover, φσ is Lgσ Lgσ +1-weakly convex and it can be written φσ = ˆφσ + K on Im(Dσ) (which is open) for some constant K ∈ R, with ˆφσ : X → R ∪ {+∞} defined by ˆφσ(x) := � gσ(Dσ −1(x))) − 1 2 ����Dσ −1(x) − x ����2 if x ∈ Im(Dσ), +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (5) Additionally ˆφσ verifies ∀x ∈ Rn, ˆφσ(x) ≥ gσ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' To get a proximal denoiser from the denoiser (3), the gradient of the learned potential gσ must be contractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In [11] the Lipschitz constant of ∇gσ is softly constrained to satisfy Lgσ < 1, by penalizing the spectral norm ����∇2gσ(x + νσ) ���� S in the denoiser training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='3 Relaxed Denoiser Once trained, the Gradient Step Denoiser Dσ = Id −∇gσ can be relaxed as in [10] with a parameter γ ∈ [0, 1] Dγ σ = γDσ + (1 − γ) Id = Id −γ∇gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (6) 4 Applying Proposition 1 with gγ σ = γgσ which has a γLgσ-Lipschitz gradient, we get that if γLg < 1, there exists a γLgσ γLgσ +1-weakly convex φγ σ such that Dγ σ = Proxφγ σ, (7) satisfying φ0 σ = 0 and φ1 σ = φσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Hence, one can control the weak convexity of the regularization function by relaxing the proximal denoising operator Dγ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 3 Plug-and-Play Proximal Gradient Descent (PnP- PGD) In this section, we give convergence results for the Prox-PnP-PGD algorithm, xk+1 = Dσ ◦ (Id −λf)(xk) = Proxφσ ◦(Id −λf)(xk), which is the PnP version of PGD, with plugged Proximal Denoiser (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The authors of [11] proposed a suboptimal convergence results as the semiconvexity of φσ was not exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Doing so, we improve the condition on the regularization parameter λ for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We present properties of smooth functions and weakly convex functions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then we show in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 the convergence of PGD in the smooth/weakly convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We finally apply this result to PnP in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 Useful inequalities We present two results relative to weakly convex functions and smooth ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We make use of the subdifferential of a proper, nonconvex function φ defined as ∂φ(x)= � v ∈ Rn, ∃(xk), φ(xk)→φ(x), vk →v, limz→xk φ(z)−φ(xk)−⟨vk,z−xk⟩ ||z−xk|| ≥0 ∀k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proposition 2 (Weakly convex functions, proof in Appendix B) For φ proper lsc and M-weakly convex with M > 0, (i) ∀x, y and t ∈ [0, 1], φ(tx + (1 − t)y) ≤ tφ(x) + (1 − t)φ(y) + M 2 t(1 − t) ||x − y||2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (8) (ii) ∀x, y, we have ∀z ∈ ∂φ(y), φ(x) ≥ φ(y) + ⟨z, x − y⟩ − M 2 ||x − y||2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (9) (iii) Three-points inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For z+ ∈ Proxφ(z), we have, ∀x φ(x) + 1 2 ||x − z||2 ≥ φ(z+) + 1 2 ����z+ − z ����2 + 1 − M 2 ����x − z+����2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (10) Lemma 1 (Descent Lemma for smooth functions) For f proper differen- tiable and with a Lf-Lipschitz gradient, we have ∀x, y f(x) ≤ f(y) + ⟨∇f(y), x − y⟩ + Lf 2 ||x − y||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (11) 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 Proximal Gradient Descent with a weakly convex func- tion We consider the following minimization problem for a smooth nonconvex func- tion f and a weakly convex function φ that are both bounded from below: min x F(x) := λf(x) + φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (12) We now show under which conditions the classical Proximal Gradient Descent xk+1 ∈ Proxτφ ◦(Id −τλ∇f)(xk) (13) converges to a stationary point of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We first show convergence of function values, and then convergence of the iterates, if F verifies the Kurdyka-Lojasiewicz (KL) property [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Large classes of functions, in particular all the proper, closed, semi-algebraic functions [1] satisfy this property, which is, in practice, the case of all the functions considered in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Theorem 1 (Convergence of PGD algorithm (13)) Assume f and φ proper lsc, bounded from below with f differentiable with Lf-Lipschitz gradient, and φ M-weakly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then for τ < 2/(λLf + M), the iterates (13) verify (i) (F(xk)) monotonically decreases and converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (ii) ||xk+1 − xk|| converges to 0 at rate mink≤K ||xk+1 − xk|| = O(1/ √ K) (iii) All cluster points of the sequence xk are stationary points of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (iv) If the sequence xk is bounded and if F verifies the KL property at the cluster points of xk, then xk converges, with finite length, to a stationary point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The proof follows standard arguments of the convergence analysis of the PGD in the nonconvex setting [4, 2, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We only demonstrate here the first point of the theorem, the rest of the proof is detailed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' [i] Relation (13) leads to xk−xk+1 τ − λ∇f(xk) ∈ ∂φ(xk+1), by definition of the proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As φ is M-weakly convex, Proposition 2 (ii) leads to φ(xk)≥φ(xk+1)+||xk − xk+1||2 τ +λ⟨∇f(xk), xk+1− xk⟩ − M 2 ||xk − xk+1||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (14) The descent Lemma 1 gives for f: f(xk+1) ≤ f(xk) + ⟨∇f(xk), xk+1 − xk⟩ + Lf 2 ||xk − xk+1||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (15) Combining both inequalities, for Fλ,σ = λf + φσ, we obtain F(xk) ≥ F(xk+1) + �1 τ − M + λLf 2 � ||xk − xk+1||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (16) Therefore, if τ < 2/(M + λLf), (F(xk)) is monotically deacreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As F is assumed lower-bounded, (F(xk)) converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='3 Prox-PnP Proximal Gradient Descent (Prox-PnP-PGD) Equipped with the convergence of PGD, we can now study the convergence of Prox-PnP-PGD, the PnP-PGD algorithm with plugged Proximal Denoiser (4): xk+1 = Dσ(Id −λf)(xk) = Proxφσ(Id −λf)(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (17) This algorithm targets stationary points of the functional Fλ,σ defined as: Fλ,σ := λf + φσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (18) The following result, obtained from Theorem 1, improves [11] using the fact that the potential φσ is not any nonconvex function but a weakly convex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Corollary 1 (Convergence of Prox-PnP-PGD (17)) Let gσ : Rn → R ∪ {+∞} of class C2, coercive, with Lgσ-Lipschitz gradient, Lgσ < 1, and Dσ := Id −∇gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let φσ be defined from gσ as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let f : Rn → R ∪ {+∞} differen- tiable with Lf-Lipschitz gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Assume f and Dσ respectively KL and semi- algebraic, and f and gσ bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then, for λLf < (Lgσ +2)/(Lgσ +1), the iterates xk given by the iterative scheme (17) verify the convergence properties (i)-(iv) of Theorem 1 for F = Fλ,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The proof of this result is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' It is a direct application of Theorem 1 using τ = 1 and the fact that φσ defined in Proposition 1 is M = Lgσ/(Lgσ + 1)-weakly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' By exploiting the weak convexity of φσ, the convergence condition λLf < 1 of [11] is here replaced by λLf < Lgσ +2 Lgσ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Even if the bound is improved, we are still limited to regularization parameters satisfying λLf < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In the next section, we propose a modification of the PGD algorithm to relax this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 4 PnP Relaxed Proximal Gradient Descent (PnP- αPGD) In this section, we study the convergence of a relaxed PGD algorithm applied to problems (1) involving a smooth convex function f and a weakly convex function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Our objective is to design a convergent algorithm in which the proximal operator is only computable for τ = 1 and the data-fidelity term constraint is less restrictive than the bound τ < 2/(M + λLf) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 αPGD algorithm We present our main result which concerns, for weakly convex functions φ, the convergence of the following α-relaxed PGD algorithm, defined for 0 < α < 1 as � � � � � qk+1 = (1 − α)yk + αxk xk+1 = Proxτφ(xk − τλ∇f(qk+1)) yk+1 = (1 − α)yk + αxk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (19a) (19b) (19c) Algorithm (19) with α = 1 exactly corresponds to the PGD algorithm (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' This scheme is reminiscent of [24] (taking α = θk and τ = 1 θkLf in Algorithm 1 of [24]), which generalizes Nesterov-like accelerated proximal gradient methods [4, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 7 As shown in [12], there is a strong connection between the proposed algorithm (19) and the Primal-Dual algorithm [5] with Bregman proximal operator [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In the convex setting, one can show that ergodic convergence is obtained with τλLf > 2 and small values α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Convergence of a close algorithm is also shown in [14] for a M-semi convex φ and a c > M-strongly convex f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' However, φσ is here nonconvex while f is only convex, so that a new convergence result needs to be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Theorem 2 (Convergence of αPGD (19)) Assume f and φ proper lsc, bounded from below, f convex differentiable with Lf-Lipschitz gradient and φ M-weakly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then2 for α ∈ (0, 1) and τ < min � 1 αλLf , α M � , the updates (19) verify (i) F(yk) + α 2τ � 1 − 1 α �2 ||yk − yk−1||2 monotonically decreases and converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (ii) ||yk+1 − yk|| converges to 0 at rate mink≤K ||yk+1 − yk|| = O(1/ √ K) (iii) All cluster points of the sequence yk are stationary points of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The proof, given in Appendix E, relies on Lemma 1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' It follows the general strategy of the proofs in [24], and also requires the convexity of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' With this theorem, αPGD is shown to verify convergence of the iterates and of the norm of the residual to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Note that we do not have here the analog of Theorem 1(iv) on the iterates convergence using the KL hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Indeed, as we detail in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1, the nonconvex convergence analysis with KL functions from [2] or [16] do not extend to our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' When α = 1, Algorithms (19) and (13) are equivalent, but we get a slightly worst bound in Theorem 2 than in Theorem 1 (τ < min � 1 λLf , 1 M � ≤ 2 λLf +M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Nevertheless, when used with α < 1, we next show that the relaxed algorithm is more relevant in the perspective of PnP with proximal denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 Prox-PnP-αPGD algorithm We can now study the Prox-PnP-αPGD algorithm obtained by taking the proximal denoiser (4) in the αPGD algorithm (19): � � � � � qk+1 = (1 − α)yk + αxk xk+1 = Dσ(xk − λ∇f(qk+1)) yk+1 = (1 − α)yk + αxk+1 (20a) (20b) (20c) This scheme targets the minimization of the functional Fλ,σ given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Corollary 2 (Convergence of Prox-PnP-αPGD (20)) Let gσ : Rn → R ∪ {+∞} of class C2, coercive, with Lg < 1-Lipschitz gradient and Dσ := Id −∇gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let φσ be the M = Lgσ/(Lgσ + 1)-weakly convex function defined from gσ as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let f be proper, convex and differentiable with Lf-Lipschitz gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Assume f and gσ bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then, if λLfM < 1 and for any α ∈ [0, 1] M < α < 1/(λLf) (21) the iterates xk given by the iterative scheme (20) verify the convergence properties (i)-(iii) of Theorem 2 for F = Fλ,σ defined in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 2As shown in the proof, a better bound can be found, but with little numerical gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 8 This PnP corollary is obtained by taking τ = 1 in Theorem 2 and using the M = (Lgσ)/(Lgσ + 1)-weakly convex potential φσ defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The existence of α ∈ [0, 1] satisfying relation (21) is ensured as soon as λLfM < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As a consequence, when M gets small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e φσ gets "more convex") λLf can get arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' This is a major advance compared to Prox-PnP- PGD that was limited (Corollary 1) to λLf < 2 even for convex φ (M = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' To further exploit this property, we now consider the relaxed denoiser Dγ σ (6) that is associated to a function φγ σ with a tunable weak convexity constant M γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Corollary 3 (Convergence of Prox-PnP-αPGD with relaxed denoiser) Let F γ λ,σ := λf +φγ σ, with the M γ = γLg γLg+1-weakly convex potential φγ σ introduced in (7) and Lg < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then, for M γ < α < 1/(λLf), the iterates xk given by the Prox-PnP-αPGD (20) with γ-relaxed denoiser Dγ σ defined in (6) verify the convergence properties (i)-(iii) of Theorem 2 for F = F γ λ,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, using the γ-relaxed denoiser Dγ σ = γDσ + (1 − γ) Id, the overall convergence condition on λ is now λ < 1 Lf γLg γLg+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Provided γ gets small, λ can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Small γ means small amount of regularization brought by denoising at each step of the PnP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Moreover, for small γ, the targeted regularization function φγ σ gets close to a convex function and it has already been observed that deep convex regularization can be sub-optimal compared to more flexible nonconvex ones [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Depending on the inverse problem, and on the necessary amount of regularization, the choice of the couple (γ, λ) will be of paramount importance for efficient restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 5 Experiments The efficiency of the proposed Prox-PnP-αPGD algorithm (20) is now demon- strated on deblurring and super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For both applications, we consider a degraded observation y = Ax∗ + ν ∈ Rm of a clean image x∗ ∈ Rn that is estimated by solving problem (1) with f(x) = 1 2 ||Ax − y||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Its gradient ∇f = AT (Ax−y) is thus Lipschitz with constant Lf = ����AT A ���� S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We use for eval- uation and comparison the 68 images from the CBSD68 dataset, center-cropped to n = 256 × 256 and Gaussian noise with 3 noise levels ν ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='05}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For deblurring, the degradation operator A = H is a convolution performed with circular boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As in [28, 10, 17, 27], we consider the 8 real-world camera shake kernels of [13], the 9 × 9 uniform kernel and the 25 × 25 Gaussian kernel with standard deviation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For single image super-resolution (SR), the low-resolution image y ∈ Rm is obtained from the high-resolution one x ∈ Rn via y = SHx+ν where H ∈ Rn×n is the convolution with anti-aliasing kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The matrix S is the standard s-fold downsampling matrix of size m × n and n = s2 × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As in [27], we evaluate SR performance on 4 isotropic Gaussian blur kernels with standard deviations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' and consider downsampled images at scale s = 2 and s = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The proximal denoiser Dσ defined in Proposition 1 is trained following [11] with Lg < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For both Prox-PnP-PGD and Prox-PnP-αPGD algorithm, we use the γ-relaxed version of the denoiser (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' All the hypotheses on f and gσ from Corollaries 1 and 3 are thus verified and convergence of Prox-PnP-PGD and Prox-PnP-αPGD are theoretically guaranteed provided the corresponding 9 conditions on λ are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Hyper-parameters γ ∈ [0, 1], λ and σ are optimized via grid-search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In practice, we found that the same choice of parameters γ and σ are optimal for both PGD and αPGD, with values depending on the amount of noise ν in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We thus choose λ ∈ [0, λlim] where for Prox- PnP-PGD λPGD lim = 1 Lf γ+2 γ+1 and for Prox-PnP-αPGD λαPGD lim = 1 Lf γ+1 γ ≥ λPGD lim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For both ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03, λ is set to its maximal allowed value λlim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Prox-PnP-αPGD is expected to outperform Prox-PnP-PGD at these noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Finally, for Prox-PnP-αPGD, α is set to its maximum possible value 1/(λLf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We numerically compare in Table 1 the presented methods Prox-PnP-PGD (that improves [11]) and Prox-PnP-PGD against three state-of-the-art deep PnP approaches: IRCNN [28], DPIR [27], and GS-PnP [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Among them, only GS-PnP has convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Both IRCNN and DPIR use PnP-HQS, the PnP version of the Half-Quadratic Splitting algorithm, with well-chosen varying stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' GS-PnP uses the gradient-step denoiser (3) in PnP-HQS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As expected, by allowing larger values for λ, we observe that Prox-PnP-αPGD outperforms Prox-PnP-PGD in PSNR at low noise level ν ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The performance gap is significant for deblurring and super-resolution with scale 2 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01, in which case only a low amount of regularization is necessary, that is to say a large λ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In these conditions, Prox-PnP-αPGD almost closes the PSNR gap between Prox-PnP-PGD and the state-of-the-art PnP methods DPIR and GS-PnP that do not have any restriction on the choice of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We assume that the remaining difference of performance is due to the γ-relaxation of the denoising operation that affects the regularizer φγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We qualitatively verify in Figure 1 (deblurring) and Figure 2 (super-resolution), on "starfish" and "leaves" images, the performance gain of Prox-PnP-αPGD over Prox-PnP-PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We also plot the evolution of Fλ,σ(xk) and ||xk+1 − xk||2 to empirically validate the convergence of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Deblurring Super-resolution scale s = 2 scale s = 3 Noise level ν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='05 IRCNN [28] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='42 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='40 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='97 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='86 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='45 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='60 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='72 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='38 DPIR [27] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='93 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='30 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='79 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='58 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='83 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='05 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='27 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='66 GS-PnP [10] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='70 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='28 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='86 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='88 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='81 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='97 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='35 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='74 Prox-PnP-PGD 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='91 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='97 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='66 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='68 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='57 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='81 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='94 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='20 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='62 Prox-PnP-αPGD 31.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='26 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='61 Table 1: PSNR (dB) results on CBSD68 for deblurring (left) and super-resolution (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' PSNR are averaged over 10 blur kernels for deblurring (left) and 4 blur kernels along various scales s for super-resolution (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 6 Conclusion In this paper, we propose a new convergent plug-and-play algorithm built from a relaxed version of the Proximal Gradient Descent (PGD) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' When used with a proximal denoiser, while the original PnP-PGD imposes restrictive conditions on the parameters of the problem, the proposed algorithms converge, with minor conditions, towards stationary points of a weakly convex functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We illustrate numerically the convergence and the efficiency of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 10 (a) Clean (b) Observed (c) Prox-PnP-PGD (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='32dB) (d) Prox-PnP-αPGD (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='62dB) (e) Fλ,σ(xk) Prox-PnP-PGD (f) Fλ,σ(xk) Prox-PnP-αPGD (g) ||xi+1 − xi||2 Figure 1: Deblurring of “starfish" blurred with the shown kernel and noise ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (a) Clean (b) Observed (c) Prox-PnP-PGD (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='19dB) (d) Prox-PnP-αPGD (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='59dB) (e) Fλ,σ(xk) Prox-PnP-PGD (f) Fλ,σ(xk) Prox-PnP-αPGD (g) ||xi+1 − xi||2 Figure 2: SR of "leaves" downscaled with the shown kernel, scale 2 and ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Acknowledgements This work was funded by the French ministry of research through a CDSN grant of ENS Paris-Saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' This study has also been carried out with financial support from the French Research Agency through the PostProdLEAP and Mistic projects (ANR-19-CE23-0027-01 and ANR-19-CE40-005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' References [1] Attouch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Bolte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Redont, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Soubeyran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=': Proximal alternating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='minimization and projection methods for nonconvex problems: An approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='11 ' metadata={'source': 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optimization: Tight convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' on Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 30(1), 149–181 (2020) [24] Tseng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=': On accelerated proximal gradient methods for convex-concave optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' submitted to SIAM Journal on Optimization 2(3) (2008) [25] Venkatakrishnan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Bouman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Wohlberg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=': Plug-and-play priors for model based reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In: IEEE GlobalSIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 945–948 (2013) [26] Wen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Chu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Qiu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' : A survey on nonconvex regularization- based sparse and low-rank recovery in signal processing, statistics, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' IEEE Access 6, 69883–69906 (2018) [27] Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Zuo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Van Gool, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Timofte, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=': Plug- and-play image restoration with deep denoiser prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' on Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (2021) [28] Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Zuo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=': Learning deep CNN denoiser prior for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' In: IEEE CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 3929–3938 (2017) 13 A Proof of Proposition 1 (Characterization of the Proximal Denoiser) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' This result is an extension of the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Under the same assumptions as ours, it is shown, using the characterization of the Proximity operator from [8], that for ˆφσ : Rn → R defined as ˆφσ(x) := � gσ(Dσ −1(x))) − 1 2 ����Dσ −1(x) − x ����2 if x ∈ Im(Dσ), +∞ otherwise (22) we have Dσ = Prox ˆφσ and ∀x ∈ Rn, ˆφσ(x) ≥ gσ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We would like to show that ˆφσ is semiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' However, as dom(ˆφσ) = Im(Dσ) is non necessarily convex, we can not obtain this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We thus propose another choice of function φσ such that Dσ = Proxφσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As we suppose that ∇gσ = Id −Dσ is L Lipschitz, Dσ is L + 1 Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' By [8, Proposition 2], we have ∀x ∈ X, Dσ(x) ∈ Proxφσ(x) (23) for some proper lsc fonction φσ : X → Rn such that x → φσ(x)+ � 1 − 1 L+1 � ||x||2 2 is convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' such that φσ is 1 − 1 L+1 = L L+1 weakly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For all y ∈ Rn, the function x → φσ(x) + 1 2 ||x − y||2 is then strongly convex and the proximal operator is single valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then ∀x ∈ X, Dσ(x) = Proxφσ(x) = Prox ˆφσ(x) (24) By [8, Corollary 5], for any C ⊂ Im(Dσ) polygonally connected, there is K ∈ R such that φσ = ˆφσ + K on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As X is convex, it is connected and, as Dσ is continuous, Im(Dσ) is connected and open, thus polygonally connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, ∃K ∈ R such that φσ = ˆφσ + K on Im(Dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ B Proof of Proposition 2 (Three points inequality for weakly convex functions) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (i) and (ii) follow from the fact that φ + M 2 ||x||2 is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We now prove (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Optimality conditions of the proximal operator z+ ∈ Proxφ(z) gives z − z+ ∈ ∂φ(z+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (25) Hence, by (ii), we have ∀x, φ(x) ≥ φ(z+) + ⟨z − z+, x − z+⟩ − M 2 ����x − z+����2 , (26) and therefore, φ(x) + 1 2 ||x − z||2 ≥ φ(z+) + 1 2 ||x − z||2 + ⟨z − z+, x − z+⟩ − M 2 ����x − z+����2 = φ(z+) + 1 2 ����x − z+����2 + 1 2 ����z − z+����2 − M 2 ����x − z+����2 = φ(z+) + 1 2 ����z − z+����2 + 1 − M 2 ����x − z+����2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ 14 C Proof of Theorem 1 (PGD for nonconvex and weakly convex optimization) We here demonstrate the last three points of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (i) We have demonstrated Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2 that (F(xk)) is monotically deacreasing and converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We call F ∗ its limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (ii) Summing (16) over k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=', m gives m � k=0 ||xk+1 − xk||2 ≤ 1 1 τ − M+Lf 2 (F(x0) − F(xm+1)) ≤ 1 1 τ − M+Lf 2 (F(x0) − F ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (27) Therefore, limk→∞ ||xk+1 − xk|| = 0 with the convergence rate γk = min 0≤i≤k ||xi+1 − xi||2 ≤ 1 k F(x0) − F ∗ 1 τ − M+Lf 2 (28) (iii) Suppose that a subsequence (xki)i is converging towards y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let’s show that x is a critical point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We had xk+1 − xk τ − λ∇f(xk+1) ∈ ∂φ(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (29) From the continuity of ∇f, we have ∇f(xki) → ∇f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As ||xk+1 − xk|| → 0, we get xki − xki−1 τ − λ∇f(xki) → −λ∇f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (30) We recall that the subdifferential of a proper, nonconvex function φ is defined as the limiting subdifferential ∂φ(x) := � v ∈ Rn, ∃xk, φ(xk) → φ(x), vk → v, lim z→xk φ(z) − φ(xk) − ⟨vk, z − xk⟩ ||z − xk|| ≥ 0 ∀k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (31) which verifies {v ∈ Rn, ∃xk, φ(xk) → φ(x), vk → v, vk ∈ ∂φ(xk)} ⊆ ∂φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (32) Therefore, if we can show that φ(xki) → φ(y), we get −λ∇f(y) ∈ ∂φ(y) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' y is a critical point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Using the fact that φ is lsc, lim inf i→∞ φ(xki) ≥ φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (33) 15 On the other hand, by weak convexity of φ, Proposition 2 (ii) gives, for zki = xki−1−xki τ − λ∇f(xki) ∈ ∂φ(xki), φ(xki) ≤ φ(y) − ⟨zki, xki − y⟩ + M 2 ||xki − y||2 ≤ φ(y) + ||zki|| ||xki − y|| + M 2 ||xki − y||2 ≤ φ(y) + �||xki−1 − xki|| τ + λ ||∇f(xki)||+ M 2 ||xki − y|| � ||xki − y|| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (34) Therefore, lim sup i→∞ φ(xki) ≤ φ(y), (35) and lim i→∞ φ(xki) = φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (36) (iv) We wish to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='9 from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We need to satisfy H1, H2, and H3 specified in [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Condition H1 corresponds to the sufficient decrease condition shown in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For condition H3, we use that, as (xk) is bounded, there exists a subsequence (xki) converging towards y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then F(xki) → F(y) has been shown in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Finally, for condition H2, from (14), we had that zk+1 + λ∇f(xk+1) = xk+1 − xk τ (37) where zk+1 ∈ ∂φ(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, ||zk+1 + λ∇f(xk+1)|| = ||xk+1 − xk|| τ (38) which gives condition H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ D Proof of Corollary 1 (Prox-PnP-PGD conver- gence) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The convergence results is obtained applying Theorem 1 for φ = φσ defined in Proposition 1 and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The condition τλLf + M < 2 from Theorem 1 becomes, with τ = 1 and the fact that φσ is M = Lgσ/(Lgσ + 1)-weakly convex λLf < 2 − M = Lgσ + 2 Lgσ + 1 (39) Point (iv) requires Fλ,σ to verify the Kurdyka-Lojasiewicz (KL) property at the cluster points of xk, and the iterates xk to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We now show these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 16 As f is supposed KL, we only need to show that φσ is KL on the open Im(Dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Indeed, cluster points of xk are fixed points of the PGD operator Dσ ◦(Id −τ∇f) and thus belong to Im(Dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We supposed Dσ to be a semi- algebraic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As mentioned in [2], by the Tarski–Seidenberg theorem, the composition and inverse of semi-algebraic mappings are semi-algebraic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Moreover, semi-algebraic scalar functions are KL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, by the definition of ˆφσ equation (5), we get that φσ is semi-algebraic and thus KL on Im(Dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let us precise here that, in practice, Dσ will be defined, as in [11], as the gradient of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' It is easy to check, using the chain rule and the fact that semi-algebraicity is stable by product, sum, inverse and composition that Dσ is a then a semi-algebraic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Using Proposition 1 with the notations of the same Proposition, we have that ∀x ∈ Rn, gσ(x) ≤ ˆφσ(x) Thus, the coercivity of gσ implies the coercivity of ˆφσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Moreover ∀x ∈ Im(Dσ), φσ = ˆφσ + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, as xk ∈ Im(Dσ), we have along the sequence that Fλ,σ = λf(xk) + φσ(xk) = λf(xk) + ˆφσ + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Therefore, by coercivity of ˆφσ (and lower boundedness of f) and the fact that Fλ,σ(xk) monotonically decreases, the sequence xk remains necessarily bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ E Proof of Theorem 2 (αPGD for convex and weakly convex optimization) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (i) and (ii) : We can write (19b) as xk+1 ∈ arg min y∈Rn φ(y) + λ⟨∇f(qk+1), y − xk⟩ + 1 2τ ||y − xk||2 ∈ arg min y∈Rn φ(y) + λf(qk+1) + λ⟨∇f(qk+1), y − qk+1⟩ + 1 2τ ||y − xk||2 ∈ arg min y∈Rn Φ(y) + 1 2τ ||y − xk||2 (40) with Φ(y) := φ(y)+λf(qk+1)+λ⟨∇f(qk+1), y −qk+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' As φ is M-weakly convex, so does Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' The three-points inequality of Proposition 2 (iii) applied to Φ thus gives ∀y ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Φ(y) + 1 2τ ||y − xk||2 ≥ Φ(xk+1) + 1 2τ ||xk+1 − xk||2 + � 1 2τ − M 2 � ||xk+1 − y||2 (41) that is to say,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' φ(y) + λf(qk+1) + λ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' y − qk+1⟩ + 1 2τ ||y − xk||2 ≥ φ(xk+1) + λf(qk+1) + λ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' xk+1 − qk+1⟩ + 1 2τ ||xk+1 − xk||2 + � 1 2τ − M 2 � ||xk+1 − y||2 (42) 17 Using relation (19c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' and the descent Lemma 1 as well as the convexity on f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' f(qk+1) + ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' xk+1 − qk+1⟩ =f(qk+1) + � ∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 1 αyk+1 + � 1 − 1 α � yk − qk+1 � = 1 α � f(qk+1) + ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' yk+1 − qk+1⟩ � + � 1 − 1 α � � f(qk+1) + ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' yk − qk+1⟩ � ≥ 1 α � f(yk+1) − Lf 2 ||yk+1 − qk+1||2 � + � 1 − 1 α � � f(qk+1) + ⟨∇f(qk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' yk − qk+1⟩ � ≥ 1 α � f(yk+1) − Lf 2 ||yk+1 − qk+1||2 � + � 1 − 1 α � f(yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (43) Since yk+1 −qk+1 = α(xk+1 −xk) (from relations (19a) and (19c)), by combining relations (42) and (43), we now have for all y ∈ Rn, φ(y) + λ � 1 α − 1 � f(yk) + 1 2τ ||y − xk||2 + λf(qk+1) + λ⟨∇f(qk+1), y − qk+1⟩ ≥ φ(xk+1)+ λ αf(yk+1)+ � 1 2τ − αλLf 2 � ||xk+1 − xk||2+ � 1 2τ − M 2 � ||xk+1 − y||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (44) Using again the convexity of f we get for all y ∈ Rn, φ(y) + λf(y) + λ � 1 α − 1 � f(yk) + 1 2τ ||y − xk||2 ≥ φ(xk+1)+ λ αf(yk+1) + � 1 2τ − αλLf 2 � ||xk+1 − xk||2+ � 1 2τ − M 2 � ||xk+1 − y||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (45) Now, the weak convexity of φ with relation (19c) gives φ(xk+1) ≥ 1 αφ(yk+1) + � 1 − 1 α � φ(yk) − M 2 (1 − α) ||yk − xk+1||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (46) Combining (45) and (46), and using F = λf + φ leads to ∀y ∈ Rn � 1 α − 1 � (F(yk) − F(y)) + 1 2τ ||y − xk||2 ≥ 1 α(F(yk+1) − F(y)) + � 1 2τ − αλLf 2 � ||xk+1 − xk||2 + � 1 2τ − M 2 � ||xk+1 − y||2 − M 2 (1 − α) ||yk − xk+1||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (47) 18 For y = yk, we get 1 α(F(yk) − F(yk+1)) ≥ − 1 2τ ||yk − xk||2 + � 1 2τ − αλLf 2 � ||xk+1 − xk||2 + � 1 2τ − M(2 − α) 2 � ||xk+1 − yk||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (48) For constant α ∈ (0, 1), using that yk − xk+1 = 1 α(yk − yk+1) yk − xk = � 1 − 1 α � (yk − yk−1) (49) (50) we get F(yk) − F(yk+1) ≥ − α 2τ � 1 − 1 α �2 ||yk − yk−1||2 + α � 1 2τ − αλLf 2 � ||xk+1 − xk||2 + 1 α � 1 2τ − M(2 − α) 2 � ||yk+1 − yk||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (51) With the assumption ταλLf < 1, the second term of the right-hand side is non-negative and therefore, F(yk) − F(yk+1) ≥ − α 2τ � 1 − 1 α �2 ||yk − yk−1||2 + 1 α � 1 2τ − M(2 − α) 2 � ||yk+1 − yk||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' = −δ ||yk − yk−1||2 + δ ||yk+1 − yk||2 + (γ − δ) ||yk+1 − yk||2 (52) with δ = α 2τ � 1 − 1 α �2 γ = 1 α � 1 2τ − M(2 − α) 2 � (53) (54) We now make use of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Lemma 2 ([3]) Let (an)n∈N and (bn)n∈N be two real sequences such that bn ≥ 0 ∀n ∈ N, (an) is bounded from below and an+1 + bn ≤ an ∀n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Then (an)n∈N is a monotonically non-increasing and convergent sequence and � n∈N bn < +∞ To apply Lemma 2 we look for a stepsize satisfying γ − δ > 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' τ < α M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (55) Therefore, hypothesis τ < min � 1 αλLf , α M � gives that (F(yk) + δ ||yk − yk−1||2) is a non-increasing and convergent sequence and that � k ||yk − yk+1||2 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' 19 Note that a slightly more precise bound can be found keeping the second term in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' For sake of completeness, we develop it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Keeping the assumption ταλLf < 1 in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We can use that xk+1 − xk = 1 α(yk+1 − yk) + � 1 − 1 α � (yk − yk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (56) Then, by convexity of the squared ℓ2 norm, for 0 < α < 1, we have ||yk+1 − yk||2 ≤ α ||xk+1 − xk||2 + (1 − α) ||yk − yk−1||2 (57) and ||xk+1 − xk||2 ≥ 1 α ||yk+1 − yk||2 + � 1 − 1 α � ||yk − yk−1||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (58) which gives finally F(yk) − F(yk+1) ≥ � α � 1 − 1 α � � 1 2τ − αLf 2 � − α 2τ � 1 − 1 α �2� ||yk − yk−1||2 + � 1 2τ − αλLf 2 + 1 α( 1 2τ − M(2 − α) 2 ) � ||yk+1 − yk||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' = −1 − α 2ατ � 1 − α2τλLf � ||yk − yk−1||2 + 1 2ατ � 1 + α − α2τλLf − τM(2 − α) � ||yk+1 − yk||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' = −δ ||yk − yk−1||2 + δ ||yk+1 − yk||2 + (γ − δ) ||yk+1 − yk||2 (59) with δ = 1 − α 2ατ � 1 − α2τλLf � γ = 1 2ατ � 1 − α2τλLf + α − τM(2 − α) � (60) (61) The condition on the stepsize becomes γ − δ > 0 ⇔ τ < 2α α3Lf + (2 − α)M (62) And the overall condition is τ < min � 1 αLf , 2α α3Lf + (2 − α)M � (63) (iii) The proof of this result is an extension of the proof proposed in Ap- pendix C in the context of the classical PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Suppose that a subsequence (yki) is converging towards y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Let’s show that y is a critical point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' From (19b), we have xk+1 − xk τ − λ∇f(qk+1) ∈ ∂φ(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (64) 20 First we show that xki+1 − xki → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We have ∀k > 1, ||xk+1 − xk|| = ���� ���� 1 αyk+1 + (1 − 1 α)yk − 1 αyk − (1 − 1 α)yk−1 ���� ���� ≤ 1 α ||yk+1 − yk|| + ( 1 α − 1) ||yk − yk−1|| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (65) From (19a), we also get ||qk+1 − qk|| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Now, let’s show that xki → y and qki → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' First using (19c), we have ||xki − y|| ≤ ||xki+1 − y|| + ||xki+1 − xki|| ≤ 1 α ||yki+1 − y|| + ( 1 α − 1) ||yki − y|| + ||xki+1 − xki|| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (66) Second, from (19a), we get in the same way qki → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' From the continuity of ∇f, we get ∇f(qki) → ∇f(y) and therefore xki − xki−1 τ − λ∇f(qki) → −λ∇f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (67) As explained in the proof Appendix C (iii), if we can also show that φ(xki) → φ(y), we get from the subdifferential characterization (32) that −λ∇f(y) ∈ ∂φ(y) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' y is a critical point of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Using the fact that φ is lsc and xki → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' lim inf i→∞ φ(xki) ≥ φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (68) On the other hand, with Equation (45) for k + 1 = ki, taking i → +∞, ||y − xki+1|| → 0, ||y − xki|| → 0, f(yki) → f(y), f(yki+1) → f(y) and we get lim sup i→∞ φ(xki) ≤ φ(y), (69) and therefore lim i→∞ φ(xki) = φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' (70) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='1 On the convergence of the iterates with the KL hypothesis In order to prove a result similar to Theorem 1 (iv) on the convergence of the iterates with the KL hypothesis, we can not directly apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='9 from [2] on F as the objective function F(xk) by itself does not decrease along the sequence but F(xk) + δ ||xk+1 − xk||2 does (where δ = α 2τ � 1 − 1 α �2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Our situation is more similar to the variant of this result presented in [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' Indeed, denoting F : Rn × Rn → R defined as F(x, y) = F(x) + δ ||x − y||2 and considering ∀k ≥ 1, the sequence zk = (yk, yk−1) with yk following our algorithm, we can easily show that zk verifies the conditions H1 and H3 specified in [16, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' However, condition H2 does not extend to our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' We plan as future work to derive a new version of [16, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content='2] that fits to our case of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} +page_content=' □ 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFST4oBgHgl3EQfJDg3/content/2301.13731v1.pdf'} diff --git a/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.pkl b/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ae9fd531017be84c21b6821d7986ebb7ed5f5ef0 --- /dev/null +++ b/m9AyT4oBgHgl3EQfyvnQ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e251ca7784abbcd042e34bcc2641c3a0a2d49b7a4b46270ef18a9df0ff71cd6c +size 57546 diff --git a/mdE1T4oBgHgl3EQfhATk/vector_store/index.faiss b/mdE1T4oBgHgl3EQfhATk/vector_store/index.faiss new file 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/dev/null +++ b/mdE3T4oBgHgl3EQf6gtE/content/tmp_files/2301.04790v1.pdf.txt @@ -0,0 +1,1679 @@ +On the Structural Generalization in Text-to-SQL +Jieyu Li, Lu Chen, Ruisheng Cao, Su Zhu, +Hongshen Xu, Zhi Chen, Hanchong Zhang and Kai Yu +X-LANCE Lab, Department of Computer Science and Engineering +MoE Key Lab of Artificial Intelligence, AI Institute +Shanghai Jiao Tong University, Shanghai, China +{oracion, chenlusz, 211314, paul2204, }@sjtu.edu.cn +{zhenchi713, xuhongshen, zhanghanchong, kai.yu}@sjtu.edu.cn +Abstract +Exploring the generalization of a text-to-SQL +parser is essential for a system to automat- +ically adapt the real-world databases. +Pre- +vious works provided investigations focusing +on lexical diversity, including the influence of +the synonym and perturbations in both natural +language questions and databases. However, +research on the structure variety of database +schema (DS) is deficient. +Specifically, con- +fronted with the same input question, the tar- +get SQL is probably represented in different +ways when the DS comes to a different struc- +ture. +In this work, we provide in-deep dis- +cussions about the structural generalization of +text-to-SQL tasks. +We observe that current +datasets are too templated to study structural +generalization. To collect eligible test data, we +propose a framework to generate novel text-to- +SQL data via automatic and synchronous (DS, +SQL) pair altering. In the experiments, signif- +icant performance reduction when evaluating +well-trained text-to-SQL models on the syn- +thetic samples demonstrates the limitation of +current research regarding structural general- +ization. According to comprehensive analysis, +we suggest the practical reason is the overfit- +ting of (NL, SQL) patterns. +1 +Introduction +Given the corresponding database, text-to-SQL (Yu +et al., 2018) aims to convert a natural language (NL) +utterance into a structured SQL program. To mea- +sure the robustness of a text-to-SQL parser for +industrial applications, the cutting edge research +focuses on the cross-domain setting, where the +databases used during training and evaluation do +not overlap. +Recently, many advanced text-to-SQL mod- +els, such as RATSQL (Wang et al., 2019) and +LGESQL (Cao et al., 2021), have been proposed +to tackle this task. Although significant progress +has been achieved considering the ultimate accu- +racy, many researchers point out that actual per- +Question: How many singers are there? +singer +id +name +age +0 +Taylor Swift +32 +SQL: SELECT count(*) FROM singer +song +id +name +singer +0 +Love Story +Taylor Swift +SQL: SELECT count(distinct singer) FROM song +people +id +name +identity +0 +Taylor Swift +singer +SQL: SELECT count(*) FROM people WHERE identity = ‘singer’ +Table +Column +Cell Value +Table +Table +Column +Column +Cell Value +Cell Value +Figure 1: Given the same question, the target SQL re- +sponds in different ways when the database schema is +different. +formances in the cross-domain setting are over- +estimated. +Suhr et al. (2020) observed a dra- +matic performance decline when evaluating a well- +trained model on another dataset. Even on the same +dataset where the collected samples all conform to +an implicit pattern, Gan et al. (2021a) discovered +that current parsers are vulnerable to the adversarial +attack from synonyms of words in user questions. +To explore the generalization capability, previous +literature mainly focused on the variety of natural +language, especially at the semantic level. How- +ever, the topological feature of database schema is +also important but is less investigated while study- +ing the generalization capability in text-to-SQL +tasks. +The database schema (DS) determines which +database elements (table/column/cell value) and +SQL clauses will be used to describe the seman- +tics of the user question in SQL query. Therefore, +the SQL queries may be completely divergent in +different databases even given the same user ques- +tion. For example, in Figure 1, the entity “singer" +can function as a column, a table or a specific cell +value of the column “identity", depending on the +arXiv:2301.04790v1 [cs.AI] 12 Jan 2023 + +ontology of the corresponding DS. Actually, in the +process of database designing, different developers +design the entities and their relationships in differ- +ent ways. The differences always represent on the +topological structure of DS. Moreover, database +contents are also considered in some cases. For +example, in Figure 1, “singer" is stored as a table +in the first database while comes to a cell value +in the third database. We named the ability to au- +tomatically adapt different structural information +representation methods the structural generaliza- +tion capability. +Nowadays, the setup of cross-domain text-to- +SQL ensures the database is completely novel in +the evaluation stage. The structural generalization +capability is supposed to be assessed. However, +we find that current cross-domain datasets are over- +templated. Models can even predict the structure +of SQL queries only with the user question. Mean- +while, when we add the DS information to the +input, the performances change marginally. Be- +sides, when we attempt to evaluate across different +datasets, the phenomena still exist. In this case, +the structural generalization capability is overesti- +mated. +In this work, we focus on studying structural +generalization and provide in-deep analysis. Be- +cause of the over-templated feature, we can not +investigate the generalization capability with cur- +rent datasets. +To provide a comprehensive ap- +praisal regarding the structural generalization of +existing text-to-SQL parsers, we propose a data- +and structure-driven framework to automatically +synthesize altered (DS, SQL) pairs given the same +input question. +The framework modifies the +DS with modest annotation cost and updates the +SQL synchronously by altering the abstract syntax +tree (AST). Inspired by the entity-relationships di- +agram (E-R Diagram) (Ling, 1985; Li and Chen, +2009), all the transformations follow the entity re- +lationships of the database to guarantee that the +modifications are reasonable. We also compared +the execution results of modified and original (DS, +SQL) pairs to ensure the framework updates the +SQL correctly. +In the experiments, we first evaluate the struc- +tural robustness by applying perturbations to +the training DS. Furthermore, we create out-of- +dataset (OOD) DS by applying perturbations to +the development DS. Then we assess the structural +generalization capability using these databases. Un- +fortunately, both the structural robustness and struc- +tural generalization are modest. +We further conduct several experiments to an- +alyze the actual reason for the modest structural +generalization. Firstly, we observe models can pre- +cisely parse the question if the transformation of +creating OOD DS does not lead to the SQL chang- +ing, otherwise, they always make mistakes. Thus, +the parsing failures are not caused by the novel +topological structures. We introduce a new con- +cept (NL, DS) pattern, which is the combination +of the semantic role of a keyword in a natural lan- +guage (NL) question and the syntax role of the +related item in the database schema (DS). We sug- +gest that current text-to-SQL algorithms always +become overfitting on (NL, DS) patterns, and it +leads to modest generalization capability. Finally, +we discuss the efficiency of data augmentation. Ex- +periment results demonstrate that only if the extra +training data provide the patterns which are rare in +the original training data but exist in the develop- +ment data. +The main contributions can be summarized: +• We propose a data- and structure-driven frame- +work which can automatically synthesize sam- +ples containing unseen (DS, SQL) patterns +with minimal human labor. +• By utilizing the plug-and-play framework, +we synthesize a testsuite and demonstrate +the poor performance of existing text-to-SQL +models regarding structural generalization. +• We analyze the reasons leading to modest +generalization towards perturbations of syn- +chronous changes in (DS, SQL) pairs. +2 +Background and Related Work +Structural +Features +in +Text-to-SQL +Tasks +Modeling the structural information in a database +and designing an efficient algorithm to decode +structured output sequences are crucial in text-to- +SQL. Several studies achieved remarkable progress +using GNN (Scarselli et al., 2008) to encode the +schema linking, which enhanced the graph struc- +ture of DS and the relationships between DS and +question tokens (Bogin et al., 2019; Lin et al., +2020; Chen et al., 2021; Hui et al., 2022; Wang +et al., 2019; Cao et al., 2021). Another line of re- +search focuses on the grammar structure of SQL. +Corresponding works proposed novel algorithms + +to precisely decode according to the syntax (Guo +et al., 2019; Rubin and Berant, 2021; Gan et al., +2021b). Recent works attempted to utilize the de- +veloped generative pre-trained language models +(Raffel et al., 2020; Lewis et al., 2020) to generate +SQL. Based on T5 (Raffel et al., 2020), Scholak +et al. (2021) proposed a rule-based post-processor +to prune syntax-illegal SQL subsequence in beam +search, and they achieved stable improvement in +the end-to-end text-to-SQL system. +Synthetic Data +Lexical +Structure +Question +Schema +Schema +SQL +Spider-Syn(Gan et al., 2021a) +✓ +� +� +� +MR-UT(Ma and Wang, 2021) +✓ +� +� +� +MR-ST(Ma and Wang, 2021) +� +� +✓ +� +ADVETA-RPL (Pi et al., 2022) +� +✓ +� +� +ADVETA-ADD (Pi et al., 2022) +� +✓ +✓ +� +Unaffected +� +� +✓ +� +Affected +� +� +✓ +✓ +Table 1: Setups of previous evaluation datasets and our +synthetic samples. The synthetic evaluation data was +modified from Spider. Unaffected and Affected are two +types of data we synthesize in this work. The mark +✓represents that the corresponding attribute is different +from that in Spider. Oppositely, we use �to note in this +table. +Robustness +of +text-to-SQL +models +Early +datasets (Dahl et al., 1994; Hemphill et al., 1990; +Zelle and Mooney, 1996; Tang and Mooney, +2000; Li and Jagadish, 2014; Yaghmazadeh et al., +2017; Iyer et al., 2017; Finegan-Dollak et al., +2018) only considered the text-to-SQL tasks on a +single database. To build a robustness text-to-SQL +model that can automatically adapt unseen domain +data, Recent works (Yu et al., 2018; Zhong et al., +2017) collected cross-domain text-to-SQL datasets. +Based on the cross-domain setup, researchers +further considered some different real-world scenes +and proposed corresponding datasets (Yu et al., +2019b,a; Wang et al., 2020). However, Suhr et al. +(2020) observed that the execution (EX.) accuracy +of a well-trained model on Spider (Yu et al., 2018) +always decreases remarkably on the unseen domain +data from other datasets1. Although Suhr et al. +(2020) depicted the reasons leading to performance +decline, in-deep discussions are necessary. To this +end, recent studies generated synthetic data under +different setups to further assess the practical +model generalization in different environments. +1Considered most related studies report the EM. accuracy +as the results, we additionally reproduce the experiments and +use the EM. accuracy as the metric and illustrate the results in +Table 2. +We summarize the characteristic of the synthetic +evaluation set in Table 1. +In respect of text, +Gan et al. (2021a) generated evaluation samples +via replacing the schema-related words in NL +questions with synonyms. Ma and Wang (2021) +substituted the aggregation-related words and +prefix phrases with synonym representations. Pi +et al. (2022) modified the column names in DS. +In respect of structure, Ma and Wang (2021) +created different DS structures by imposing +perturbations. Pi et al. (2022) added adversarial +columns in DS. However, the golden SQLs in both +of their synthetic datasets remain unchanged when +applying perturbations. In this work, we consider +both changed and unchanged golden SQLs to +provide a comprehensive appraisal regarding +structural generalization. +Dataset +RATSQL +LGESQL +Spider (Yu et al., 2018) +69.57 +70.11 +SParC (Yu et al., 2019b) +42.20 +43.59 +Spider-Syn (Gan et al., 2021a) +49.81 +50.93 +Academic (Li and Jagadish, 2014) +6.26 +7.36 +GeoQuery (Zelle and Mooney, 1996) +7.51 +7.86 +IMDB (Yaghmazadeh et al., 2017) +18.96 +18.74 +Restaurant (Tang and Mooney, 2000) +0.00 +0.53 +Scholar (Iyer et al., 2017) +0.18 +0.24 +Yelp (Yaghmazadeh et al., 2017) +6.01 +7.51 +Table 2: Models are trained on Spider, while evaluated +on other datasets. The databases of SParC and Spider- +Syn are similar to Spider. +3 +Eligible Evaluation Data +To evaluate the generalization capability of text-to- +SQL models, test data providing novel database +schema (DS) structure are necessary. However, +current text-to-SQL datasets are not eligible be- +cause of the over-templated features (Section 3.1). +Therefore, we propose a data- and structure-driven +generation framework to synthesize relevant data to +assess the generalization capability (Section 3.2). +3.1 +Current Datasets are Undesirable +To verify that current text-to-SQL datasets are over- +templated, we conduct a syntax role prediction ex- +periment. +Syntax Role Prediction aims to predict which +SQL syntax roles are mentioned in the query, in- +cluding the SQL keywords, nested structure, and +aggregation clause. The metric used in this exper- +iment is joint accuracy, which means the case is +treated as correct if and only if all the syntax roles +are predicted. + +Train. +Test. +Test. Setup +w/o. DB Schema +w. DB Schema +Spider Train. +Spider Dev. +Spider-like Cross-Domain +86.08 +87.34 ↑1.26 +Spider Train. +Spider-Syn Dev. +Spider-like Cross-Domain +85.59 +84.72 ↓0.87 +Spider-Syn Train. +Spider-Syn Dev. +85.69 +85.40 ↓0.29 +Spider Train. +SParC Dev. +Spider-like Cross-Domain +74.31 +74.48 ↑0.17 +SParC Train. +SParC Dev. +66.92 +66.50 ↓0.42 +Spider Train. +Academic +Single Domain +92.27 +89.50 ↓2.77 +GeoQuery +51.42 +45.57 ↓5.85 +IMDB +90.83 +93.58 ↑2.75 +Restaurants +75.20 +89.60 ↑12.40 +Scholar +67.66 +71.00 ↑3.34 +Yelp +96.40 +93.69 ↓2.71 +Table 3: Experiment results of syntax role prediction. w/o. DB Schema represents a vanila model using BERT-base +to encode user questions. W. DB Schema represents the model using RAT encoder to process the user questions +and database schema. +In this experiment, we compare the perfor- +mances of whether contains database schema in the +inputs. As the results in Table 3 shown, the model +can directly predict the approximate structure of the +target SQL only with the user question most of the +time, even though the databases for training and for +testing are not overlapping. Meanwhile, the perfor- +mance differences between using and without using +DS demonstrate that the DS information is helpless +for predicting the SQL structure. Therefore, we +suspect that current datasets are too templated to +evaluate the generalization capability using them. +To this end, we need to synthesize eligible evalua- +tion data. +3.2 +Evaluation Data Generation +To assess the structural generation capability, we +propose a data- and structure-driven generation +framework to synthesize relevant data. The syn- +thetic data in this paper are modified from Spi- +der (Yu et al., 2018) 2 which is the most popular +cross-domain text-to-SQL dataset. It contains 8659 +training examples and 1034 validation examples +across 146 databases. The test dataset is unseen +and contains 2147 samples with 40 databases. +For a given sample, we synthesize a new sample +via altering the DS while keeping the question con- +stant. In order to obtain a reasonable DS, we con- +struct the entity-relationship graph of the given DS +and apply graph-based transformations. Moreover, +we synchronously update the SQL by modifying +the abstract syntax tree. We show more details in +Appendix A +2https://yale-lily.github.io//spider. +In this work, we use four different transforma- +tions in DS. Figure 2 illustrates the examples of +each transformation, and we show a brief introduc- +tion below: +• Entity to Attribute (E2A) merges two tables +into one. +• Concept to Attribute (C2A) converts the +concept3 of an entity, which represents via +table name in DS, to its attribute. +• Named to Unnamed (N2U) replaces the ta- +ble corresponding to a relationship with for- +eign keys. +• Unnamed to Named (U2N) replaces a for- +eign key with a relationship table. +Table 4 shows the total number of each kind of +synthetic data synthesized via different E-R trans- +formations. We evaluate the synthetic quality by +comparing the execution results of the original and +synthetic (DS, SQL) pairs. Over 90.43% gener- +ated samples kept consistent execution results on +average. +Trans. +Train. +Dev. +Affected +Unaffected +Affected +Unaffected +E2A +3035 +9466 +493 +1477 +C2A +2659 +4271 +379 +445 +U2N +2969 +12910 +114 +376 +N2U +2605 +48507 +303 +4484 +Table 4: Statistics of generated data for four transfor- +mations. +3It refers to the definition of concept node in the knowledge +graph. + +Entity ➔ Attribute +Concept ➔ Attribute +Named ➔ Unnamed +Unnamed ➔ Named +Original +Figure 2: Examples of the DS synthesized via different transformations. The dotted lines denote foreign keys (from +foreign key to primary key) +4 +Generalization Evaluation +In this section, we conduct experiments to evalu- +ate the following capability of current text-to-SQL +models: +• C1: The robustness of current text-to-SQL +parsers when applying a perturbation on the +database schema. (Section 4.2) +• C2: The practical generalization capability of +current text-to-SQL parsers when it comes to +novel databases? (Section 4.3) +4.1 +Experiment Setup +In this work, we experiment with two grammar- +based SOTA text-to-SQL parsers, RATSQL (Wang +et al., 2019) and LGESQL (Cao et al., 2021). Be- +sides, we also experiment with the T5-based end-to- +end text-to-SQL parser, including the methods of +decoding with and without PICARD(Scholak et al., +2021). The evaluation metric we use to report the +results is exact set match accuracy (EM). Results +are averaged over three trials to reduce variance. +Equivalent Test Set (ETS) To precisely evalu- +ate the model robustness, we construct an equiva- +lent test set for the given dataset, which contains +the same number of samples. We restrict that each +sample in the original dataset matches exactly one +synthetic variant in the ETS. If a sample can not +generate a variant, we will add the duplication in +the ETS. Furthermore, to reduce the influence of +hardness, we utilize a heuristic algorithm to modu- +late the ETS so that its distribution is closed to the +original dataset. +4.2 +Structural Robustness +To evaluate C1, we construct the equivalent test +set (ETS) for the training set of Spider. The training +data in this experiment is the Spider training set. +We compare the performances on two test sets, the +Spider training set, and the corresponding ETS. +Model +Test Data +EM. Acc. +RATSQL +Spider Train. +98.19 +Spider Train. ETS +60.81↓37.82 +LGESQL +Spider Train. +98.94 +Spider ETS +62.73↓36.21 +T5 +Spider Train. +99.03 +Spider Train. ETS +57.78↓41.25 +T5+Picard +Spider Train. +98.52 +Spider Train. ETS +66.48 ↓32.04 +Table 5: Exactly match (EM.) accuracy on the evalua- +tion data synthesized from the Spider training set. +Experiment results illustrated in Table 5 indi- +cate that the perturbation applied to the database +schema will disturb the parsing process. The mod- +els can not precisely infer the representation of the +SQL query when confronting novel DS structures +despite the questions and the other parts of the DS +being the same as they appeared in the training +phrase. In this case, we suggest that the structural +robustness of current text-to-SQL models is mod- +est. +4.3 +Practical Structural Generalization +To evaluate C2, we construct the ETS for the de- +velopment set of Spider. The training data in this +experiment is also the Spider training set while we +compare the performances on the Spider develop- +ment set and the corresponding ETS. +Experiment results in Table 6 illustrate that the +parsers seem to perform well on completely novel +DS, however, the performances dramatically de- +cline once perturbations are applied on these novel +DS. These phenomena demonstrate that the practi- + +singer +id +name +0 +Taylor Swift +song +id +name +album +album_year +sing +1 +All Too Well +Red +2012 +singer_id +song_id +0 +1people +album +id +identity +name +id +name +year +0 +Taylor Swift +singer +2 +Red +2012 +sing +song +singer_id +song_id +id +name +aid +0 +1 +1 +All Too Well +2singer +album +id +id +name +name +year +0 +Taylor Swift +2 +Red +2012 +song +id +singer +name +aid +1 +0 +All Too Well +2singer +album +id +id +name +name +year +0 +Taylor Swift +2 +Red +2012 +sing +album_song +song +singer_id +song_id +aid +sid +id +name +0 +1 +2 +1 +1 +All Too Wellsinger +album +id +id +name +name +year +0 +Taylor Swift +2 +Red +2012 +sing +song +singer_id +song_id +id +name +aid +0 +1 +1 +All Too Well +2Model +Test Data +EM. Acc. +RATSQL +Spider Dev. +69.83 +Spider Dev. ETS +44.68↓25.15 +LGESQL +Spider Dev. +70.57 +Spider Dev. ETS +45.10↓25.47 +T5 +Spider Dev. +59.09 +Spider Dev. ETS +25.63↓33.46 +T5+Picard +Spider Dev. +66.89 +Spider Dev. ETS +36.87↓33.02 +Table 6: EM. accuracy on the evaluation data synthe- +sized from the Spider development set. +cal structure generalization capability is also mod- +est, which is similar to the structural robustness. +Therefore, we suspect that current text-to-SQL +parsers can not automatically infer the SQL pat- +tern according to the DS. We will discuss the truly +reason caused this issue in the next section. +5 +Discussion about Structural +Generalization +In this section, we discuss the structure general- +ization of text-to-SQL by answering the following +questions: +• Q1: What is the actual function of database +schema input? (Section 5.1) +• Q2: What is the actual reason causing the +modest generalization capability? (Section +5.2) +• Q3: Does data augmentation can increase the +generalization capability? (Section 5.3) +5.1 +Function of Database Schema Input +To answer Q1, we first verify that the database +schema (DS) information is independent with the +process of constructing SQL patterns. Reviewing +the experiments in Section 4, models always make +mistakes when facing out-of-dataset (OOD) DS. To +estimate whether the OOD structure confuses the +parsers, we consider the evaluation data containing +OOD DS while keeping the SQL query unchanged. +Setup: Different from the evaluation data used in +Section 4, we generate the data with different DS +but the same SQL. For each piece of data, the DS +transformations are applied on untapped parts so +that the SQL will not be influenced. Similarly, we +also construct the equivalent test set (ETS) for the +Spider development set with this kind of synthetic +data. The training data in this experiment is the +training set of Spider. +Model +Test Data +EM. Acc. +RATSQL +Spider Dev. +69.83 +Spider Dev. ETS +67.67↓2.16 +LGESQL +Spider Dev. +70.57 +Spider Dev. ETS +67.41↓3.16 +T5 +Spider Dev. +59.09 +Spider Dev. ETS +57.20 ↓1.89 +T5+Picard +Spider Dev. +66.89 +Spider Dev. ETS +64.06 ↓2.83 +Table 7: EM. accuracy on the evaluation data synthe- +sized from the Spider training set. The SQL in syn- +thetic data is consistent with the original. +Results shown in Table 7 demonstrate that the +OOD structure does not influence the inference +process. Reviewing the syntax role prediction ex- +periments discussed in Section 3.1, we suggest that +current text-to-SQL models construct SQL query +via sentence pattern of the user question rather than +the actual structure of DS. We suspect that the func- +tion of DS input is providing the correct presenta- +tion of the SQL non-keywords (table name, column +name, and value). The efficiency of using schema +linking provides a strong signal on target database +item. Once the explicit relationships between these +SQL non-keywords and the presentations in ques- +tion are destroyed, models will make mistakes on +selecting correct schema item. However, the SQL +structure is always predicted in correct ways(Gan +et al., 2021a). +5.2 +(NL, DS) Pattern +To answer Q2, we first introduce the concept of +(NL, DS) pattern. Then we will illustrate how +the (NL, DS) pattern influences the generalization +capability. The (NL, DS) pattern represents the +combination of a natural language (NL) role and a +database schema (DS) role. +NL Role: As the examples in Figure 3 shown, we +assume that some words in the NL question de- +scribe the key information of the query. In this sec- +tion, we simply split these keywords into two cate- +gories, target and condition, which we call them the +NL role of these words. target represents the query- +ing target and condition represents the constraint +of the target. The NL roles are DS-independent, in +the other words, they only depend on the semantics +of the NL question. +DS Role: For a DS, some elements link the key- +words in the given question, such as the word +“singer" in the first case, and all of them play a + +Question: How many people whose identity is singer ? +SQL: SELECT count(*) FROM people WHERE identity = ‘singer’ +target +Table +condition +condition +Column +Cell Value +Question: How many people whose identity is singer ? +target +condition +condition +Table +SQL: SELECT count(*) FROM singer +Padding +Padding +Figure 3: Different combinations of NL role and DS +role determine different SQL sketches. +unique role in the given DS. We define the DS role +as table, column, cell value and a padding role to +link the non-schema-related keywords, for instance, +the word “people" in the second case. +(NL, DS) Pattern: For each of these elements, we +named the combination of an NL role and a DS role +as an (NL, DS) pattern, which determines the syn- +tax role in SQL. For example, the element “singer" +in the first case functions the NL role condition +and the DS role cell value, in this case, it locates +in the WHERE clause. However, when the DS role +comes to table, as shown in the second case, the +element “singer" will locate in the FROM clause. For +the given NL question and DS, the structure of the +SQL query depends on the containing (NL, DS) +patterns. +We assume that the modest generalization ca- +pability is because of the over-fitting of (NL, DS) +patterns. Unseen (NL, DS) patterns lead to failed +parsing. We first evaluate the performance on the +samples synthesized via different E-R transforma- +tions. The experiment results are illustrated in Ta- +ble 4. We notice that models make mistakes on +almost all the samples generated using C2A E-R +transformation. Actually, C2A is a special transfor- +mation because it must create an (NL, DS) pattern, +(target, cell value). This pattern always represents +a condition in the WHERE clause. However, we find +that the condition is always deficient. Therefore, +we randomly sample 100 pieces of data from the +training set and the synthetic C2A data to evalu- +ate whether (target, cell value) is an unseen pattern. +Table 8 shows the statistic results of manually calcu- +lating the distribution of (NL, SQL) patterns. The +combination (target, cell value) is not contained in +the training set but exists as unseen patterns when +it comes to C2A samples. We additionally enu- +merate some typical error cases in Appendix. The +DS Role +Train. +C2A. +Target +Condition +Target +Condition +Table +9.09% +42.62% +5.00% +15.18% +Column +81.82% +24.04% +83.33% +21.43% +Cell Value +0.00% +25.68% +11.67% +58.04% +Padding +9.09% +7.65% +0.00% +5.36% +Table 8: Distribution of (NL, DS) pattern in the training +set and C2A samples. +Figure 4: Training without extra data, evaluating on +different synthetic samples. +examples demonstrate that models tend to parse ac- +cording to experiences so that they make mistakes +on novel patterns. In this case, we suggest that +the actual reason causing the modest generalization +capability is the (NL, DS)-pattern-wise over-fitting. +5.3 +Convert Unseen to Seen +In this section, we answer Q3 by evaluating the +efficiency of training with extra synthetic data. +Setup: We conduct experiments to on both original +and synthetic evaluation data. As the data augmen- +tation, we train models with the original training set +of Spider and additional synthetic data with a 1:0.2 +ratio. We consider two kinds of extra synthetic +training data in these experiments. The one is the +data containing novel database schema (DS) and +different SQL queries (compared with the original +data). They are similar to the evaluation data we +used to build the ETS in Section 4. We named these +data Affected. The other are the data containing +novel DS while the same SQL queries, which is +similar to the data from the Dev. ETS in Section +5.1. We named them Unaffected. The synthetic +evaluation data we used in this section is Affected. +We report our results in Table 9. Experiment +results in the upper block illustrate that either af- + +E2A: 48.88 +80.00 +70.00 +60.00 +50.00 +40.00 +30.00 +Original +C2A +20.00 +1.32 +69.83 +10.00 +0.00 +U2N +N2U +59.65 +28.38Test Data +Training Data +RATSQL +LGESQL +T5 +T5+PICARD +Spider Dev. +Spider Train. +69.83 +70.57 +59.09 +66.89 +Spider Train. + Affected +70.12 ↑0.29 +69.89 ↓0.68 +58.22 ↓0.87 +67.01 ↑0.12 +Spider Train. + Unaffected +70.38 ↑0.55 +70.05 ↓0.52 +58.99 ↓0.10 +66.85 ↓0.04 +Spider Dev. ETS +Spider Train. +44.68 +45.10 +25.63 +36.87 +Spider Train. + Affected +67.21 ↑22.53 +67.57 ↑22.47 +56.12 ↑30.49 +62.08 ↑25.79 +Spider Train. + Unaffected +45.23 ↑0.55 +45.17 ↑0.07 +25.57 ↓0.06 +37.09 ↑0.22 +Table 9: Results of using extra synthetic samples in the training stage. +fected or unaffected extra training data can improve +the performance on the original development set. +The reason is that extra training data do not provide +the (NL, DS) patterns which are rare in the origi- +nal training set but appear in the original develop- +ment set. Actually, the problem of over-templated +demonstrates that it is hard to find the aforemen- +tioned patterns. Unlikely, the transformations ap- +plied in this work either do not guarantee these +patterns are created. +Experiment results in the lower block of Table 9 +show that the affected extra training data is helpful +to improve the performance on the synthetic evalu- +ation data. However, the usage of unaffected data +can not. The reason is that the former provides the +(NL, DS) patterns which are rare in the original +training set while are contained in the ETS. On the +other hand, the latter do not provide any of these +patterns because the perturbations are applied on +untapped parts of DS in unaffected data. There- +fore, we suggest that specific data augmentation +can enhance the model despite it can not be verified +on current datasets. This experiment amplifies the +improvement of data augmentation by increasing +the overlap of (NL, DS) patterns between the extra +training data and the synthetic evaluation data. +6 +Not Only in Cross-Domain +Actually, the problem of (NL, DS)-pattern-wise +overfitting is not the specific problem that only ex- +ists in cross-domain text-to-SQL. Modest structural +generalization is just one of the phenomena under +a cross-domain setup. Single-domain text-to-SQL +also has the same problem. +From the view of (NL, DS) patterns, the defi- +ciency of patterns in the training stage leads to the +appearance of unseen patterns in the test stage, and +further cause performance decline. However, leav- +ing out patterns is inevitable during the data collec- +tion process. Annotators can neither ensure to ask +questions in all possible sentence patterns nor guar- +antee that all combinations of schema items are +considered. For instance, as the example illustrated +in Figure 1, confronting the third DS, annotators +may not come up with the question about “singer", +or they may ask in the way of How many people +whose identity is a singer?. 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In Proceedings of the national con- +ference on artificial intelligence, pages 1050–1055. +Victor Zhong, Caiming Xiong, and Richard Socher. +2017. +Seq2sql: +Generating structured queries +from natural language using reinforcement learning. +arXiv preprint arXiv:1709.00103. +A +Generation Framework +The overview of the generation framework is +shown in Figure 5. For a given sample, we synthe- +size a new sample via altering the DS while keeping +the question constant. In order to obtain a reason- +able DS, we construct the entity-relationship graph +of the given DS and apply graph-based transforma- +tions, which we introduce in Section A.1 and Sec- +tion A.2 respectively. Moreover, we synchronously +update the SQL by modifying the abstract syntax +tree, and we show more details in Section A.3. +A.1 +Entity-Relationship Graph +A relational database organizes data in predefined +relationships, which are represented as the struc- +tural relationships among tables and columns. To +clearly describe reasonable relationships, develop- +ers always utilize Entity-Relationship Diagram (E- +R Diagram) (Ling, 1985; Li and Chen, 2009) to +define the relationships among the raw data, which +is helpful to design the database structure. Inspired +by ER Diagram, we attempt to modify the DS fol- +lowing the entity relationships so that the rationality +of the altered DS can be ensured. To this end, we +introduce the definition of Entity-Relationship (E- +R) Graph in this paper, which evolves from E-R +Diagram while leaving out the attributes vertexes +to emphasize the topology feature 4. The vertex in +E-R Graph represents an entity, and the edge repre- +sents the relationship between the entities that its +terminal vertexes correspond. Both the vertex and +the edge function as a table in DS. For example, +as shown in Figure 5, each of the table people, the +table author, and the table novel corresponds to +a vertex in E-R Graph, and the table write corre- +sponds to an edge. +Thus, to construct the E-R Graph, we manually +annotate a binary tag for each table in DS to dis- +tinguish between entity and relationship. We label +relationship following two principles and label the +others as entity: +4The attributes node in E-R Diagram refers to a column in +the database. To emphasize the topology feature, we replace +Diagram with Graph. + +FROM +author +write +novel +FROM +author +novel +write +author +novel +aid +id +age +name +nid +id +name +people +pid +id +author +novel +age +name +aid +name +people +pid +id +id +E-R Transformation +people +novel +author +people +novel +author +AST Updating +Sample( X, D, Y ): +Question( X ): Who is the author of Harry Potter ? +Database( D ) +SQL( Y ): SELECT people.name FROM people +JOIN author ON people.id = author.pid JOIN write +ON author.id = write.aid JOIN novel ON write.nid = +novel.id WHERE novel.name = ‘Harry Potter’ +Database(D) +Database(D’) +Sample( X, D’, Y’ ) +Question(X): Who is the author of Harry Potter ? +Database( D’ ) +SQL( Y’ ): SELECT people.name FROM people +JOIN author ON people.id = author.pid JOIN novel +ON author.id = novel.aid WHERE novel.name = +‘Harry Potter’ +E-R Graph of D +AST of Y +AST of Y’ +E-R Graph of D +Figure 5: Overview of the generation framework we proposed. The upper part illustrates the original sample, and +the lower part illustrates the synthetic sample. The red rectangles and red lines denote the modified parts. For the +E-R graph, the dotted lines denote unnamed relationships, and solid lines denote named relationships. +Bridge Structure: The given table should con- +tain exactly two foreign keys. +Semantic Convention: The table name should +be the combination of two entities such as the rela- +tionship Customer_Addresses combining Customer +and Address. Apart from that, the phrase obeys hu- +man language conventions is also considered. For +instance, the relationship visit linking visitor and +museum. +A.2 +E-R Transformation +E-R transformation is the graph transformation in +the E-R graph. There are ten kinds of E-R trans- +formation, containing five operations applied on +vertexes or edges. We assume the databases that +store the same data in the different schema can +transform between each other via a sequence of +E-R transformations. We illustrate all kinds of E-R +transformations and the corresponding transforma- +tions in DS in Appendix. However, some trans- +formations are insecure. For example, the usage +of delete edge transformation will lead to informa- +tion loss. Besides, some transformations rely on +strict annotation criteria and costly manual labeling. +For instance, whether a table can be split into two +need rigorous judgment according to the semantic +environment. In this work, we use three E-R trans- +formations with no need for additional annotations, +and they totally correspond to four different trans- +formations in DS. Figure 2 illustrates the examples +of each transformation, and we show more details +below. +Entity to Attribute (E2A) corresponds to a +kind of merge vertexes E-R transformation. For +a pair of vertexes in the E-R graph, we split them +as a source entity and a target entity. The target en- +tity corresponded table is the only one that contains +the foreign keys of the source entity corresponded +table in the DS. Both of the vertexes can be treated +as the source entity as long as the combination is +eligible. As the example shown in Figure 2, for +the attributes in the source entity, we convert them +into the new attributes in the target entity. To avoid +semantic loss, we rename the attributes following +rules. Besides, we utilize a series of rules to rec- +ognize a special column as the agent of the entity, +such as the column name, and it will be used to +replace the foreign key. +Concept to Attribute (C2A) corresponds to a +kind of modify vertex E-R transformation. Different +from the column-wise modification, we focus on +altering the role of the table. We attempt to convert +the concept5 of an entity, which represents via table +name in DS, to its attribute. Firstly, we detect a +high-level category of the entity using a pre-trained +NER model (Qi et al., 2020). In the example shown +in Figure 2, people is the high-level category of +singer. Then, we create an additional attribute +to store the concept by rules. In this case, we use +5It refers to the definition of concept node in knowledge +graph. + +the new column identity to record the concept +singer. +Named to Unnamed (N2U) corresponds to a +kind of modify edge E-R transformation. We name +the relationship represented by a table as Named, +and that by foreign keys as Unnamed. For instance, +in the original DS illustrated in Figure 2, the table +sing is a named relationship and the foreign key +aid in the table song represents an unnamed re- +lationship. We change the type of relationship by +creating a foreign key of one table in the other table, +as the example shows. +Unnamed to Named (U2N) also corresponds to +a kind of modify edge E-R transformation, which is +the reversed transformation of Named to Unnamed. +We create a relationship table and name it with +the combination of two target table names to store +the relationship. Then, we build the connection by +transferring the foreign key in the table and creating +another foreign key in it, as the example in Figure +2 shows. +A.3 +AST Updating +To update the SQL precisely, we construct the AST +of the given SQL following grammar rules and al- +ter the SQL by modifying the AST. For each E-R +transformation, we detect related subtrees in the +AST and apply the corresponding rule to update the +subtrees. For instance, we add an additional con- +dition subtree in the corresponding WHERE subtrees +while applying concept to attribute transformation. +Finally, we parse the altered SQL with the modified +AST. +In this work, we consider two type of synthetic +data, affected and unaffected. Affected samples +contain different SQL compared with the original +data, and the unaffected contain the same. We +distinguish these two types according to whether +the SQL involves a DS element that is influenced by +the transformation. And the AST updating module +is only used to synthesize affected data. +B +All kinds of E-R Transformations +Table 10 illustrates all kind of E-R transformations +and the corresponding transformation in DS. +C +Errors Cases in Synthetic Evaluation +Data +Examples in Figure 6, Figure 7, and Figure 8 illus- +trate that models tend to predict following familiar +sketch. +Operation +Transformation in DS +Vertexes +modify +inner modification of a table +merge +merge two tables +split +split an table in two +add +add a table +delete +delete a table +Edges +modify +conversion between table and foreign keys +merge +merge two tables/foreign keys +split +split a table in foreign keys +add +add a table or a foreign key +delete +delete a table or a foreign key +Table 10: All kinds of transformation and the necessity +of labels. Considered the cost of manual annotation, we +only choose three of them in this work. + +Figure 6: An example of failure prediction. +Figure 7: An example of failure prediction. +Figure 8: Another example of failure prediction. + +singers +singer +Singer ID +Name +SELECT +FROM : + singer +1 +Liliane Bettencourt +people +Singer_ ID +Identity +Name +SELECT +FROM +WHERE +singer +1 +Taylor Swift + singer +SELECT +FROMname +battles +battle +id +name +bulgarian_commander +result +6 +Battle of Boruy +Boril +Bulgarian victory +SELECT +FROM + name +WHERE +ship +lost_in_battle +id +name +6 +3 +Mary +ship +SELECT +battle +FROM +id +name +battle + battle_bulgarian_commander +battle_result +WHERE +Battle of Boruy +Battle of Boruy +Boril +Bulgarian victory +6 +SELECT +FROM +WHEREvisited +visitor +ID +Name +Age +Fernando Gago +36 +5 +SELECT +FROM +JOIN +visit +ON visitor.D = visit.Visitor_D JOIN +ON visit.Museum_D = +Museum ID +Visitor ID +museum.Museum_ID GROUP BY +1 +5 +museum +Museum ID +Name +1 +Plaza Museum +SELECT +FROM +JOIN +museum +visitor +ID +ON Visitor.ID = museum.Visitor_ID + Museum _ID + Name +Visitor_ID +Name +Age +GROUPBY + Plaza Museum +5 +5 + Fernando Gago +36 +SELECT +FROM +JOIN +GROUP BY \ No newline at end of file diff --git a/mdE3T4oBgHgl3EQf6gtE/content/tmp_files/load_file.txt b/mdE3T4oBgHgl3EQf6gtE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75541f14ec825ecf176ded24114bb19496cbda9c --- /dev/null +++ b/mdE3T4oBgHgl3EQf6gtE/content/tmp_files/load_file.txt @@ -0,0 +1,994 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf,len=993 +page_content='On the Structural Generalization in Text-to-SQL Jieyu Li, Lu Chen, Ruisheng Cao, Su Zhu, Hongshen Xu, Zhi Chen, Hanchong Zhang and Kai Yu X-LANCE Lab, Department of Computer Science and Engineering MoE Key Lab of Artificial Intelligence, AI Institute Shanghai Jiao Tong University, Shanghai, China {oracion, chenlusz, 211314, paul2204, }@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='cn {zhenchi713, xuhongshen, zhanghanchong, kai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='yu}@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='cn Abstract Exploring the generalization of a text-to-SQL parser is essential for a system to automat- ically adapt the real-world databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Pre- vious works provided investigations focusing on lexical diversity, including the influence of the synonym and perturbations in both natural language questions and databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, research on the structure variety of database schema (DS) is deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Specifically, con- fronted with the same input question, the tar- get SQL is probably represented in different ways when the DS comes to a different struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we provide in-deep dis- cussions about the structural generalization of text-to-SQL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We observe that current datasets are too templated to study structural generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To collect eligible test data, we propose a framework to generate novel text-to- SQL data via automatic and synchronous (DS, SQL) pair altering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In the experiments, signif- icant performance reduction when evaluating well-trained text-to-SQL models on the syn- thetic samples demonstrates the limitation of current research regarding structural general- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' According to comprehensive analysis, we suggest the practical reason is the overfit- ting of (NL, SQL) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 1 Introduction Given the corresponding database, text-to-SQL (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018) aims to convert a natural language (NL) utterance into a structured SQL program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To mea- sure the robustness of a text-to-SQL parser for industrial applications, the cutting edge research focuses on the cross-domain setting, where the databases used during training and evaluation do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Recently, many advanced text-to-SQL mod- els, such as RATSQL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019) and LGESQL (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021), have been proposed to tackle this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Although significant progress has been achieved considering the ultimate accu- racy, many researchers point out that actual per- Question: How many singers are there?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' singer id name age 0 Taylor Swift 32 SQL: SELECT count(*) FROM singer song id name singer 0 Love Story Taylor Swift SQL: SELECT count(distinct singer) FROM song people id name identity 0 Taylor Swift singer SQL: SELECT count(*) FROM people WHERE identity = ‘singer’ Table Column Cell Value Table Table Column Column Cell Value Cell Value Figure 1: Given the same question, the target SQL re- sponds in different ways when the database schema is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' formances in the cross-domain setting are over- estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2020) observed a dra- matic performance decline when evaluating a well- trained model on another dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Even on the same dataset where the collected samples all conform to an implicit pattern, Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2021a) discovered that current parsers are vulnerable to the adversarial attack from synonyms of words in user questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To explore the generalization capability, previous literature mainly focused on the variety of natural language, especially at the semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' How- ever, the topological feature of database schema is also important but is less investigated while study- ing the generalization capability in text-to-SQL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The database schema (DS) determines which database elements (table/column/cell value) and SQL clauses will be used to describe the seman- tics of the user question in SQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Therefore, the SQL queries may be completely divergent in different databases even given the same user ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For example, in Figure 1, the entity “singer" can function as a column, a table or a specific cell value of the column “identity", depending on the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='04790v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='AI] 12 Jan 2023 ontology of the corresponding DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Actually, in the process of database designing, different developers design the entities and their relationships in differ- ent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The differences always represent on the topological structure of DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Moreover, database contents are also considered in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For example, in Figure 1, “singer" is stored as a table in the first database while comes to a cell value in the third database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We named the ability to au- tomatically adapt different structural information representation methods the structural generaliza- tion capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Nowadays, the setup of cross-domain text-to- SQL ensures the database is completely novel in the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The structural generalization capability is supposed to be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, we find that current cross-domain datasets are over- templated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Models can even predict the structure of SQL queries only with the user question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Mean- while, when we add the DS information to the input, the performances change marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Be- sides, when we attempt to evaluate across different datasets, the phenomena still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this case, the structural generalization capability is overesti- mated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we focus on studying structural generalization and provide in-deep analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Be- cause of the over-templated feature, we can not investigate the generalization capability with cur- rent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To provide a comprehensive ap- praisal regarding the structural generalization of existing text-to-SQL parsers, we propose a data- and structure-driven framework to automatically synthesize altered (DS, SQL) pairs given the same input question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The framework modifies the DS with modest annotation cost and updates the SQL synchronously by altering the abstract syntax tree (AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Inspired by the entity-relationships di- agram (E-R Diagram) (Ling, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Li and Chen, 2009), all the transformations follow the entity re- lationships of the database to guarantee that the modifications are reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We also compared the execution results of modified and original (DS, SQL) pairs to ensure the framework updates the SQL correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In the experiments, we first evaluate the struc- tural robustness by applying perturbations to the training DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Furthermore, we create out-of- dataset (OOD) DS by applying perturbations to the development DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Then we assess the structural generalization capability using these databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Un- fortunately, both the structural robustness and struc- tural generalization are modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We further conduct several experiments to an- alyze the actual reason for the modest structural generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Firstly, we observe models can pre- cisely parse the question if the transformation of creating OOD DS does not lead to the SQL chang- ing, otherwise, they always make mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Thus, the parsing failures are not caused by the novel topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We introduce a new con- cept (NL, DS) pattern, which is the combination of the semantic role of a keyword in a natural lan- guage (NL) question and the syntax role of the related item in the database schema (DS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We sug- gest that current text-to-SQL algorithms always become overfitting on (NL, DS) patterns, and it leads to modest generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Finally, we discuss the efficiency of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Ex- periment results demonstrate that only if the extra training data provide the patterns which are rare in the original training data but exist in the develop- ment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The main contributions can be summarized: We propose a data- and structure-driven frame- work which can automatically synthesize sam- ples containing unseen (DS, SQL) patterns with minimal human labor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' By utilizing the plug-and-play framework, we synthesize a testsuite and demonstrate the poor performance of existing text-to-SQL models regarding structural generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We analyze the reasons leading to modest generalization towards perturbations of syn- chronous changes in (DS, SQL) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2 Background and Related Work Structural Features in Text-to-SQL Tasks Modeling the structural information in a database and designing an efficient algorithm to decode structured output sequences are crucial in text-to- SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Several studies achieved remarkable progress using GNN (Scarselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2008) to encode the schema linking, which enhanced the graph struc- ture of DS and the relationships between DS and question tokens (Bogin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Another line of re- search focuses on the grammar structure of SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Corresponding works proposed novel algorithms to precisely decode according to the syntax (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Rubin and Berant, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Recent works attempted to utilize the de- veloped generative pre-trained language models (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020) to generate SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Based on T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020), Scholak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2021) proposed a rule-based post-processor to prune syntax-illegal SQL subsequence in beam search, and they achieved stable improvement in the end-to-end text-to-SQL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Synthetic Data Lexical Structure Question Schema Schema SQL Spider-Syn(Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021a) ✓ � � � MR-UT(Ma and Wang, 2021) ✓ � � � MR-ST(Ma and Wang, 2021) � � ✓ � ADVETA-RPL (Pi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2022) � ✓ � � ADVETA-ADD (Pi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2022) � ✓ ✓ � Unaffected � � ✓ � Affected � � ✓ ✓ Table 1: Setups of previous evaluation datasets and our synthetic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The synthetic evaluation data was modified from Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Unaffected and Affected are two types of data we synthesize in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The mark ✓represents that the corresponding attribute is different from that in Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Oppositely, we use �to note in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Robustness of text-to-SQL models Early datasets (Dahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Hemphill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Zelle and Mooney, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Tang and Mooney, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Li and Jagadish, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Yaghmazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Finegan-Dollak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018) only considered the text-to-SQL tasks on a single database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To build a robustness text-to-SQL model that can automatically adapt unseen domain data, Recent works (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017) collected cross-domain text-to-SQL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Based on the cross-domain setup, researchers further considered some different real-world scenes and proposed corresponding datasets (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2020) observed that the execution (EX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=') accuracy of a well-trained model on Spider (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018) always decreases remarkably on the unseen domain data from other datasets1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Although Suhr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2020) depicted the reasons leading to performance decline, in-deep discussions are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To this end, recent studies generated synthetic data under different setups to further assess the practical model generalization in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 1Considered most related studies report the EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' accuracy as the results, we additionally reproduce the experiments and use the EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' accuracy as the metric and illustrate the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We summarize the characteristic of the synthetic evaluation set in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In respect of text, Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2021a) generated evaluation samples via replacing the schema-related words in NL questions with synonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Ma and Wang (2021) substituted the aggregation-related words and prefix phrases with synonym representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Pi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2022) modified the column names in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In respect of structure, Ma and Wang (2021) created different DS structures by imposing perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Pi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (2022) added adversarial columns in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, the golden SQLs in both of their synthetic datasets remain unchanged when applying perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we consider both changed and unchanged golden SQLs to provide a comprehensive appraisal regarding structural generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Dataset RATSQL LGESQL Spider (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='11 SParC (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019b) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='20 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='59 Spider-Syn (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021a) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='81 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='93 Academic (Li and Jagadish, 2014) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='36 GeoQuery (Zelle and Mooney, 1996) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='86 IMDB (Yaghmazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='96 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='74 Restaurant (Tang and Mooney, 2000) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='53 Scholar (Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='24 Yelp (Yaghmazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2017) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='51 Table 2: Models are trained on Spider, while evaluated on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The databases of SParC and Spider- Syn are similar to Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 3 Eligible Evaluation Data To evaluate the generalization capability of text-to- SQL models, test data providing novel database schema (DS) structure are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, current text-to-SQL datasets are not eligible be- cause of the over-templated features (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Therefore, we propose a data- and structure-driven generation framework to synthesize relevant data to assess the generalization capability (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1 Current Datasets are Undesirable To verify that current text-to-SQL datasets are over- templated, we conduct a syntax role prediction ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Syntax Role Prediction aims to predict which SQL syntax roles are mentioned in the query, in- cluding the SQL keywords, nested structure, and aggregation clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The metric used in this exper- iment is joint accuracy, which means the case is treated as correct if and only if all the syntax roles are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Setup w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' DB Schema w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' DB Schema Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider-like Cross-Domain 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='08 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='34 ↑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='26 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider-Syn Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider-like Cross-Domain 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='59 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='72 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='87 Spider-Syn Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider-Syn Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='69 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='40 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='29 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' SParC Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider-like Cross-Domain 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='48 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='17 SParC Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' SParC Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='92 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='50 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='42 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Academic Single Domain 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='27 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='50 ↓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='77 GeoQuery 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='42 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 ↓5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='85 IMDB 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='58 ↑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='75 Restaurants 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='20 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='60 ↑12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='40 Scholar 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='66 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 ↑3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='34 Yelp 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='40 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='69 ↓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='71 Table 3: Experiment results of syntax role prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' DB Schema represents a vanila model using BERT-base to encode user questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' DB Schema represents the model using RAT encoder to process the user questions and database schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this experiment, we compare the perfor- mances of whether contains database schema in the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' As the results in Table 3 shown, the model can directly predict the approximate structure of the target SQL only with the user question most of the time, even though the databases for training and for testing are not overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Meanwhile, the perfor- mance differences between using and without using DS demonstrate that the DS information is helpless for predicting the SQL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Therefore, we suspect that current datasets are too templated to evaluate the generalization capability using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To this end, we need to synthesize eligible evalua- tion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 Evaluation Data Generation To assess the structural generation capability, we propose a data- and structure-driven generation framework to synthesize relevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The syn- thetic data in this paper are modified from Spi- der (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2018) 2 which is the most popular cross-domain text-to-SQL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' It contains 8659 training examples and 1034 validation examples across 146 databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The test dataset is unseen and contains 2147 samples with 40 databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For a given sample, we synthesize a new sample via altering the DS while keeping the question con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In order to obtain a reasonable DS, we con- struct the entity-relationship graph of the given DS and apply graph-based transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Moreover, we synchronously update the SQL by modifying the abstract syntax tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We show more details in Appendix A 2https://yale-lily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='io//spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we use four different transforma- tions in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 2 illustrates the examples of each transformation, and we show a brief introduc- tion below: Entity to Attribute (E2A) merges two tables into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Concept to Attribute (C2A) converts the concept3 of an entity, which represents via table name in DS, to its attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Named to Unnamed (N2U) replaces the ta- ble corresponding to a relationship with for- eign keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Unnamed to Named (U2N) replaces a for- eign key with a relationship table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Table 4 shows the total number of each kind of synthetic data synthesized via different E-R trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We evaluate the synthetic quality by comparing the execution results of the original and synthetic (DS, SQL) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Over 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='43% gener- ated samples kept consistent execution results on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Affected Unaffected Affected Unaffected E2A 3035 9466 493 1477 C2A 2659 4271 379 445 U2N 2969 12910 114 376 N2U 2605 48507 303 4484 Table 4: Statistics of generated data for four transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 3It refers to the definition of concept node in the knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Entity ➔ Attribute Concept ➔ Attribute Named ➔ Unnamed Unnamed ➔ Named Original Figure 2: Examples of the DS synthesized via different transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The dotted lines denote foreign keys (from foreign key to primary key) 4 Generalization Evaluation In this section, we conduct experiments to evalu- ate the following capability of current text-to-SQL models: C1: The robustness of current text-to-SQL parsers when applying a perturbation on the database schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2) C2: The practical generalization capability of current text-to-SQL parsers when it comes to novel databases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1 Experiment Setup In this work, we experiment with two grammar- based SOTA text-to-SQL parsers, RATSQL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2019) and LGESQL (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Be- sides, we also experiment with the T5-based end-to- end text-to-SQL parser, including the methods of decoding with and without PICARD(Scholak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The evaluation metric we use to report the results is exact set match accuracy (EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Results are averaged over three trials to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Equivalent Test Set (ETS) To precisely evalu- ate the model robustness, we construct an equiva- lent test set for the given dataset, which contains the same number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We restrict that each sample in the original dataset matches exactly one synthetic variant in the ETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' If a sample can not generate a variant, we will add the duplication in the ETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Furthermore, to reduce the influence of hardness, we utilize a heuristic algorithm to modu- late the ETS so that its distribution is closed to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 Structural Robustness To evaluate C1, we construct the equivalent test set (ETS) for the training set of Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The training data in this experiment is the Spider training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We compare the performances on two test sets, the Spider training set, and the corresponding ETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Model Test Data EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' RATSQL Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='19 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='81↓37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='82 LGESQL Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='94 Spider ETS 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='73↓36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='21 T5 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='03 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='78↓41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='25 T5+Picard Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='52 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='48 ↓32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='04 Table 5: Exactly match (EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=') accuracy on the evalua- tion data synthesized from the Spider training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Experiment results illustrated in Table 5 indi- cate that the perturbation applied to the database schema will disturb the parsing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The mod- els can not precisely infer the representation of the SQL query when confronting novel DS structures despite the questions and the other parts of the DS being the same as they appeared in the training phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this case, we suggest that the structural robustness of current text-to-SQL models is mod- est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3 Practical Structural Generalization To evaluate C2, we construct the ETS for the de- velopment set of Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The training data in this experiment is also the Spider training set while we compare the performances on the Spider develop- ment set and the corresponding ETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Experiment results in Table 6 illustrate that the parsers seem to perform well on completely novel DS, however, the performances dramatically de- cline once perturbations are applied on these novel DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' These 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='Test Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' RATSQL Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='68↓25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='15 LGESQL Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='10↓25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='47 T5 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='63↓33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='46 T5+Picard Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='89 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='87↓33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='02 Table 6: EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' accuracy on the evaluation data synthe- sized from the Spider development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' cal structure generalization capability is also mod- est, which is similar to the structural robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Therefore, we suspect that current text-to-SQL parsers can not automatically infer the SQL pat- tern according to the DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We will discuss the truly reason caused this issue in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 5 Discussion about Structural Generalization In this section, we discuss the structure general- ization of text-to-SQL by answering the following questions: Q1: What is the actual function of database schema input?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1) Q2: What is the actual reason causing the modest generalization capability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2) Q3: Does data augmentation can increase the generalization capability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1 Function of Database Schema Input To answer Q1, we first verify that the database schema (DS) information is independent with the process of constructing SQL patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Reviewing the experiments in Section 4, models always make mistakes when facing out-of-dataset (OOD) DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To estimate whether the OOD structure confuses the parsers, we consider the evaluation data containing OOD DS while keeping the SQL query unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Setup: Different from the evaluation data used in Section 4, we generate the data with different DS but the same SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For each piece of data, the DS transformations are applied on untapped parts so that the SQL will not be influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Similarly, we also construct the equivalent test set (ETS) for the Spider development set with this kind of synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The training data in this experiment is the training set of Spider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Model Test Data EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' RATSQL Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='67↓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='16 LGESQL Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='41↓3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='16 T5 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='20 ↓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='89 T5+Picard Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='89 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='06 ↓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 Table 7: EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' accuracy on the evaluation data synthe- sized from the Spider training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The SQL in syn- thetic data is consistent with the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Results shown in Table 7 demonstrate that the OOD structure does not influence the inference process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Reviewing the syntax role prediction ex- periments discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1, we suggest that current text-to-SQL models construct SQL query via sentence pattern of the user question rather than the actual structure of DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We suspect that the func- tion of DS input is providing the correct presenta- tion of the SQL non-keywords (table name, column name, and value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The efficiency of using schema linking provides a strong signal on target database item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Once the explicit relationships between these SQL non-keywords and the presentations in ques- tion are destroyed, models will make mistakes on selecting correct schema item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, the SQL structure is always predicted in correct ways(Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 (NL, DS) Pattern To answer Q2, we first introduce the concept of (NL, DS) pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Then we will illustrate how the (NL, DS) pattern influences the generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The (NL, DS) pattern represents the combination of a natural language (NL) role and a database schema (DS) role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' NL Role: As the examples in Figure 3 shown, we assume that some words in the NL question de- scribe the key information of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this sec- tion, we simply split these keywords into two cate- gories, target and condition, which we call them the NL role of these words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' target represents the query- ing target and condition represents the constraint of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The NL roles are DS-independent, in the other words, they only depend on the semantics of the NL question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' DS Role: For a DS, some elements link the key- words in the given question, such as the word “singer" in the first case, and all of them play a Question: How many people whose identity is singer ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' SQL: SELECT count(*) FROM people WHERE identity = ‘singer’ target Table condition condition Column Cell Value Question: How many people whose identity is singer ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' target condition condition Table SQL: SELECT count(*) FROM singer Padding Padding Figure 3: Different combinations of NL role and DS role determine different SQL sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' unique role in the given DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We define the DS role as table, column, cell value and a padding role to link the non-schema-related keywords, for instance, the word “people" in the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' (NL, DS) Pattern: For each of these elements, we named the combination of an NL role and a DS role as an (NL, DS) pattern, which determines the syn- tax role in SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For example, the element “singer" in the first case functions the NL role condition and the DS role cell value, in this case, it locates in the WHERE clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, when the DS role comes to table, as shown in the second case, the element “singer" will locate in the FROM clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For the given NL question and DS, the structure of the SQL query depends on the containing (NL, DS) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We assume that the modest generalization ca- pability is because of the over-fitting of (NL, DS) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Unseen (NL, DS) patterns lead to failed parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We first evaluate the performance on the samples synthesized via different E-R transforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The experiment results are illustrated in Ta- ble 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We notice that models make mistakes on almost all the samples generated using C2A E-R transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Actually, C2A is a special transfor- mation because it must create an (NL, DS) pattern, (target, cell value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' This pattern always represents a condition in the WHERE clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, we find that the condition is always deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Therefore, we randomly sample 100 pieces of data from the training set and the synthetic C2A data to evalu- ate whether (target, cell value) is an unseen pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Table 8 shows the statistic results of manually calcu- lating the distribution of (NL, SQL) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The combination (target, cell value) is not contained in the training set but exists as unseen patterns when it comes to C2A samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We additionally enu- merate some typical error cases in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The DS Role Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' C2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Target Condition Target Condition Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='62% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='18% Column 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='82% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='04% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='33% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='43% Cell Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='68% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='67% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='04% Padding 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='65% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='36% Table 8: Distribution of (NL, DS) pattern in the training set and C2A samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 4: Training without extra data, evaluating on different synthetic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' examples demonstrate that models tend to parse ac- cording to experiences so that they make mistakes on novel patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this case, we suggest that the actual reason causing the modest generalization capability is the (NL, DS)-pattern-wise over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3 Convert Unseen to Seen In this section, we answer Q3 by evaluating the efficiency of training with extra synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Setup: We conduct experiments to on both original and synthetic evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' As the data augmen- tation, we train models with the original training set of Spider and additional synthetic data with a 1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We consider two kinds of extra synthetic training data in these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The one is the data containing novel database schema (DS) and different SQL queries (compared with the original data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' They are similar to the evaluation data we used to build the ETS in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We named these data Affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The other are the data containing novel DS while the same SQL queries, which is similar to the data from the Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We named them Unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The synthetic evaluation data we used in this section is Affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We report our results in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Experiment results in the upper block illustrate that either af- E2A: 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='88 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 Original C2A 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='32 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00 U2N N2U 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='65 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='38Test Data Training Data RATSQL LGESQL T5 T5+PICARD Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='83 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='89 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' + Affected 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='12 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='29 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='89 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='68 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='22 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='87 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='01 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='12 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' + Unaffected 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='38 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='55 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='05 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='52 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='99 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='10 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='85 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='04 Spider Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ETS Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='68 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='10 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='63 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='87 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' + Affected 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='21 ↑22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='53 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 ↑22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='47 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='12 ↑30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='49 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='08 ↑25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='79 Spider Train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' + Unaffected 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='23 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='55 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='17 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='07 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='57 ↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='06 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='09 ↑0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='22 Table 9: Results of using extra synthetic samples in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' fected or unaffected extra training data can improve the performance on the original development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The reason is that extra training data do not provide the (NL, DS) patterns which are rare in the origi- nal training set but appear in the original develop- ment set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Actually, the problem of over-templated demonstrates that it is hard to find the aforemen- tioned patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Unlikely, the transformations ap- plied in this work either do not guarantee these patterns are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Experiment results in the lower block of Table 9 show that the affected extra training data is helpful to improve the performance on the synthetic evalu- ation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, the usage of unaffected data can not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The reason is that the former provides the (NL, DS) patterns which are rare in the original training set while are contained in the ETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' On the other hand, the latter do not provide any of these patterns because the perturbations are applied on untapped parts of DS in unaffected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' There- fore, we suggest that specific data augmentation can enhance the model despite it can not be verified on current datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' This experiment amplifies the improvement of data augmentation by increasing the overlap of (NL, DS) patterns between the extra training data and the synthetic evaluation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 6 Not Only in Cross-Domain Actually, the problem of (NL, DS)-pattern-wise overfitting is not the specific problem that only ex- ists in cross-domain text-to-SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Modest structural generalization is just one of the phenomena under a cross-domain setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Single-domain text-to-SQL also has the same problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' From the view of (NL, DS) patterns, the defi- ciency of patterns in the training stage leads to the appearance of unseen patterns in the test stage, and further cause performance decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, leav- ing out patterns is inevitable during the data collec- tion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Annotators can neither ensure to ask questions in all possible sentence patterns nor guar- antee that all combinations of schema items are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For instance, as the example illustrated in Figure 1, confronting the third DS, annotators may not come up with the question about “singer", or they may ask in the way of How many people whose identity is a singer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='. In this case, automati- cally addressing unseen patterns is also essential in single-domain text-to-SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 7 Conclusion We find that current text-to-SQL datasets are too tamplated to investigate generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To this end, we constructed a generation framework to synthesize text-to-SQL data for evaluation in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Experiment results illustrate that the model generalization is modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Furthermore, the analysis illustrates that the problem is caused by the overfitting of (NL, DS) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Finally, we demonstrate that when adding extra training data to convert unseen patterns to seen patterns in the evaluation stage, the problem will improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' References Ben Bogin, Matt Gardner, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Global reasoning over database structures for text-to- sql parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} 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Asso- ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Yujian Gan, Xinyun Chen, Qiuping Huang, Matthew Purver, John R Woodward, Jinxia Xie, and Peng- sheng Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Towards robustness of text- to-sql models against synonym substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='01065.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Lan- guage Processing (EMNLP-IJCNLP), pages 1962– 1979, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Tao Yu, Rui Zhang, Kai Yang, Michihiro Yasunaga, Dongxu Wang, Zifan Li, James Ma, Irene Li, Qingn- ing Yao, Shanelle Roman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Spider: A large-scale human-labeled dataset for complex and cross-domain semantic parsing and text-to-sql task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='08887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Tao Yu, Rui Zhang, Michihiro Yasunaga, Yi Chern Tan, Xi Victoria Lin, Suyi Li, Heyang Er, Irene Li, Bo Pang, Tao Chen, Emily Ji, Shreya Dixit, David Proctor, Sungrok Shim, Jonathan Kraft, Vin- cent Zhang, Caiming Xiong, Richard Socher, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' SParC: Cross-domain se- mantic parsing in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Com- putational Linguistics, pages 4511–4523, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' John M Zelle and Raymond J Mooney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Learn- ing to parse database queries using inductive logic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In Proceedings of the national con- ference on artificial intelligence, pages 1050–1055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Victor Zhong, Caiming Xiong, and Richard Socher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Seq2sql: Generating structured queries from natural language using reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' arXiv preprint arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='00103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' A Generation Framework The overview of the generation framework is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For a given sample, we synthe- size a new sample via altering the DS while keeping the question constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In order to obtain a reason- able DS, we construct the entity-relationship graph of the given DS and apply graph-based transforma- tions, which we introduce in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1 and Sec- tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Moreover, we synchronously update the SQL by modifying the abstract syntax tree, and we show more details in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='1 Entity-Relationship Graph A relational database organizes data in predefined relationships, which are represented as the struc- tural relationships among tables and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To clearly describe reasonable relationships, develop- ers always utilize Entity-Relationship Diagram (E- R Diagram) (Ling, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Li and Chen, 2009) to define the relationships among the raw data, which is helpful to design the database structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Inspired by ER Diagram, we attempt to modify the DS fol- lowing the entity relationships so that the rationality of the altered DS can be ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To this end, we introduce the definition of Entity-Relationship (E- R) Graph in this paper, which evolves from E-R Diagram while leaving out the attributes vertexes to emphasize the topology feature 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The vertex in E-R Graph represents an entity, and the edge repre- sents the relationship between the entities that its terminal vertexes correspond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Both the vertex and the edge function as a table in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For example, as shown in Figure 5, each of the table people, the table author, and the table novel corresponds to a vertex in E-R Graph, and the table write corre- sponds to an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Thus, to construct the E-R Graph, we manually annotate a binary tag for each table in DS to dis- tinguish between entity and relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We label relationship following two principles and label the others as entity: 4The attributes node in E-R Diagram refers to a column in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To emphasize the topology feature, we replace Diagram with Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' FROM author write novel FROM author novel write author novel aid id age name nid id name people pid id author novel age name aid name people pid id id E-R Transformation people novel author people novel author AST Updating Sample( X, D, Y ): Question( X ): Who is the author of Harry Potter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Database( D ) SQL( Y ): SELECT people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='name FROM people JOIN author ON people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='id = author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='pid JOIN write ON author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='id = write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='aid JOIN novel ON write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='nid = novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='id WHERE novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='name = ‘Harry Potter’ Database(D) Database(D’) Sample( X, D’, Y’ ) Question(X): Who is the author of Harry Potter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Database( D’ ) SQL( Y’ ): SELECT people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='name FROM people JOIN author ON people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='id = author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='pid JOIN novel ON author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='id = novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='aid WHERE novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='name = ‘Harry Potter’ E-R Graph of D AST of Y AST of Y’ E-R Graph of D Figure 5: Overview of the generation framework we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The upper part illustrates the original sample, and the lower part illustrates the synthetic sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The red rectangles and red lines denote the modified parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For the E-R graph, the dotted lines denote unnamed relationships, and solid lines denote named relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Bridge Structure: The given table should con- tain exactly two foreign keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Semantic Convention: The table name should be the combination of two entities such as the rela- tionship Customer_Addresses combining Customer and Address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Apart from that, the phrase obeys hu- man language conventions is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For instance, the relationship visit linking visitor and museum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='2 E-R Transformation E-R transformation is the graph transformation in the E-R graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' There are ten kinds of E-R trans- formation, containing five operations applied on vertexes or edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We assume the databases that store the same data in the different schema can transform between each other via a sequence of E-R transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We illustrate all kinds of E-R transformations and the corresponding transforma- tions in DS in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' However, some trans- formations are insecure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For example, the usage of delete edge transformation will lead to informa- tion loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Besides, some transformations rely on strict annotation criteria and costly manual labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For instance, whether a table can be split into two need rigorous judgment according to the semantic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we use three E-R trans- formations with no need for additional annotations, and they totally correspond to four different trans- formations in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 2 illustrates the examples of each transformation, and we show more details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Entity to Attribute (E2A) corresponds to a kind of merge vertexes E-R transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For a pair of vertexes in the E-R graph, we split them as a source entity and a target entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' The target en- tity corresponded table is the only one that contains the foreign keys of the source entity corresponded table in the DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Both of the vertexes can be treated as the source entity as long as the combination is eligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' As the example shown in Figure 2, for the attributes in the source entity, we convert them into the new attributes in the target entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' To avoid semantic loss, we rename the attributes following rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Besides, we utilize a series of rules to rec- ognize a special column as the agent of the entity, such as the column name, and it will be used to replace the foreign key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Concept to Attribute (C2A) corresponds to a kind of modify vertex E-R transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Different from the column-wise modification, we focus on altering the role of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We attempt to convert the concept5 of an entity, which represents via table name in DS, to its attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Firstly, we detect a high-level category of the entity using a pre-trained NER model (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In the example shown in Figure 2, people is the high-level category of singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Then, we create an additional attribute to store the concept by rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this case, we use 5It refers to the definition of concept node in knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' the new column identity to record the concept singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Named to Unnamed (N2U) corresponds to a kind of modify edge E-R transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We name the relationship represented by a table as Named, and that by foreign keys as Unnamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For instance, in the original DS illustrated in Figure 2, the table sing is a named relationship and the foreign key aid in the table song represents an unnamed re- lationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We change the type of relationship by creating a foreign key of one table in the other table, as the example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Unnamed to Named (U2N) also corresponds to a kind of modify edge E-R transformation, which is the reversed transformation of Named to Unnamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We create a relationship table and name it with the combination of two target table names to store the relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Then, we build the connection by transferring the foreign key in the table and creating another foreign key in it, as the example in Figure 2 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='3 AST Updating To update the SQL precisely, we construct the AST of the given SQL following grammar rules and al- ter the SQL by modifying the AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For each E-R transformation, we detect related subtrees in the AST and apply the corresponding rule to update the subtrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' For instance, we add an additional con- dition subtree in the corresponding WHERE subtrees while applying concept to attribute transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Finally, we parse the altered SQL with the modified AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' In this work, we consider two type of synthetic data, affected and unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Affected samples contain different SQL compared with the original data, and the unaffected contain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' We distinguish these two types according to whether the SQL involves a DS element that is influenced by the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' And the AST updating module is only used to synthesize affected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' B All kinds of E-R Transformations Table 10 illustrates all kind of E-R transformations and the corresponding transformation in DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' C Errors Cases in Synthetic Evaluation Data Examples in Figure 6, Figure 7, and Figure 8 illus- trate that models tend to predict following familiar sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Operation Transformation in DS Vertexes modify inner modification of a table merge merge two tables split split an table in two add add a table delete delete a table Edges modify conversion between table and foreign keys merge merge two tables/foreign keys split split a table in foreign keys add add a table or a foreign key delete delete a table or a foreign key Table 10: All kinds of transformation and the necessity of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Considered the cost of manual annotation, we only choose three of them in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 6: An example of failure prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 7: An example of failure prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' Figure 8: Another example of failure prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='singers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='singer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='Singer ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE3T4oBgHgl3EQf6gtE/content/2301.04790v1.pdf'} +page_content='SELECT ' metadata={'source': 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a/odE3T4oBgHgl3EQfjQqW/content/tmp_files/2301.04587v1.pdf.txt b/odE3T4oBgHgl3EQfjQqW/content/tmp_files/2301.04587v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..90150af497955abd47abcaec4e71b8e8c2eaa0c9 --- /dev/null +++ b/odE3T4oBgHgl3EQfjQqW/content/tmp_files/2301.04587v1.pdf.txt @@ -0,0 +1,2230 @@ +1 +Electric Vehicles Security and Privacy: +Challenges, Solutions, and Future Needs +Alessandro Brighente, Member, IEEE, Mauro Conti, Fellow, IEEE, Denis Donadel, +Raadha Poovendran, Fellow, IEEE, Federico Turrin, and Jianying Zhou, Senior Member, IEEE, +Abstract—Electric Vehicles (EVs) share common technologies +with classical fossil-fueled cars, but they also employ novel +technologies and components (e.g., Charging System and Battery +Management System) that create an unexplored attack surface +for malicious users. Although multiple contributions in the lit- +erature explored cybersecurity aspects of particular components +of the EV ecosystem (e.g., charging infrastructure), there is still +no contribution to the holistic cybersecurity of EVs and their +related technologies from a cyber-physical system perspective. +In this paper, we provide the first in-depth study of the +security and privacy threats associated with the EVs ecosystem. +We analyze the threats associated with both the EV and the +different charging solutions. Focusing on the Cyber-Physical +Systems (CPS) paradigm, we provide a detailed analysis of all the +processes that an attacker might exploit to affect the security and +privacy of both drivers and the infrastructure. To address the +highlighted threats, we present possible solutions that might be +implemented. We also provide an overview of possible future +directions to guarantee the security and privacy of the EVs +ecosystem. Based on our analysis, we stress the need for EV- +specific cybersecurity solutions. +Index Terms—Electric Vehicles, Cyber-Physical Systems, Se- +curity, Privacy +I. INTRODUCTION +T +HE recent climate crisis demands green alternatives to +replace technologies with high environmental impact. +Among the others, fossil-fueled transportation is one of the +significant causes of greenhouse gases. Electric Vehicles (EVs) +have been proposed as a green alternative, where electric bat- +teries are employed as a power source. During the last years, +the number of people opting for the EV alternative increased +up to the point where the market share of new EV sales +reached more than 50% in countries such as Iceland (55.6%) +and Norway (82.7%) [1]. The adoption of EVs is further +expected to increase in the next years. In fact, governments are +incentivizing the adoption of EVs thanks to the deployment of +a large number of Electric Vehicle Supply Equipment (EVSE) +in public charging infrastructures [2] and planning to ban +sales of fossil-fueled vehicles [3]. Furthermore, technology +advancements remove the current barriers against consumers’ +A. Brighente, M. Conti, D. Donadel, and F. Turrin are with the Department +of Mathematics and HIT Research Center, University of Padova, 35131 +Padova, Italy (e-mail: alessandro.brighente@unipd.it; conti@math.unipd.it;). +R. Poovendran is with the Department of Electrical and Computer Engi- +neering, University of Washington, 98195 Seattle, USA. +J. Zhou is School of Information Systems Technology and Design, Sin- +gapore University of Technology and Design, 487372, Singapore. (email : +jianying zhou@sutd.edu.sg). +Manuscript received X X, 2022; revised X X, 2022. +adoption of EVs, providing extended driving range and seam- +less charging [4]. +The increasing number of EVs demands a thorough analysis +of the security of both vehicles and infrastructure operations. +Like traditional vehicles, EVs are equipped with many Elec- +tronic Control Units (ECUs), sensors and actuators that mea- +sure, process, and control the different stimuli inside and out- +side the vehicle. However, EVs include additional components. +Indeed, an EV integrates components to govern the hardware +and software dedicated to managing electric energy smartly. +These components are, for instance, the Battery Management +System (BMS) and the charging system. +Different studies have already proven the impact of potential +cybersecurity attacks on automotive systems. For instance, +Miller and Valasek [5] proved the feasibility of hijacking a +vehicle by remotely controlling it through the infotainment +system. Furthermore, most existing vehicles exploit Controller +Area Network (CAN) as in-vehicle network architecture, +which has already been proved as non-secure [6] and, there- +fore, may be vulnerable to potential cyberattacks. Lastly, +privacy shall also be guaranteed to prevent malicious users +from obtaining sensitive information on the driver, such as +her location or habits. It is essential to include security and +privacy features by design to prevent these and other attacks. +Researchers investigated vehicle security, focusing on the +different aspects of in-car communications [7], [8]. However, +EVs are equipped with specific components that provide +fundamentally different attack surfaces and exploitation points. +For instance, EVs are equipped with electric batteries to power +the vehicle components ranging from the infotainment system +to the acceleration pedal, together with systems to manage the +electrical power as shown in Figure 1. +Therefore, analyzing the in-vehicle threats associated with +these components is essential. Furthermore, the power supply +must be regulated by dedicated hardware, not classical vehi- +cles. Researchers discussed how the EV charging infrastructure +could be exploited by attackers [9], however, neglecting the +in-vehicle threats. +Contribution. In this paper, we examine the security and pri- +vacy issues of EVs from a Cyber-Physical System (CPS) point +of view. Given the high demand for EVs and the increasing +number of deployed charging facilities, it is fundamental to +guarantee the security and privacy of both vehicles and users. +Many literature contributions discuss solely technical aspects +of the EV ecosystem without focusing on security issues. +Other security-focused works study a single system compo- +nent (e.g., the vehicle’s internal bus, the smart grid, or the +arXiv:2301.04587v1 [cs.CR] 11 Jan 2023 + +2 +communication protocols) without comprehensively analyzing +the whole environment. We provide a general overview of EV +functioning, focusing on their core components to build the +basic knowledge needed to analyze the possible threat vectors. +We then discuss possible attacks and countermeasures specific +for EV and underline the existing security solutions for fuel +vehicles that are also effective in EVs. With the bird-eye on the +CPS concept, we are not only able to discuss the issues related +to the exchange of information between the different involved +entities, but also the side channels that may leak sensitive +information or that could lead to hazardous behavior impacting +on users’ safety. We hence shed light on the unresolved chal- +lenges of EVs ecosystems, providing interested researchers +with possible directions worth investigating to guarantee the +security and privacy of the overall EV ecosystem. We also +consider future directions such as the Wireless Power Transfer +(WPT) charging of EVs, which has only been developed on +small-scale testbeds at the time of writing. We believe that +delving into this emerging system’s security and privacy issues +will help future developers design and implement secure-by- +design WPT solutions for EV. +We summarize the contribution of this work as follows. +• We examine the peculiar components that differentiate +EVs from fossil-fueled vehicles and provide an overview +of their role and how they exchange information. +• We provide an overview of the different technologies +employed to charge electric vehicles, comprising both +wired and wireless charging. We present the available +standards for each of them and describe their basic +functioning. +• We provide an in-depth discussion of the security and +privacy issues of the EVs ecosystem. We analyze the +threats related to the in-vehicle network and the threats +related to the charging process. We particularly focus on +their effects on the peculiar EV components and analyze +them from a CPS point of view. +• We analyze and compare possible countermeasures pro- +posed in the literature for each of the presented attacks, +even grasping from other similar areas. +• We outline future directions for research in the EV +cybersecurity domain. +Organization. The rest of the paper is organized as follows. +In Section II, we review the related literature. In Section III, +we describe the EV components the charging infrastructure. +In Section IV, we then discuss the in-vehicle security and +privacy threats, and those related to the charging infrastruc- +ture in Section V. Along with the threats, we also present +possible countermeasures. Then, we discuss the possible future +direction in Section VI, and lastly we conclude the paper in +Section VII. +II. RELATED LITERATURE ON EV SECURITY +Automotive cyber-security requires standardization to allow +for security guarantees and interoperability. Schmittner et +al. [7] reviewed the available standards, including designing +and validation aspects. These standards, however, do not +consider the peculiar features of EVs. Scalas et al. [8] provided +Fig. 1. Main components of an EV. Green components are EV specific. +an overview of the cybersecurity requirements for the future of +the automotive industry, focusing on in-vehicle components. +They discussed several technologies and attacks but were +not specific for EVs. Furthermore, different works present +technical reviews of the EV ecosystem [10], [11]. However, +none of them consider the security aspects. +Some contributions in the literature focused on specific +components of EVs. For instance, Khalid et al. [12] focus on +the BMS, discussing the lack of a cybersecurity standard to +guarantee its security and providing an overview of the possi- +ble standardization framework that could be adopted to achieve +this goal. Chandwani et al. [13] presented an overview of +the cybersecurity threats associated with the onboard charging +system of EVs. Despite providing an accurate analysis of the +security of this component, their contribution does not consider +how these attacks can impact the other peculiar components +of the EV. These contributions do not provide a general +overview of the EV ecosystem. Furthermore, they do not +discuss the threats associated with privacy. Acharya et al. [14] +provide the first discussion on how EVs can be considered +as CPS. The authors discuss how different attacks can be +conducted inside the car and during communication with the +power supplier. However, they do not consider the specific +components of the EVs such as the BMS. Jin et al. [15] focus +on the CPS system represented by the power electronics in +EVs. However, they do not consider how these attacks may +impact the other components of the EV and did not discuss the +issues related to WPT. Most of the literature related to EVs’ +cybersecurity focus on the charging infrastructure and process. +Gottumukkala et al. [9] provide an overview of the CPS threats +associated with a wired EV charging infrastructure. Antoun +et al. [16] discuss the security threats associated with the +negotiation and actuation of a charging session investigating +the communications between the multiple involved entities. +They presented different charging scenarios, neglecting the +WPT option. +Vehicles can be interconnected with one another to form +the internet of vehicles. This is also feasible with EVs, +which imposes additional security challenges. Fraiji et al. [21] +discuss the cybersecurity threats associated with the internet +of electric vehicles, discussing the threats associated with the +communication with the multiple involved entities being part + +ECU +ECU +LIN bus +LIN Master +ECU +Battery +BMS +Controller +Pack +Motor +CANbus +ECU +LIN Master +Onboard +Charger +LINbus +ECL3 +Reference +BMS +Onboard Charger +Battery Pack +Controller +Electric Motor +Wired Charging +Wireless Charging +Khalid et al. [12] +Chandwani et al. [13] +Acharya et al. [14] +Ye et al. [15] +Sripad et al. [17] +Gottumukkala et al. [9] +Antoun et al. [16] +Garofalaki et al. [18] +Van Auben et al. [19] +Babu et al. [20] +Our paper +of the road infrastructure. The cybersecurity focus is on the +communication links, therefore neglecting the impact of the +peculiar EVs’ components. +Garofalaki et al. [18] present a detailed survey on Open +Charge Point Protocol (OCPP) and the corresponding threat +and vulnerabilities on the Vehicle-to-Grid (V2G) ecosystem +due to its adoption. Similarly [19] overview the main protocols +for EV charging adopted in the Netherlands and analyze their +security features, while Babu et al. [20] analyzed the security +of the main protocols proposed for the EV environment with +a particular focus on the payment methods and the authentica- +tion solutions. Differently from these works, instead of focus- +ing on the protocols, we focus on the entire EV architecture, +highlighting the main security and privacy challenges in this +typology of CPS. +Table II compares the related works on EV security with +our contribution. We can see that most of the contributions +focus on vehicle-to-grid communications in the wired case. +However, none of the available papers focus on intertwining +the different cyber-physical aspects of EVs. Therefore, our +paper provides a more complete analysis of the security and +privacy challenges for EVs. +III. ELECTRIC VEHICLES FROM A CYBER-PHYSICAL +SYSTEM PERSPECTIVE +In this section, we analyze EVs from a CPS perspective. We +emphasize those components that differentiate EVs from gas- +fueled vehicles. In particular, we first describe the traditional +vehicle architecture in Section III-A. Then, we present the +main components of an EV in Section III-B, showing how +it differs from traditional vehicles. Lastly, we provide an +overview of the EV charging infrastructure in Section III-C. +A. Traditional Vehicle Architecture +Nowadays, vehicles contain dozens of different micro- +computers, called Electronic Control Units (ECUs), running +millions of lines of code [22]. Each ECU is responsible for +controlling a mechanical (e.g., brakes) or electrical (e.g., light) +component of a modern vehicle. Depending on the component +it has to manage, an ECU generally employs a wide range +of microcontrollers, from simple 8-bit RISC controllers to +more complex 32-bit multicore processors. ECU are typically +implemented with ad-hoc firmware, even if complex ECUs +may run complete operating systems: the infotainment system, +for instance, usually runs a Linux-based kernel. In order +to provide more flexibility during updates, more advanced +solutions envisage the implementation of multiple ECUs on +a single FPGA board [23]. +Communications among ECUs that reside in the vehicle +pass through wires that connect multiple components. The +two mostly implemented technologies are CAN and Local +Interconnect Network (LIN). CAN represents the main net- +work that allows for cost-effective wiring, self-diagnosis and +error correction [24]. The CAN bus consists of two wires and +implements a distributed architecture, where car modules (i.e., +the ECUs) share messages upon winning a contention phase. +However, CAN has been designed to be a reliable solution, +neglecting possible security ad privacy shortcomings. +The LIN bus is a supplement to CAN [25]. In particular, it +connects a smaller number of ECUs (one master and up to 16 +slave nodes) and offers a drastically cheaper implementation +at the cost of lower performance and reliability. A LIN master +node is typically a gateway to CAN, and multiple LIN buses +can communicate via the CAN bus. The LIN bus can be used +to control, among the others, sensors and actuators for steering +wheels, comfort, powertrain, engine, air conditioning, doors, +and seats. +Besides CAN and LIN, also other technologies such +as FlexRay [26] and Media Oriented Systems Transport +(MOST) [27] are currently used for automotive networks. To +overcome some of these technologies’ limitations and ease +their interoperability, automotive Ethernet has recently been +introduced as a possible solution [28]. Given the CPS nature +of our investigation, we do not prefer one technology over +another, as these all represent communication means for the +exchange of information inside the EV. We refer the interested +reader to [28] for a discussion on automotive Ethernet security. +Modern vehicles also include mechanisms to update the +internal software. This service is generally implemented with +the aid of external device plug (e.g., USB flash drive) or +Over-the-Air (OTA) software update [29] (e.g., via Internet +connection). Furthermore, many vehicles nowadays include +complex entertainment systems, which may expand the vul- +nerable surface, exposing new connections (e.g., Bluetooth) +and operating systems (e.g., Android). + +4 +B. Electric Vehicle Specific Components +EVs share most of the architecture with fuel-based vehicles. +However, they comprise a different set of hardware modules +that manage how the vehicle generates power and how to +generate motion. In particular, an EV comprises the following +components [30], depicted in Figure 1. +Battery. +The battery is where the charge is stored in the +form of Direct Current (DC). It provides the power needed to +operate the EV components. Batteries are usually combined +in packs and connected in series or parallel to increase the +voltage and Amper/hour they can deliver to the EV. Batteries +suitably combined are enclosed into a metal casing to prevent +damage. The case usually includes a cooling system to avoid +damage due to batteries overheating. +Battery Management System. +This module manages all +operations regarding the battery. It manages the current output +and the charging and discharging of the battery by keeping it +in a safe operating area. Hence, it regulates the electricity flow +through the battery. The BMS is unique for each EV model, +and may be designed according to various topologies, i.e., +modular, centralized or distributed [12]. The BMS monitors +each battery in the pack and measures each cell’s voltage, +current, and temperature. It is instructed with a threshold +limit for each of them and disconnects the load if values +exceed the threshold value. Furthermore, the BMS measures +the State of Charge (SoC) and state of health of the battery. +The BMS communicates with the human-machine interface to +report information on this information. All the information are +exchanged via CAN or LIN bus. +Battery Charger/Onboard Charger. +This component pro- +vides an interface between the charging system and the EV +battery. As soon as an Alternate Current (AC) charging process +begins, the charger converts the input voltage to DC and +passes it to the battery for storage. For high power DC +charging, the conversion phase is done on the charging column. +Furthermore, it prevents possible damages to the battery or +the supply system (e.g., overheating) by limiting the power +flow [13]. +Controller. The controller handles the flow of current from +the battery to the EV associated with all operations, ranging +from motors-related operations to powering the infotainment +system. It receives the input from the driver to control the +acceleration, brake pressure, and driving mode and converts +the energy in the battery from DC to AC to control the +EV accordingly. On the other hand, the EV may generate +electricity due to, e.g., regenerative braking. In this case, the +controller converts the generated AC to DC such that the +energy can be stored in the battery. +Electric Motor. +The motor is powered by the EV battery, +which provides the electricity needed to turn it and move the +EV. The electric motor communicates with sensors and actua- +tors in the EV that control the amount of thrust required [31]. +There exist many implementations of electric motors. The +most commonly used for EVs are AC induction due to their +lower cost implementation thanks to the absence of permanent +magnets. +These components characterize an EV and differentiate it +from other types of vehicles. In particular, the conventional +motor is replaced with an electric one, and a battery pack re- +places the fuel tank. Notice that all the components mentioned +above need to share messages inside and outside the vehicle +to guarantee the correct functioning. An attacker might exploit +some of these messages to create inconsistencies on the EV +status or to cause damages to both the vehicle and driver. We +provide a detailed security analysis based on these components +in Section IV. +C. Electric Vehicles Charging Infrastructure +The EV needs to charge its battery periodically to provide +power to its components. To this aim, the EV shall be +connected to a charging infrastructure with whom it negotiates +a charging session. According to the negotiated session, the +infrastructure then delivers the needed energy to the EV. +Charging may happen either in public areas (e.g., shopping +malls or offices) or at a private site (e.g., home). To prevent +possible malfunctioning, the charging infrastructure must be +carefully managed. This is particularly true when considering +a scenario where handling a massive number of EVs may lead +to blackout and other grid malfunctions [32]. +Charging an EV differs from other devices, such as smart- +phones or laptops, as it requires dedicated hardware and a +drastically larger energy supply. Indeed, if many EVs are +concurrently charging, there can be grid overloading, leading +to malfunctions and local blackouts. To avoid these issues, the +grid must employ a communication channel with the EV to +negotiate to charge parameters that respect the vehicle’s battery +requirements without overloading the grid. V2G refers to the +technology enabling this communication type. There are two +solutions to manage a charging session: wired and wireless. +While the former is more diffused and widely implemented +nowadays, the latter is still in the initial stage and under +development. Unfortunately, there is no unique world standard +to regulate this communication channel. Instead, different +manufacturers implement different standards based on the +technologies used for the charging process. For instance, +CHAdeMO [33] (Japan) or GB/T [34] (China) can be used +only with wired charging, while ISO 15118 (Europe, North +America) also supports WPT [35], [36]. +1) Wired Charging: With this setting, the EV is connected +to an EVSE through a cable that transmits both the control +signals ad the charging current. In turn, EVSEs negotiate with +power grids for the energy needed to charge the vehicle, +based on both EV and grid requirements. However, these +basic functions are integrated by every charging standard, +which employs different communication methods. Low current +charging levels, such as AC Level 1 or AC Level 2, require a +simple control channel which is generally provided by a Pulse- +Width Modulation (PWM) communication. More advanced +charging, such as DC charging, needs better management of +the energy provided by a High-Level Communication (HLC) +provided by protocols such as CAN or Power Line Commu- +nication (PLC). These technologies enable the development +of additional services, such as the automatizing of the billing +process [37], [38], or the download of firmware updates [39]. +In case of a lack of automated authentication solutions, EVSE + +5 +may be equipped with RFID readers through which users can +authenticate and pay for the service. EVSEs can be deployed +at private or public premises: private charging columns are +generally less advanced and support less charging level with +respect to public EVSE. +There are mainly two protocols supporting the HLCs be- +tween EV and EVSE during DC charging sessions. The first +one, employed by Combined Charging System (CCS), is the +ISO 15118 [38], which modulates data over the control pilot +pin using PLC. The second one, CHAdeMO [33] employs a +CAN channel for the communication. +The physical connection between EV and EVSE may be +implemented with different plugs according to different stan- +dards. In particular, we can classify EVSEs according to +different levels. Figure 2 shows the different charger levels +together with their lead characterization. +(a) Level 2 +(b) Level 3 +Fig. 2. Different types of EV chargers. L1 = AC line 1; N = AC line neutral, +P1 and P2 = proximity lines, PE = ground. +Level 1 and Level 2 EVSEs exploit a five-leads connector +implementing the SAE J1772 protocol [40], as shown in +Figure 2(a). This connector exploits two leads to deliver the +charging current, two leads for pilot signals, and one lead +for ground (or protective earth). The two current leads plus +the ground one are used by the EVSE for metering and +computing the session cost. The two pilot lines have two +different functions. The first one, the control pilot, is used +to exchange information with the EV during the charging +session. The signals exchanged through the control pilot either +control the amount of current delivered to the EV [41] or +are used to check the connection status and remove power +from the adapter in case of disconnection to prevent the user +injuries [9]. The second pilot line is the proximity pilot, used +by the EV to check whether a proper physical connection has +been established with the EVSE. +Level 3 EVSEs, i.e., those allowing for fast charging, +are based on different implementations and are showed in +Figure 2(b). The first is the CCS expansion of the SAE J1772, +which allows for direct current exchange for fast charging. +Furthermore, it implements PLC to exchange information +between the EV, the EVSE, and the smart grid [40]. The +second implementation is the Japanese CHAdeMO [42], which +implements a fast charging protocol. Besides delivering power, +this implementation allows for data exchange via the CAN bus +protocol. Thanks to this type of connection, it is possible to +avoid applying power to the connector in case of a non-safe +connection or to exchange information related to the battery +SoC. Furthermore, CHAdeMO allows V2G communication, +where the EV battery is later used as energy storage to +provide service to the grid [33]. Other protocols exist, such as +the proprietary protocol employed by Tesla vehicles and the +Chinese GB/T, which will probably be replaced by Chaoji, an +evolution of CHAdeMO [10]. +The main differences between Level 3 and Level 2 chargers +lie in the higher number of leads in Level 3, and in the +implemented circuitry which converts AC to DC, which is +inside the charging columns for Level 3, while it is onboard +in the EV for Level 2. Furthermore, Level 3 charging includes +richer communication capabilities thanks to the support of +HLC. +2) Wireless Power Transfer: Charging via WPT allows +charging an EV’s battery without physically connecting the ve- +hicle to the charging infrastructure. In WPT, a source (powered +by the grid) generates a time-varying electromagnetic field that +triggers the generation of a current at the receiver’s (EV’s) +side. This current is generated thanks to a coil mounted on the +EV’s side that receives the transmitted electromagnetic field +and, due to Faraday’s law of induction, generates an AC [43]. +Via WPT, it is possible to create multiple charging scenarios +depending on the mobility of the EV [44]. In fact, thanks to the +absence of a physical connection, EVs can be either charged +while parked or while driving in a dynamic scenario [20]. +The static scenario is similar to the one previously described +in Section III-C1, where a user books a charging session +and receives the power from the grid while parked at a +charging facility. Instead, the dynamic case requires a suitably +designed infrastructure composed of multiple sequential WPT +transmitters. Figure 3 shows a pictorial representation of a +dynamic WPT system for EVs. The street is equipped with +multiple WPT transmitters deployed underneath the street. +These transmitters are connected to the grid that provides the +power needed to charge the EV. +Fig. 3. Representation of a WPT system for EVs. +Dynamic WPT can be further divided into two categories: +quasi-dynamic and fully-dynamic [45]. In the former case, + +L1 +N +P1 +P2 +PEPE +L1 +N +P1 +P2 +PE +DC+ +DC- +X +DC+ +DC-6 +charging is limited to the cases where the EV is not moving, +e.g., while waiting at stops or traffic lights. In fully-dynamic +WPT, charging is continuously delivered to the EV as long as +it drives near transmitting coils. In both dynamic scenarios, the +challenge is to guarantee that transmitters are activated only +when needed to avoid energy waste and that only legitimate +users access the emitted power. In fact, due to the absence of +a medium, users could steal power by driving close to an EV +that paid for the charging session. We discuss all the security +problems related to WPT in Section IV. +As specified in the ISO 15118 standard [35], the connection +between the vehicle and the charging column during a static +WPT scenario uses WiFi (IEEE 802.11). The vehicle can +connect before being correctly parked or when it is already +over the coil. If needed, the EVSE provides the EV with +fine positioning messages to help the driver correctly place +the vehicle to reduce energy dispersion. After establishing +the connection, the two entities communicate similarly to +wired cases. Some modifications are introduced to adapt to +the wireless scenario, including the WPT charging mode and +the fine positioning messages. +Due to their novelty, dynamic and quasi-dynamic charging +are not yet covered by approved and widely adopted standards. +Some research works [46] adopt Dedicated Short-Range Com- +munications (DSRC) to create a channel between vehicles and +the Road Side Units (RSUs), which are in charge of controlling +a portion of the road coils. +Another possible solution can be to extend WPT to deliver +also information along with power. In fact, Wireless Infor- +mation and Power Transfer (WIPT) represents a technology +that might be exploited for electric vehicle [47]. WIPT can +be adopted to implement a system similar to that exploited +in wired EV charging, where control signals are exchanged +through the pilot line and the charging current. In WIPT, +control signals can be coded into the time-varying electromag- +netic field to deliver power and check the connection’s status +simultaneously. Furthermore, this solution can be exploited to +authenticate EVs and solve part of the security challenges +in WPT. Although not yet discussed in the literature, we +believe that WIPT represents a suitable line of research for +EV charging technologies. +IV. IN-VEHICLE SECURITY AND PRIVACY CHALLENGES +This section discusses the security and privacy challenges +related to the components and protocols used inside EVs. First +discuss in Section IV-A the challenges related to the battery +and the BMS. Then, we discuss the challenges related to the +controller and charger in Section IV-B. The security of CAN +bus has been extensively studied in the literature, as it does +not envision secure by design solutions [48]. However, how +these attacks may impact EVs has never been studied. Since +all in-vehicle messages are exchanged through CAN and LIN +buses, we discuss how their vulnerabilities can be exploited to +impact those components specific to the EVs. We summarize +in Table IV the in-vehicle security and privacy challenges +together with their effects, impact severeness, and possible +countermeasures. +A. Battery and BMS +The battery pack is a sensitive component of an EV. In +case of malfunctions, it may catch fire and even explode [49], +[50]. Such situations can severely harm the passengers and +create financial damage to the owner and a reputation loss +to the manufacturer. Less severe cyberattacks can, however, +create financial damage, for instance, by reducing the battery’s +lifespan, forcing the owner to a premature battery replace- +ment [17]. +The battery pack is managed by the BMS, which handles +communication with the other ECUs via the vehicle bus. +Again, this channel has been proven to be vulnerable to +many cyberattacks [48], [24]. In the following, we discuss +how cyberattacks impact EVs, extend their effects to the CPS +domain, and highlight their effect on the battery and BMS. +a) Denial of Service: The BMS is responsible for re- +porting information on the battery status and managing the +energy delivery. An attacker might flood the BMS controller +by forging and sending a vast number of requests, in similar +ways to what may happen with Denial of Service (DoS) attacks +against websites [51]. An overload of the BMS may slow +responses to legitimate requests or even prevent the BMS +from sending response messages completely. This may lead +to multiple effects depending on the information requested +to the BMS and how the requester device reacts to the +absence of a response. In fact, this might cause damage +to the battery if power is not properly removed in case of +abnormal behavior or physical tampering by the attacker. A +DoS may target sensor measurements, such as temperature, +and it may prevent the activation of cooling mechanisms, +forcing the battery into critical temperatures, which may be +irreversible [52]. Furthermore, this attack may also prevent the +user from obtaining information on the amount of charge left, +causing range anxiety and possibly jeopardizing the drivers’ +safety in case of a sudden EV stop. +Flow control might prevent the BMS from handling many +fake requests. In this context, source authentication may pro- +vide information regarding the legitimacy of the sender [53]. A +solution for flow control may be given by an adapted version +of time-lock puzzles [54]. Furthermore, rate limiting can help +mitigate against DoS attacks [55], while intrusion detection +strategies can help in identifying the attack before it creates +damage [56]. Redundancy on the controllers can also help in +mitigating severe DoS attacks against the BMS [57]. +b) Tampering: An attacker might physically tamper with +the battery and the BMS. Depending on the specific tampered +component, an attacker may be able to cause a short circuit +that may lead to catastrophic events such as the start of +a fire that might harm both the vehicle and the passenger. +This consideration holds for battery and BMS, as they both +manage high voltages. Tampering may also lead to less severe +consequences, such as the BMS being unable to communicate +with the battery or to deliver the full power to the battery +during charging. These attacks may also include detaching or +cutting cables. +As a possible countermeasure, the battery and BMS shall +include an anomaly detection system to prevent applying a + +7 +TABLE I +SUMMARY OF IN-VEHICLE CHALLENGES. +Component +Attack Type +Effect +Impact +Possible Solutions +DoS +Prevent energy delivery +Prevent information reception +Increase energy consumption +Physically damage the battery +Medium +Flow control +Time-lock puzzles +Rate limiting +Intrusion detection +Tampering +Short circuit +Prevent energy delivery +High +Anomaly detection +Tamper-proof hardware +Malicious Code Injection +Modify BMS response to command +Collect sensitive information +Medium +Authentication +Remote attestation +Intrusion detection +Battery +and +BMS +Spoofing, Replaying, and MitM +Report false information to the driver +Report false information to the other ECUs +Physically damage the battery +Disrupt charging process +Excessive discharging +Overcharging +High +Identity management +Authentication +Intrusion detection +Redundancy +Timestamps +Integrity protection +MitM +Report false information +Isolate charger components +Modify control signals +Increase energy consumption +High +Anomaly detection +Intrusion detection +Intrusion prevention +Integrity Protection +DoS +Prevent the exchange of energy +Medium +Cookies +Time-lock puzzles +Rate limiting +Intrusion detection +Spoofing, and Replaying +Report false information +Physical damage +Increase energy consumption +High +Intrusion detection +Identity management +Timestamps +Malicious Code Injection +Modify EV response to commands +Collect sensitive information +Remote control/hijack +High +Authentication +Remote attestation +Intrusion detection +Tampering +Impair the charging process +Power loss and overvoltage +High +Anomaly detection +Tamper-proof hardware +Controller +and +Charger +Eavesdropping, and Side Channels +Track the user +Profile users’ preferences +Low +Differential privacy +Encryption +voltage to tampered components and causing the aforemen- +tioned damages [58]. Another solution may be the physical +protection of these components with tamper-proof hardware. +For instance, in case of physical tampering, the battery should +be designed so that it cannot receive or deliver power [59]. +The SAE J2464 standard contains safety measures that can +also be effective against tampering [60]. +c) Malicious Code Injection: The battery pack is man- +aged by the BMS, which is a piece of hardware with firmware +onboard. Attackers may try to reverse engineer the software +to discover vulnerabilities and build exploit against them [61]. +To patch bugs, EVs’ software may be updated over the air or +via the charging cable [39], thus easing the update process for +manufacturers and users. However, this represents a security +challenge, as software updates need to access the overall +EV network [29]. A malicious user may inject malware via +software update to gain control of the BMS [29]. By having +partial or complete control over it, the attacker may thus +impact the normal functioning of the vehicle. For instance, +the malware may prevent the BMS from requesting energy +from the battery, causing a blackout in the EV. Contrarily, the +BMS may be forced into requesting more energy than needed +to speed up the discharging process. Furthermore, thanks to the +malware, the attacker may measure other sensitive information +of the driver, which may lead to privacy leakages. +To prevent code injection and its effects, access to the +EV’s internal network shall be strictly regulated. Possible +solutions include the use of external source authentication. +In case of a successful injection, it is fundamental to iden- +tify and mitigate its effect. To this aim, remote attestation +and its collective extension may be used to validate the in- +vehicle components [62]. Furthermore, anomaly and intrusion +detection techniques may help identify attacks to the in- +vehicle network [56]. Injection of malicious updates can be +detected by integrity verification on the new software, possibly +employing a blockchain [63]. +d) Spoofing, Replaying, and Man-in-the-Middle: An at- +tacker may spoof or modify messages to report to the driver + +8 +false information on the battery SoC, thus impairing a safe +drive. The attacker may also report incorrect information to +the charging infrastructure by impersonating the BMS or +modifying information in the middle of the communication. +This may cause the charging process to provoke damage +to the battery or the EV circuitry. Furthermore, an attacker +may report false information to prevent the correct exchange +of energy from the battery to the BMS, for instance, by +lowering the current demand and preventing the exchange +of a sufficient amount of power from the battery to the EV. +By requiring excessive power, an attacker can discharge the +battery faster than expected in a battery exhaustion attack [64] +or may force the battery to overcharge, leading to a massive +shortening of the battery lifetime [17]. Finally, through Man- +in-the-Middle (MitM), an attacker may modify the voltage +values of the battery pack, leading to over-discharging and +consequent battery degradation [65]. +To prevent these attacks, the battery and BMS should be +given an identity, and all messages shall provide source authen- +tication and integrity protection. The cryptographic material +shall be embedded in these devices, with examples in trusted +platform modules [66] or physical unclonable functions [67], +and shall not be disclosed during communications to prevent +MitM attacks. An intrusion detection system can help in +identifying ongoing attack [64]. Redundant controllers can be +employed to enhance the resilience of the BMS against adver- +sarial attacks during charging [57]. Kim et al. [68] proposed +to employ blockchain to provide authentication and access +control in the communication between the BMS and the other +devices inside the EV. Strategies to mitigate the effect of an +attacker who has gained direct access to the vehicle’s bus have +been proposed [48]. Some works have considered peculiar +features of EV to detect spoofing attacks: Guo et al. [69] +proposed a physically-guided machine learning method to +detect replay and false data injection attacks on the bus. Their +system, tested in a Hardware-In-the-Loop (HIL) simulation +testbed, could identify the attacks with an accuracy of more +than 98%. Finally, to prevent replaying attacks, designers can +consider the addition of timestamps to packets and signals +transmitted [70]. +B. Controller, Charger, and Electric Motors +The controller and charger are fundamental elements that +communicate with the BMS to exchange power to recharge +the battery and to feed the EV components with energy. +The charger communicates with the EVSE to negotiate the +parameters of the charge. Moreover, it manages the energy +received and forward it to the battery pack according to the +BMS requirements. If bidirectional charging is available to +EV, the charger may also deliver energy from the EV battery +to the charging column upon request [71]. The controller +manages the energy delivered from the battery to the other +components. Some of these components are powered by the +battery also in petrol-based vehicles, such as the infotainment +system or the lights. Others, such as the electric motors, are +instead specific to EV. The controller sends energy to them +following the driver’s input, such as the torque pedal pressure. +This section discusses how an attacker may impair their correct +functioning. +a) Man-in-the-Middle: Modifying the data in the bus +may disrupt the regular operation of the charger since the +control signals are usually transmitted through this channel. +An attacker may isolate certain charger components (e.g., the +load relay), leading to a surge in the DC voltage. These attacks +can damage the battery causing degradation in the performance +and shortening the lifetime of the battery [72]. An attacker +may also modify the signals managing the electric motors by +adding noise or other mutations to the original signal. This +attack can damage the correct functionality of the motors and +put the driver in dangerous situations [73]. +Mitigation techniques can be applied using algorithms that +can detect the attack in almost real-time by monitoring the +physical properties of the vehicle, such as sensor data [13], +[72]. To make the receiver aware of possible MitM attacks +targeting certain packets, integrity protection mechanisms must +be in place. Furthermore, intrusion detection and prevention +systems that monitor the data exchanged on the bus can also +be implemented to strengthen the defense mechanism [64], +[74]. +b) Denial of Service: The operations handled by the +charger and controller heavily rely on the sensors reporting +information on the charging status. A malicious user can +generate a large number of requests to the sensors reporting +data or overload the charger and control modules by flooding +them with packets, thus preventing the receipt of legitimate +messages. If not properly handled, this attack may cause the +controller to stop receiving correct state information, impairing +the overall state control. +Flow control may prevent controllers and motors from +handling a large number of fake requests similarly to the +BMS, for instance, by employing an adapted version of time- +lock puzzles [54]. Source authentication might be employed +to verify the sender’s lawfulness [53], while rate limiting can +help mitigate against DoS attacks [55]. Furthermore, intrusion +detection can be adopted to identify ongoing DoS attacks [56]. +c) Spoofing, and Replaying: +An attacker may spoof +sensor identities to create multiple packets with legitimate +identifications. By exploiting the same concept, an attacker +may also report false information to the charger and con- +troller. Therefore, the controller may take actions based on +false data. This may cause damage to the hardware, possi- +bly impairing the whole charging system [13]. False data, +if correctly crafted, may also impact the electric motors’ +functionality. For instance, they could force a stop of the +motors by sending false control signals. Encryption can be +a countermeasure to spoofing, preventing a malicious user +from freely creating new packets. However, replay attacks +can be employed to send correct sensor measurements or +actuator updates previously recorded from the bus. An attacker +may also spoof the information from the infotainment system, +acceleration pedal, or other energy-hungry devices in the EV. +The malicious entity may demand a power amount higher than +the truly needed one, thus causing higher energy consumption +and shortening the battery’s lifespan causing the driver to +charge the EV frequently. The controller also handles the + +9 +information regarding acceleration and breaking. An attacker +may spoof the related sensors to report false state changes to +the electricity supplier. For instance, an attacker may spoof +the gas pedal and prevent the receipt of the amount of power +needed by the driver to speed up. This may cause safety issues, +for instance, when the driver needs to surpass another vehicle. +As already depicted, encryption can only prevent certain +kinds of spoofing attacks, but it is insufficient to mitigate +replay attacks. To prevent the latter, a combination of unique +identifiers and timestamps can be adopted [70]. Identity man- +agement may be another fundamental countermeasure against +these threats [53]. In fact, the controller and charger need +to have, by design, access to the identities of all legitimate +components. The identification of attacks is possible using +intrusion and anomaly detection techniques [56]. +d) Malicious Code Injection: Similarly to the BMS case, +controllers, chargers, and motors also contain software, which +may often require updates. However, software updates repre- +sent a security challenge since it needs access to the overall EV +network [29]. A malicious user may force the installation of a +malicious software update to gain control of some components +of the EV’s internal network [29]. The attacker may thus +impact the safety of the driver. For instance, the malware may +cause the EV to respond to the driver commands oppositely +(e.g., decelerating while pushing on the gas pedal) and may +also propagate to all the EV’s components. Furthermore, +thanks to the malware, the attacker may measure sensitive +information on the driver (e.g., location) or profile the driver. +A further threat is due to the implementation of controllers +and ECUs via Field Programmable Gate Array (FPGA). In +this case, an attacker may be able to inject malicious software +into the central management system via communication lines +by manipulating the FPGA controller [13]. +Countermeasures are similar to the BMS case. The access +to the EV’s internal network shall be strictly regulated, for +instance, using strong authentication mechanisms. Remote +attestation and its collective extension may be used to validate +the in-vehicle components [62], while blockchain can have a +role in the verification of new software [63]. Finally, intrusion +detection techniques may help the identification of ongoing +attacks [56]. +e) Tampering: An attacker might physically tamper one +of the controllers, charger, or motor components. In this case, +for instance, the attacker may prevent the charger from cor- +rectly detecting the presence of a power source (either wired +or wireless), thus impairing the possibility of charging the +vehicle. This is the case for proximity sensors. For instance, +the attacker may attach a shield to the pin on the EV side +such that it cannot correctly communicate with the proximity +pilot line. Furthermore, an attacker may tamper with the +power converter to degrade its quality, causing power losses +or overvoltage. +To prevent physical tampering, controllers, chargers, and +motors may implement anomaly detection frameworks to +detect the application of a voltage in non-safe situations and +react to the attack [58]. Furthermore, these devices can be +designed as tamper-proof, so they will stop functioning in +case of tampering [59]. Although this may impair the vehicle’s +functioning, it allows for safeguarding the user’s safety. +f) Eavesdropping, and Side Channels: The current ex- +changed during the charging process leaks features that can be +exploited for user tracking and profiling [75], [76]. An attacker +may attach a module to the charger and controller to collect +the current exchanged during the charging process and extract +those features, thanks to the absence of encryption methods. +For the same reason, an attacker may also eavesdrop on the +information exchanged between the controller, the charger, +the motors, and the BMS to launch the attacks mentioned +above. An adversary may also analyze the power exchanged +between the controller and the infotainment system to obtain +users’ sensitive information, such as preferences, habits, and +passwords. This attack has been shown in other scenarios [77], +such as smartphone charging, where users’ activities can also +be detected in case of encrypted traffic [78]. Therefore, it is +fundamental to include methods to prevent malicious users +from accessing the communication among these entities. +To guarantee the user’s privacy, the current exchanged may +be altered via a noisy signal that hides the original signal’s +features similarly to differential privacy in other contexts [79]. +On the receiver side, the components shall be able to guarantee +that the input current does not cause any damage to the +circuitry. These solutions may hold for all the involved sources +of current. Cryptographic methods may not always represent +a viable solution, as they would add computational overhead +to a possibly safety-critical system. Furthermore, they do not +represent a solution to side channels, which are challenging to +mitigate [80]. +V. EV CHARGING SECURITY AND PRIVACY CHALLENGES +This section discusses the security and privacy issues related +to the EV charging process. In particular, we discuss the chal- +lenges associated with wired charging in Section V-A. Then, +we discuss the challenges associated with WPT and WIPT in +Section V-B. For both technologies, we also discuss possible +solutions and countermeasures. In Table V, we summarize +all the security and privacy challenges associated with the +charging process, their effects, impact severeness, and possible +countermeasures. +A. Wired Charging Challenges +In the following, we focus on attacks targeting a wired +charging scenario, which is the most common way at the +moment of writing. Some of the attacks are specific to +cases where HLC is available (e.g., MitM, spoofing), while +others are suitable for every type of wired charging, such as +tampering or side channel analysis. +a) Tampering Attacks: In this attack, a malicious user +physically tampers with the devices involved in the charging +process. In particular, an attacker might manipulate the pilot +lines and tamper with the proximity sensor to prevent an EV +from deeming a secure connection and hence prevent charging. +Furthermore, this can also impact users’ safety, as it might +be possible to detach the cable before removing the current. +By observing electromagnetic leaks or operations in the chip +components both in the EVSE and EV, an attacker might infer + +10 +TABLE II +SUMMARY OF EV CHARGING CHALLENGES. +System +Attack +Effect +Impact +Possible Solutions +Tampering +Prevent charging +Cause a shock to the driver +Get sensitive information +High +Tamper-proof hardware +Inconsistencies handler +Energy repudiation +Cheat on billing +Steal energy from the system +Low +Aggregate signature schemes +Blockchain for energy transactions +DoS +Prevent EV charging +Disruption of the charging service +Medium +Identity verification +Authentication +Intrusion detection +MitM +Prevent proper charging +Modify charging parameters +High +Integrity protection +Encryption +Spoofing, and Replaying +Create charging state inconsistencies +Steal energy from another EV +Medium +Identity management +Authentication +Encryption +Timestamps +Relaying +Steal energy from another EV +Medium +Distance bounding +Fingerprinting +Eavesdropping +Steal sensitive EV information +Medium +Encryption +Wired Charging +Side Channels, and Information Leaking +Track user +Profile user’s preferences +Low +Differential privacy +Secondary batteries +Overpower +Damage to EV battery +High +Energy-efficient overvoltage protection +Anomaly detection +Jamming, and DoS +Prevent EV charging +Medium +Channel hopping +Identity verification +Authentication +Intrusion detection +Freeride attack +Steal energy from the system +Low +Authentication +Blockchain +Energy repudiation +Cheat on billing +Steal energy from the system +Low +Aggregate signature schemes +Blockchain for energy transactions +Spoofing, and Replaying +Create charging state inconsistencies +Steal energy from another EV +Medium +Authentication +Encryption +Physical layer authentication +Timestamps +Relaying +Steal energy from another EV +High +Distance bounding +MitM +Prevent proper charging +Modify charging parameters +Medium +Integrity protection +Encryption +Physical layer authentication +Eavesdropping +Steal sensitive EV information +Medium +Encryption +WPT +Side Channels, and Information Leaking +Track user +Profile users’ preferences +Medium +Differential privacy +Secondary batteries +sensitive information on the user, such as private keys used +for billing purposes [9]. Furthermore, by tampering with the +charging cable, an attacker might prevent the proper charging +of the victim EV or steal energy from an EV in charge by +connecting additional cables [81], [82]. +As possible countermeasures, tamper-proof hardware may +represent a viable solution [59], [83]. Thanks to these devices, +the attack may be limited to the car functioning without im- +pacting the users’ safety. Lastly, proper inconsistency handling +mechanisms may be implemented to check that all involved +components report the same physical status. +b) Energy Charging Repudiation: A malicious user may +report to the EVSE that the EV’s battery did not receive any +power by exploiting the behavior of the pilot line and the +feedback associated with it. In this situation, the attacker may +be able to charge a smaller amount compared to the amount of +energy effectively used. If bidirectional charging is available, +an attacker may pretend to have sold more energy than it has +actually sold, thus stealing money from the energy provider. +A possible countermeasure to energy repudiation is the use +blockchain technology to handle transactions and guarantee +traceability and non-repudiation [84], [85]. Aggregate signa- +ture schemes from different physical components can represent +another possible mitigation to the problem [86]. +c) Denial of Charging: A malicious actor may try to +prevent a vehicle from charging. It may be done at the data +level by modifying values on the packets exchanged during the +handshake between the EV and EVSE [87]. In some cases, +DoS can also be performed remotely, exploiting unshielded +cables, which are often used for the recharge [88]. DoS may +also be launched against more than one vehicle, trying to +compromise a portion of the grid. A greedy attacker may +falsify the information on the battery’s SoC, such that s/he can +demand an energy amount higher than needed, thus preventing + +11 +other users from benefiting from the service. The number +of users that can simultaneously charge their EVs and the +energy effectively delivered each moment depends on the +grid’s capacity. If the grid capacity is limited, the attacker can +successfully launch this attack and prevent other users from +charging. +Possible countermeasures to DoS attacks include low- +complexity authentication services in all the packets ex- +changed such that the EVSE can rapidly decide whether to +accept or discard a request. Identity-based traffic filtering +may be combined with a physical state update related to the +charge level of a certain user to prevent multiple malicious +requests. Intrusion detection can be employed to detect ongo- +ing DoS attacks which may generate strange communication +patterns [56]. Furthermore, enforcing physical security by +adopting shielded cables can prevent some kinds of DoS and +eavesdropping attacks [88]. +d) Man-in-the-Middle: When operating charging modes +employing HLCs, such as CHAdeMo or ISO 15118, the +EV and EVSE exchange data through network packets. A +MitM attack can be employed to modify the content of this +communication. It may be a consequence of tampering if the +malicious actor can insert a device on the pilot line between +the vehicle and the charging column. In some cases, MitM +can be performed from other charging columns attacking the +SECC Discovery Protocol [89]. A malicious actor may exploit +this channel to manipulate the exchanged information and +create inconsistencies in the recharging process. For instance, +an attacker who can modify packets on the fly may prevent +proper charging by modifying request and response parame- +ters. Further attacks can be launched starting from MitM, such +as malware injections or DoS [87]. +To identify modified data, integrity protection can be added +to packets [90]. Another possible countermeasure is encryp- +tion. Novel versions or ISO 15118 mandates the usage of +Transport Layer Security (TLS) for all the communications +between the vehicle and charging column, even if in real life, +data are often exchanged in plaintext [81]. +e) Spoofing, and Replaying: An attacker might interact in +the communication link between the vehicle and the charging +column by injecting packets spoofing other devices’ identities. +For instance, a malicious user can spoof the identity of an ECU +and report false information on the battery SoC. Furthermore, +an attacker may inject false information by spoofing the +identity of an EVSE and stealing sensitive information from +an EV. For example, in the case of automatic billing based +on the EV features, a malicious user can extrapolate those +features from an EV and store them for later use to bill the +victim. The same concept can also be applied to other types +of connectors, as long as billing is based on automatic feature +recognition [37]. +Possible countermeasures to these attacks include using a +proper identity management scheme, authentication, and data +encryption [91]. Authentication systems shall include infor- +mation related to the charging status of the EV or the energy +delivered by the EVSE to help guarantee the consistency +between the reported information and the actual physical state. +It is important to consider that encryption cannot prevent the +replaying of packets, which may instead be enforced with +unique identifiers and timestamps [70]. +f) Relaying: A relay attack is possible if an attacker has +access to the network traffic and can relay it to a nearby +charging column. By relaying information, a malicious user +can manipulate the billing system. For instance, a malicious +user can relay the data between two neighboring EVSEs to +bill a closely-located victim user for a charging session [82]. +If bidirectional charging is available, a malicious user can sell +the energy of a victim’s vehicle and get paid for it. +The location information of EV and EVSE may be exploited +to prevent relay attacks, e.g., employing distance bounding +protocols [92], [82]. Furthermore, the physical features of +the EV may be exploited to design dedicated authentication +protocols [93], [94]. +g) Eavesdropping: An attacker may be able to read +the information exchanged between the vehicle and charging +columns in different ways, similarly to what was presented +before for MitM attacks. With access to all the network traffic, +a malicious entity can steal sensitive information from the user, +from simple charging parameters to credit card numbers. +To protect against eavesdropping, encryption can be applied. +As already explained, novel versions of ISO 15118 mandate +the usage of TLS for all the communications between the +vehicle and the charging column, even if real-life data are still +often exchanged in plaintext [81]. It is important to recall that +even if the exchanged data are encrypted, side channel analysis +is possible to extract some users’ preferences, as presented in +the following section. +h) Side Channels, and Information Leakages: An at- +tacker in control of an EVSE may be able to track and +profile users who authenticate to the EVSE even if data are +encrypted. It may rely on different information, such as the +MAC address of the EV or the certificate employed by Plug +and Charge [38]. However, in Level 1 and Level 2 charging, +these kinds of data are unavailable since no HLC is generated +between the two entities. In that case, an attacker may rely on +other features, such as the exact voltage of the control pilot +pin or the duration of the handshake at the beginning of the +charging process [94]. Another side channel that may transfer +information is the effective current exchange. This does not +convey information in a network sense, i.e., it does not +involve the creation of packets with the sender’s and receiver’s +information. Therefore, no encryption method is applied to this +signal, which is transmitted in plaintext. However, it has been +shown that it is possible to profile users by extracting features +from the charging current [75], [76]. In particular, the charging +current contains features peculiar to each EV, allowing for +EV tracking and user profiling based on the current demand. +Therefore, it is fundamental to manipulate the current signal +to prevent these attacks. +Countermeasures to privacy threats shall not undermine +the efficiency of the charging process. Therefore, possible +solutions must allow the involved parties to retrieve sufficient +information, e.g., to the SoC. Differential privacy methods may +represent a viable solution [95]. An alternative is represented +by the use of secondary batteries to create a connection +between the EV and EVSE, similarly to what was discussed + +12 +in [75], [76]. When HLC is available, MAC address random- +ization may represent a good mitigation technique to reduce +the profiling power of an attacker. +B. WPT Challenges +Due to the exposure of the wireless medium, WPT incurs in +a large number of safety, security, and privacy issues. In fact, +it is likely that WPT signals to impact more vehicles and that +an attacker gets access to the signals or information wirelessly +exchanged [96]. In this section, we review and extend the +taxonomy of the possible attacks to WPT presented in [96] +and adapted it to the EV case. +a) Overpower attack: The wireless medium’s intrinsic +vulnerability makes it possible that a single EV receives +both its signal and the signal intended for another vehicle. +For instance, if two cars are closely located, and both are +charging their batteries via WPT, they will receive more power +than expected. This is even more likely when considering +fully-dynamic WPT, where vehicles move and cannot hence +guarantee that a reasonable safety space is kept between them. +The excessive received power might harm some components +of the BMS or the battery if a proper overvoltage regulator +is not deployed. Furthermore, an attacker might exploit this +concept to launch an overvoltage attack to damage the EV’s +components. +Possible countermeasures include implementing overvoltage +protection mechanisms at the EV’s side. Such mechanisms +shall, however, guarantee the efficiency of the charging process +to avoid requiring excessive charging times. Anomaly detec- +tion methods can also be applied to detect the reception of +abnormal power values or other anomalies in the charging pro- +cess. Design choices can help mitigate overpower attacks. For +instance, the distance between coils must be designed to make +overpower attack unfeasible or, at least, more complicated. +b) Jamming, and Denial of Service: In the case of +WIPT, the reception of multiple signals might cause exces- +sive interference at the receiver’s side, thus preventing the +correct reception of messages. Due to the openness of the +WPT medium, an attacker might be able to simultaneously +jam multiple EVs by sending random WIPT messages and +degrading the channel quality up to the point where messages +are not correctly received. Furthermore, this concept can be +exploited to prevent a successful charging negotiation phase, +thus preventing a connection between the EV and the charging +system. This represents a DoS attack. Similarly, an attacker +may launch a jamming attack against the charging column’s +WiFi access point, preventing legitimate users from connecting +and using the service. An attacker may also target a portion +of the energy grid by continuously sending charging requests. +If many users engage in this session, they might prevent other +users from benefiting from the service availability. Although +feedback mechanisms to report on the SoC of the receiver +might be implemented to automatically detach an EV when +fully charged, an expert attacker might be able to craft +feedback packets to avoid showing full battery’s SoC. +Possible solutions include frequency hopping mechanisms, +where channels are selected according to different strate- +gies to avoid using a channel under jamming attack [97]. +Low-complexity authentication services in all the packets +exchanged such that the EVSE can rapidly decide whether +to accept or discard a request can help in preventing DoS +attacks. To detect a DoS attack, intrusion detection systems +can be deployed [56]. Identity-based traffic filtering may be +combined with a physical state update related to the charge +level of a specific user to prevent multiple malicious requests. +c) Freeriding attack: As previously mentioned, a user +might connect to public infrastructure and pay for charging via +WPT. Due to the openness of the WPT medium, a malicious +user could exploit the proximity to a vehicle in charge to steal +energy and charge his/her EV. A similar scenario envisions the +collusion of multiple EV owners when a single one registers +for the service and multiple users share the bill and benefit +from the charging process. These attacks are feasible in all +types of dynamic WPT models; the only requirement is a short +inter-EV distance. This attack is challenging to detect, as it +does not impact the legitimate channel. In fact, although a +second EV might be connected to the charging channel, the +main channel will not face any performance degradation, thus +making it unfeasible to detect the attack. +WPT sessions need to be authenticated to prevent other +users from benefiting from a charging session they are not pay- +ing for. Furthermore, authentication procedures might include +the physical features of the involved devices and the amount +of power transferred. The blockchain solutions proposed by +Jiang et al. [98] may be adapted to the EV case to guarantee +security against this attack. +d) Energy repudiation: WPT is less efficient compared to +its wired counterpart, as the wireless medium is characterized +by losses due to both attenuation and the relative position of +the transmitter and receiver devices. Therefore, part of the +transmitted energy may be lost during the charging process. +A fair system requires that users pay for the actually received +energy. Therefore the billing system needs to compare the +transmitted power with the received one. However, this might +create security issues. In fact, a malicious user might continu- +ously report a received power value smaller than the true one +or report zero received energy. This is commonly known as +a repudiation attack, where the user denies benefiting from a +service. +To guarantee the correctness of the reported power us- +age information, possible solutions might include the use of +aggregate signature schemes from different physical compo- +nents [86] or the blockchain technology [85], [98]. +e) Spoofing, and Replaying: +A malicious user who +knows the standard employed or which is able to eavesdrop on +the communication can easily craft malicious packets. Based +on the crafted information, this class of attacks may have +different impacts on the system. For instance, an attacker may +use the identifier of another vehicle to negotiate a charging +session that the victim will pay for. Furthermore, a malicious +actor can craft packets declaring weird SoC and spoof other +vehicles’ identifiers to create inconsistencies in the charging +process. +The use of authentication and integrity protection mecha- +nisms can be effective countermeasures against spoofing. In +this context, using physical layer authentication may help in + +13 +designing suitable protocols [99]. In the context of WPT, the +transmission frequency can be regulated to encrypt information +and guarantee that only the legitimate party can receive +power [100]. This also represents a possible solution to the +attacks aforementioned in this section. Finally, timestamps can +be added to identify multiple sending of the same packet in a +replaying attack [70]. +f) Eavesdropping: Due to the exposure of the wireless +medium, an attacker may easily intercept WPT packets. These +packets may contain different types of information, such as the +vehicle identifier, SoC information, or billing information. +Possible solutions include the use of cryptographic tech- +niques to hide information. The newest release of ISO- +15118 [36] mandates TLS on every communication. +g) Relaying: An attacker may relay information from a +victim vehicle to the access point of the attacker’s charging +column to steal energy [82]. This kind of attack work even +if the traffic is encrypted since the data is only relayed and +not modified. With respect to the wired counterpart, where +the attacker has to tamper in some way with the charging +column, a wireless relay attack does not need any hardware +modification. +To protect against relay attacks, a distance bounding proto- +col can be employed to assess if a malicious entity is relaying +the network flow [82]. +h) Man-in-the-Middle attack: With respect to wireless +charging, when dealing with WPT, the interception and for- +warding of communication flow are easier due to the openness +of the medium [101]. At the same time, directional jamming +can be employed in some cases to prevent the receiver +from getting both the original and the modified data. If the +communication flow is unencrypted or the cryptography is +weak, an attacker can launch a MitM attack to modify on- +the-fly packets. For instance, a malicious user might modify +the information sent by the victim (i.e., report full SoC) after +establishing a connection with the service provider. To perform +such an attack, a malicious entity may set up a fake access +point and use it to relay the communication to the legitimate +one, gaining the ability to modify packets at will. +Partial solutions include the previously mentioned solutions, +such as encryption, authentication, and integrity protection +mechanisms. In the context of WPT, the transmission fre- +quency can be regulated to encrypt information and guarantee +that only the legitimate party can receive power [100]. Further- +more, physical layer authentication may enhance the security +of the authentication process [102]. +i) Side Channels, and Information Leaking: Although +WPT signals might be encrypted or avoid sensitive reporting +information on the user, the power signal can be exploited for +profiling purposes. This attack has been proven feasible for +smartphones, where the WPT signal analysis reveals informa- +tion on the user’s activity [103]. This might also be the case for +EVs, where an attacker can infer different types of information. +This attack is similar to the profiling performed in the wired +case, where it might be possible to track a user and obtain +information on her habits and power demands. Preventing this +attack represents a challenging task, as it cannot be detected, +and data encryption is not sufficient [104]. +A possible solution is represented by differential privacy, +where data is corrupted with a noisy pattern that might prevent +inferring sensitive users’ data [79]. Furthermore, as for the +wired counterpart, secondary batteries may prevent the attacker +from inferring sensitive users’ information. +VI. FUTURE DIRECTIONS +Looking at the impact of the different attacks in Tables IV +and V, we can conclude that many security issues related to +the cyber-physical nature of EVs may impact the safety of the +driver. We notice that some of the attacks and countermeasures +discussed can also be applied to other EV assets. However, +due to space limitations and to avoid being repetitive, we +only discussed those we considered to be the most interesting. +Nevertheless, we summarize all the attacks and countermea- +sures in the comprehensive Table III. Although many threats +concern the charging infrastructure, the most severe in terms +of safety are related to the in-vehicle network. In fact, the +electric component of EVs may be tampered with or impaired +to electroshock the user. Furthermore, the increasing cyber +nature of the EVs’ components leads to challenges regarding +the coherency of the information coming from the cyber and +the physical worlds. Lastly, the increasing interest in the +application of the WPT technology to EVs impose significant +challenges that still need to be properly addressed from a CPS +point of view. +Based on our analysis, we foresee the following future +directions and needs in the field of EV CPS security. +• Denial of Service (DoS) represents one of the most chal- +lenging threats in EVs security. It is known as difficult +to prevent, and almost every component of the EV can +suffer from it. Compromised internal components can +attack other ECU to compromise the in-vehicle network, +but DoS attacks can be launched from charging columns +to EV during charging, or vice-versa. In certain cases, +DoS can have an impact not only on a single vehicle +but can compromise EVSEs in a certain geographical +area. To mitigate this risk, not only do all the vehicle +entities need to be associated with an identity, but their +allowed flow of information (and hence generated traffic) +should depend on the vehicle’s physical situation. In fact, +it might be that a specific ECU needs to send messages +at a higher rate when the vehicle is experiencing certain +physical stimuli. At the same time, all ECUs shall be +guaranteed a sufficient amount of resources. Therefore, +future protections against DoS for in-vehicle networks +should account for the physical factors and the possible +impact on the whole electric grid. +• The potential tampering with the EV components might +represent a significant threat to the user’s safety and +may also have repercussions on other elements of the +vehicle system. All vehicle components shall be equipped +with anomaly detection capabilities or should prevent +the application of a voltage or current flow in case of +tampering. Possible future solutions might include the +collective verification of multiple components to make +tampering with a single unit ineffective. + +14 +TABLE III +SUMMARY TABLE WITH ATTACKS AND COUNTERMEASURES FOR EACH ASSET. THE FIRST ROW INDICATE THE ASSETS INTERESTED BY EACH ATTACK, +WHILE THE FOLLOWING ROWS POINT OUT WHICH COUNTERMEASURE IS EFFECTIVE AGAINST EACH ATTACK. +� : BMS AND BATTERY; � : CONTROLLERS, CHARGER AND MOTORS; � : WIRED CHARGING; AND �: WPT. +DoS +Tampering +MitM +Replaying +Spoofing +Malware +Overpower +Freeride +Jamming +Repudiation +Eavesdropping +Side-Channels +Relaying +Affected assets +� � � � +� � � +� � � � +� � � � +� � � � +� � +� +� +� +� � +� � � � +� � � � +� � +Aggregate signature +� � +Anomaly detection +� � +� +Authentication +� � +� � � � +� � � � +� � +� +Blockchain +� +� � +Channel hopping +� +Cookies +� � +Differential privacy +� � +� � � � +Distance bounding +� � +Encryption +� � � � +� � � � +� � � � +Fingerprinting +� +Flow control +� � +Intrusion detection/prevention +� � � � +� � +� � +� � +� � +Identity management +� � +� � +Identity verification +� � +� � +� � +Inconsistencies handler +� +Integrity protection +� � � � +Overvoltage protection +� +Physical layer security +� +� +� +Rate limiting +� � +Redundancy +� +� +� +� +Remote Attestation +� � +Secondary battery +� � +Tamper-proof hardware +� � � +Time-lock puzzles +� � +Timestamps +� � � � +Countermeasure +Attack +• The increasing attackers’ capabilities impose additional +challenges in guaranteeing the cyber-security of EVs. +In fact, an attacker might be able to combine multiple +attacks to impair the EV functioning. To strengthen the +defense mechanisms, it is essential to implement in EVs +frameworks collecting information from multiple sources, +combining the cyber and the physical world. For instance, +verifying the message integrity might employ data from +different sensors and actuators to increase the difficulty +of information manipulation. Similarly, intrusion detec- +tion techniques might combine network data exchanged +through the bus with physical signals from sensors to +model better the state of the EV. This will also help +prevent attacks related to malicious ECUs controlling ac- +tuators for mechanical operations (e.g., steering). Future +work should consider the EV-specific components such +as the battery or the charger as data sources regarding +the vehicle’s state. For instance, the charge and discharge +curves of batteries can be modeled by computers with +discrete confidence [105], [106], [107]. A simple applica- +tion of these simulations is a reference to identify packets +declaring modified SoC. +• One of the strengths of EVs compared to previous genera- +tions of vehicles relies on the software managing all their +operations. However, this implies that vehicles are more +subject to cybersecurity attacks. Some of these attacks +may include malware injection into some of the EV’s +components. To this aim, collective remote attestation can +be used to verify the integrity of all the EV’s components +and prevent possible safety threats. Remote attestation +measures should, however, account for the resource- +limited nature of EVs’ components and the time-critical +nature of the exchanged information. +• WPT is one of the promising technological solutions to +alleviate the range anxiety of drivers fearing not reaching +their destination with the available charge. Thanks to the +charging while driving paradigm, EVs can be charged +during their operation. However, deploying the required +public infrastructure poses many security challenges both +to the operators and the users. Some examples include +the billing process and the openness of the wireless +medium. WPT related challenges heavily rely on the +cyber-physical nature of the overall infrastructure. There- +fore, security solutions in this area should account for the +coherency of information from the cyber and the physical +domains. +VII. CONCLUSION +The increasing market for EVs demands an in-depth anal- +ysis of EV technology’s security and privacy challenges. In +this paper, we provided an overview of the components of an +EV, focusing on their characteristic components. We provided +the basic information needed to understand how in-vehicle +communication networks work and which devices need to +communicate with one another. We then discussed how an +EV battery could be charged via wire and WPT. 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Park, “A study on mitm (man in +the middle) vulnerability in wireless network using 802.1 x and eap,” +in 2008 International Conference on Information Science and Security +(ICISS 2008). +IEEE, 2008, pp. 164–170. +[102] L. Y. Paul, J. S. Baras, and B. M. Sadler, “Physical-layer authentica- +tion,” IEEE Transactions on Information Forensics and Security, vol. 3, +no. 1, pp. 38–51, 2008. +[103] A. La Cour, K. Afridi, and G. E. Suh, “Wireless charging power side- +channel attacks,” arXiv preprint arXiv:2105.12266, 2021. +[104] B. Saltaformaggio, H. Choi, K. Johnson, Y. Kwon, Q. Zhang, X. Zhang, +D. Xu, and J. Qian, “Eavesdropping on fine-grained user activities +within smartphone apps over encrypted network traffic,” in 10th +USENIX Workshop on Offensive Technologies (WOOT 16), 2016. +[105] R. C. Kroeze and P. T. Krein, “Electrical battery model for use in +dynamic electric vehicle simulations,” in 2008 IEEE Power Electronics +Specialists Conference, 2008, pp. 1336–1342. +[106] G. Salda˜na, J. I. San Mart´ın, I. Zamora, F. J. Asensio, and O. O˜nederra, +“Analysis of the current electric battery models for electric vehicle +simulation,” Energies, vol. 12, no. 14, p. 2750, 2019. +[107] M. Chen and G. Rincon-Mora, “Accurate electrical battery model +capable of predicting runtime and i-v performance,” IEEE Transactions +on Energy Conversion, vol. 21, no. 2, pp. 504–511, 2006. +Alessandro Brighente is assistant professor at the +University of Padova. He was visiting researcher at +Nokia Bell Labs, Stuttgart, Germany in 2019 and +University of Washington, Seattle, USA, in 2022. +He served as TPC for several conferences, including +Globecom, VTC, and WWW. He is guest editor +for IEEE Transactions on Industrial Informatics and +program chair of DevSecOpsRA, co-located with +EuroS&P. His current research interests include se- +curity and privacy in cyber-physical systems, ve- +hicular networks, blockchain, and communication +systems. +Mauro Conti is Full Professor at the University of +Padua, Italy. He is also affiliated with TU Delft and +University of Washington, Seattle. He obtained his +Ph.D. from Sapienza University of Rome, Italy, in +2009. After his Ph.D., he was a Post-Doc Researcher +at Vrije Universiteit Amsterdam, The Netherlands. +In 2011 he joined as Assistant Professor the Uni- +versity of Padua, where he became Associate Pro- +fessor in 2015, and Full Professor in 2018. He has +been Visiting Researcher at GMU, UCLA, UCI, +TU Darmstadt, UF, and FIU. He has been awarded +with a Marie Curie Fellowship (2012) by the European Commission, and +with a Fellowship by the German DAAD (2013). His research is also +funded by companies, including Cisco, Intel, and Huawei. His main research +interest is in the area of Security and Privacy. In this area, he published +more than 400 papers in topmost international peer-reviewed journals and +conferences. He is Area Editor-in-Chief for IEEE Communications Surveys & +Tutorials, and has been Associate Editor for several journals, including IEEE +Communications Surveys & Tutorials, IEEE Transactions on Dependable +and Secure Computing, IEEE Transactions on Information Forensics and +Security, and IEEE Transactions on Network and Service Management. He +was Program Chair for TRUST 2015, ICISS 2016, WiSec 2017, ACNS 2020, +and General Chair for SecureComm 2012, SACMAT 2013, CANS 2021, and +ACNS 2022. He is Senior Member of the IEEE and ACM. He is a member +of the Blockchain Expert Panel of the Italian Government. He is Fellow of +the Young Academy of Europe. +Denis Donadel received his MSc in Telecommu- +nication Engineering from the University of Padua, +Italy, in 2020. He is now a Ph.D. Student in Brain, +Mind and Computer Science (BMCS) at the Univer- +sity of Padua where he joined the SPRITZ Security +and Privacy Research Group under the supervision +of Prof. Mauro Conti. Together with his academic +course, Denis is also working with Omitech SRL as +part of his high apprenticeship program. During the +2021 Summer, he was granted the New Generation +Internet (NGI) Explorers grant to support a collabo- +ration with the University of Washington (Seattle, USA). His research interests +lie primarily in Cyber-Physical Systems security, focusing particularly on +Vehicles Security and Critical Infrastructures Security. + +18 +Radha Poovendran is Professor of the Department +of Electrical & Computer Engineering at the Univer- +sity of Washington. He is the founding director of the +Network Security Lab and is a founding member and +associate director of research for the UW’s Center +for Excellence in Information Assurance Research +and Education. He has also been a member of the +advisory boards for Information Security Educa- +tion and Networking Education Outreach at UW. +In collaboration with NSF, he served as the chair +and principal investigator for a Visioning Workshop +on Smart and Connected Communities Research and Education in 2016. +Poovendran’s research focuses on wireless and sensor network security, +adversarial modeling, privacy and anonymity in public wireless networks and +cyber-physical systems security. He co-authored a book titled Submodularity +in Dynamics and Control of Networked Systems and co-edited a book titled +Secure Localization and Time Synchronization in Wireless Ad Hoc and +Sensor Networks. Poovendran is a Fellow of IEEE and has received various +awards including Distinguished Alumni Award, ECE Department, University +of Maryland, College Park, 2016; NSA LUCITE Rising Star 1999; NSF +CAREER 2001; ARO YIP 2002; ONR YIP 2004; PECASE 2005; and Kavli +Fellow of the National Academy of Sciences 2007. +Federico +Turrin +received the Master’s Degree +in Computer Engineering from the University of +Padova, Italy, in 2019, where he is currently pur- +suing the interdisciplinary Ph.D. in Brain, Mind, +and Computer science, since October 2019. He has +been visiting researcher at SUTD Singapore in 2022. +His research interests lie primarily in Cyber-Physical +System Security with a particular focus on Industrial +Control Systems Security, Vehicles Security, and +Anomaly detection. +Jianying Zhou is a professor and co-center director +for iTrust at Singapore University of Technology +and Design (SUTD). He received PhD in Infor- +mation Security from Royal Holloway, University +of London. His research interests are in applied +cryptography and network security, cyber-physical +system security, mobile and wireless security. He +has published 300 referred papers at international +conferences and journals with 13,000 citations, and +received ESORICS’15 best paper award. He has 2 +technologies being standardized in ISO/IEC 29192-4 +and ISO/IEC 20009-4, respectively. He is a co-founder & steering committee +co-chair of ACNS. He is also steering committee chair of ACM AsiaCCS, +and steering committee member of Asiacrypt. He has served 200 times in +international cyber security conference committees (ACM CCS & AsiaCCS, +IEEE CSF, ESORICS, RAID, ACNS, Asiacrypt, FC, PKC etc.) as general +chair, program chair, and PC member. He has also been in the editorial +board of top cyber security journals including IEEE Security & Privacy, IEEE +TDSC, IEEE TIFS, Computers & Security. He is an ACM Distinguished +Member. He received the ESORICS Outstanding Contribution Award in 2020, +in recognition of contributions to the community. + diff --git a/odE3T4oBgHgl3EQfjQqW/content/tmp_files/load_file.txt b/odE3T4oBgHgl3EQfjQqW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c695498d0c64851cdba11ba14e50aacd06463bbb --- /dev/null +++ b/odE3T4oBgHgl3EQfjQqW/content/tmp_files/load_file.txt @@ -0,0 +1,1719 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf,len=1718 +page_content='1 Electric Vehicles Security and Privacy: Challenges, Solutions, and Future Needs Alessandro Brighente, Member, IEEE, Mauro Conti, Fellow, IEEE, Denis Donadel, Raadha Poovendran, Fellow, IEEE, Federico Turrin, and Jianying Zhou, Senior Member, IEEE, Abstract—Electric Vehicles (EVs) share common technologies with classical fossil-fueled cars, but they also employ novel technologies and components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', Charging System and Battery Management System) that create an unexplored attack surface for malicious users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Although multiple contributions in the lit- erature explored cybersecurity aspects of particular components of the EV ecosystem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', charging infrastructure), there is still no contribution to the holistic cybersecurity of EVs and their related technologies from a cyber-physical system perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this paper, we provide the first in-depth study of the security and privacy threats associated with the EVs ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We analyze the threats associated with both the EV and the different charging solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Focusing on the Cyber-Physical Systems (CPS) paradigm, we provide a detailed analysis of all the processes that an attacker might exploit to affect the security and privacy of both drivers and the infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To address the highlighted threats, we present possible solutions that might be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We also provide an overview of possible future directions to guarantee the security and privacy of the EVs ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Based on our analysis, we stress the need for EV- specific cybersecurity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Index Terms—Electric Vehicles, Cyber-Physical Systems, Se- curity, Privacy I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' INTRODUCTION T HE recent climate crisis demands green alternatives to replace technologies with high environmental impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Among the others, fossil-fueled transportation is one of the significant causes of greenhouse gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Electric Vehicles (EVs) have been proposed as a green alternative, where electric bat- teries are employed as a power source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' During the last years, the number of people opting for the EV alternative increased up to the point where the market share of new EV sales reached more than 50% in countries such as Iceland (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='6%) and Norway (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='7%) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The adoption of EVs is further expected to increase in the next years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, governments are incentivizing the adoption of EVs thanks to the deployment of a large number of Electric Vehicle Supply Equipment (EVSE) in public charging infrastructures [2] and planning to ban sales of fossil-fueled vehicles [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, technology advancements remove the current barriers against consumers’ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Brighente, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Conti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Donadel, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Turrin are with the Department of Mathematics and HIT Research Center, University of Padova, 35131 Padova, Italy (e-mail: alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='brighente@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' conti@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Poovendran is with the Department of Electrical and Computer Engi- neering, University of Washington, 98195 Seattle, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Zhou is School of Information Systems Technology and Design, Sin- gapore University of Technology and Design, 487372, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' (email : jianying zhou@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Manuscript received X X, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' revised X X, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' adoption of EVs, providing extended driving range and seam- less charging [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The increasing number of EVs demands a thorough analysis of the security of both vehicles and infrastructure operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Like traditional vehicles, EVs are equipped with many Elec- tronic Control Units (ECUs), sensors and actuators that mea- sure, process, and control the different stimuli inside and out- side the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, EVs include additional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Indeed, an EV integrates components to govern the hardware and software dedicated to managing electric energy smartly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These components are, for instance, the Battery Management System (BMS) and the charging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Different studies have already proven the impact of potential cybersecurity attacks on automotive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, Miller and Valasek [5] proved the feasibility of hijacking a vehicle by remotely controlling it through the infotainment system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, most existing vehicles exploit Controller Area Network (CAN) as in-vehicle network architecture, which has already been proved as non-secure [6] and, there- fore, may be vulnerable to potential cyberattacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Lastly, privacy shall also be guaranteed to prevent malicious users from obtaining sensitive information on the driver, such as her location or habits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It is essential to include security and privacy features by design to prevent these and other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Researchers investigated vehicle security, focusing on the different aspects of in-car communications [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, EVs are equipped with specific components that provide fundamentally different attack surfaces and exploitation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, EVs are equipped with electric batteries to power the vehicle components ranging from the infotainment system to the acceleration pedal, together with systems to manage the electrical power as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, analyzing the in-vehicle threats associated with these components is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, the power supply must be regulated by dedicated hardware, not classical vehi- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Researchers discussed how the EV charging infrastructure could be exploited by attackers [9], however, neglecting the in-vehicle threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this paper, we examine the security and pri- vacy issues of EVs from a Cyber-Physical System (CPS) point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Given the high demand for EVs and the increasing number of deployed charging facilities, it is fundamental to guarantee the security and privacy of both vehicles and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Many literature contributions discuss solely technical aspects of the EV ecosystem without focusing on security issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Other security-focused works study a single system compo- nent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', the vehicle’s internal bus, the smart grid, or the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='04587v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='CR] 11 Jan 2023 2 communication protocols) without comprehensively analyzing the whole environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provide a general overview of EV functioning, focusing on their core components to build the basic knowledge needed to analyze the possible threat vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We then discuss possible attacks and countermeasures specific for EV and underline the existing security solutions for fuel vehicles that are also effective in EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' With the bird-eye on the CPS concept, we are not only able to discuss the issues related to the exchange of information between the different involved entities, but also the side channels that may leak sensitive information or that could lead to hazardous behavior impacting on users’ safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We hence shed light on the unresolved chal- lenges of EVs ecosystems, providing interested researchers with possible directions worth investigating to guarantee the security and privacy of the overall EV ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We also consider future directions such as the Wireless Power Transfer (WPT) charging of EVs, which has only been developed on small-scale testbeds at the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We believe that delving into this emerging system’s security and privacy issues will help future developers design and implement secure-by- design WPT solutions for EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We summarize the contribution of this work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We examine the peculiar components that differentiate EVs from fossil-fueled vehicles and provide an overview of their role and how they exchange information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provide an overview of the different technologies employed to charge electric vehicles, comprising both wired and wireless charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We present the available standards for each of them and describe their basic functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provide an in-depth discussion of the security and privacy issues of the EVs ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We analyze the threats related to the in-vehicle network and the threats related to the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We particularly focus on their effects on the peculiar EV components and analyze them from a CPS point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We analyze and compare possible countermeasures pro- posed in the literature for each of the presented attacks, even grasping from other similar areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We outline future directions for research in the EV cybersecurity domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In Section II, we review the related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In Section III, we describe the EV components the charging infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In Section IV, we then discuss the in-vehicle security and privacy threats, and those related to the charging infrastruc- ture in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Along with the threats, we also present possible countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Then, we discuss the possible future direction in Section VI, and lastly we conclude the paper in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' RELATED LITERATURE ON EV SECURITY Automotive cyber-security requires standardization to allow for security guarantees and interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Schmittner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [7] reviewed the available standards, including designing and validation aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These standards, however, do not consider the peculiar features of EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Scalas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [8] provided Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Main components of an EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Green components are EV specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' an overview of the cybersecurity requirements for the future of the automotive industry, focusing on in-vehicle components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' They discussed several technologies and attacks but were not specific for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, different works present technical reviews of the EV ecosystem [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, none of them consider the security aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some contributions in the literature focused on specific components of EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, Khalid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [12] focus on the BMS, discussing the lack of a cybersecurity standard to guarantee its security and providing an overview of the possi- ble standardization framework that could be adopted to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Chandwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [13] presented an overview of the cybersecurity threats associated with the onboard charging system of EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Despite providing an accurate analysis of the security of this component, their contribution does not consider how these attacks can impact the other peculiar components of the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These contributions do not provide a general overview of the EV ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, they do not discuss the threats associated with privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [14] provide the first discussion on how EVs can be considered as CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The authors discuss how different attacks can be conducted inside the car and during communication with the power supplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, they do not consider the specific components of the EVs such as the BMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [15] focus on the CPS system represented by the power electronics in EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, they do not consider how these attacks may impact the other components of the EV and did not discuss the issues related to WPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Most of the literature related to EVs’ cybersecurity focus on the charging infrastructure and process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Gottumukkala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [9] provide an overview of the CPS threats associated with a wired EV charging infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Antoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [16] discuss the security threats associated with the negotiation and actuation of a charging session investigating the communications between the multiple involved entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' They presented different charging scenarios, neglecting the WPT option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Vehicles can be interconnected with one another to form the internet of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This is also feasible with EVs, which imposes additional security challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Fraiji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [21] discuss the cybersecurity threats associated with the internet of electric vehicles, discussing the threats associated with the communication with the multiple involved entities being part ECU ECU LIN bus LIN Master ECU Battery BMS Controller Pack Motor CANbus ECU LIN Master Onboard Charger LINbus ECL3 Reference BMS Onboard Charger Battery Pack Controller Electric Motor Wired Charging Wireless Charging Khalid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [12] Chandwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [13] Acharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [14] Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [15] Sripad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [17] Gottumukkala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [9] Antoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [16] Garofalaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [18] Van Auben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [19] Babu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [20] Our paper of the road infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The cybersecurity focus is on the communication links, therefore neglecting the impact of the peculiar EVs’ components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Garofalaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [18] present a detailed survey on Open Charge Point Protocol (OCPP) and the corresponding threat and vulnerabilities on the Vehicle-to-Grid (V2G) ecosystem due to its adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Similarly [19] overview the main protocols for EV charging adopted in the Netherlands and analyze their security features, while Babu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [20] analyzed the security of the main protocols proposed for the EV environment with a particular focus on the payment methods and the authentica- tion solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Differently from these works, instead of focus- ing on the protocols, we focus on the entire EV architecture, highlighting the main security and privacy challenges in this typology of CPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Table II compares the related works on EV security with our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We can see that most of the contributions focus on vehicle-to-grid communications in the wired case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, none of the available papers focus on intertwining the different cyber-physical aspects of EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, our paper provides a more complete analysis of the security and privacy challenges for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ELECTRIC VEHICLES FROM A CYBER-PHYSICAL SYSTEM PERSPECTIVE In this section, we analyze EVs from a CPS perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We emphasize those components that differentiate EVs from gas- fueled vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, we first describe the traditional vehicle architecture in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Then, we present the main components of an EV in Section III-B, showing how it differs from traditional vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Lastly, we provide an overview of the EV charging infrastructure in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Traditional Vehicle Architecture Nowadays, vehicles contain dozens of different micro- computers, called Electronic Control Units (ECUs), running millions of lines of code [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Each ECU is responsible for controlling a mechanical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', brakes) or electrical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', light) component of a modern vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Depending on the component it has to manage, an ECU generally employs a wide range of microcontrollers, from simple 8-bit RISC controllers to more complex 32-bit multicore processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ECU are typically implemented with ad-hoc firmware, even if complex ECUs may run complete operating systems: the infotainment system, for instance, usually runs a Linux-based kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In order to provide more flexibility during updates, more advanced solutions envisage the implementation of multiple ECUs on a single FPGA board [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Communications among ECUs that reside in the vehicle pass through wires that connect multiple components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The two mostly implemented technologies are CAN and Local Interconnect Network (LIN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' CAN represents the main net- work that allows for cost-effective wiring, self-diagnosis and error correction [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The CAN bus consists of two wires and implements a distributed architecture, where car modules (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', the ECUs) share messages upon winning a contention phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, CAN has been designed to be a reliable solution, neglecting possible security ad privacy shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The LIN bus is a supplement to CAN [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, it connects a smaller number of ECUs (one master and up to 16 slave nodes) and offers a drastically cheaper implementation at the cost of lower performance and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A LIN master node is typically a gateway to CAN, and multiple LIN buses can communicate via the CAN bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The LIN bus can be used to control, among the others, sensors and actuators for steering wheels, comfort, powertrain, engine, air conditioning, doors, and seats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Besides CAN and LIN, also other technologies such as FlexRay [26] and Media Oriented Systems Transport (MOST) [27] are currently used for automotive networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To overcome some of these technologies’ limitations and ease their interoperability, automotive Ethernet has recently been introduced as a possible solution [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Given the CPS nature of our investigation, we do not prefer one technology over another, as these all represent communication means for the exchange of information inside the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We refer the interested reader to [28] for a discussion on automotive Ethernet security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Modern vehicles also include mechanisms to update the internal software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This service is generally implemented with the aid of external device plug (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', USB flash drive) or Over-the-Air (OTA) software update [29] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', via Internet connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, many vehicles nowadays include complex entertainment systems, which may expand the vul- nerable surface, exposing new connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', Bluetooth) and operating systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', Android).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Electric Vehicle Specific Components EVs share most of the architecture with fuel-based vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, they comprise a different set of hardware modules that manage how the vehicle generates power and how to generate motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, an EV comprises the following components [30], depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The battery is where the charge is stored in the form of Direct Current (DC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It provides the power needed to operate the EV components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Batteries are usually combined in packs and connected in series or parallel to increase the voltage and Amper/hour they can deliver to the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Batteries suitably combined are enclosed into a metal casing to prevent damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The case usually includes a cooling system to avoid damage due to batteries overheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Battery Management System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This module manages all operations regarding the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It manages the current output and the charging and discharging of the battery by keeping it in a safe operating area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Hence, it regulates the electricity flow through the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The BMS is unique for each EV model, and may be designed according to various topologies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', modular, centralized or distributed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The BMS monitors each battery in the pack and measures each cell’s voltage, current, and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It is instructed with a threshold limit for each of them and disconnects the load if values exceed the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, the BMS measures the State of Charge (SoC) and state of health of the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The BMS communicates with the human-machine interface to report information on this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' All the information are exchanged via CAN or LIN bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Battery Charger/Onboard Charger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This component pro- vides an interface between the charging system and the EV battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As soon as an Alternate Current (AC) charging process begins, the charger converts the input voltage to DC and passes it to the battery for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For high power DC charging, the conversion phase is done on the charging column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, it prevents possible damages to the battery or the supply system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', overheating) by limiting the power flow [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The controller handles the flow of current from the battery to the EV associated with all operations, ranging from motors-related operations to powering the infotainment system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It receives the input from the driver to control the acceleration, brake pressure, and driving mode and converts the energy in the battery from DC to AC to control the EV accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' On the other hand, the EV may generate electricity due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', regenerative braking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this case, the controller converts the generated AC to DC such that the energy can be stored in the battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Electric Motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The motor is powered by the EV battery, which provides the electricity needed to turn it and move the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The electric motor communicates with sensors and actua- tors in the EV that control the amount of thrust required [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' There exist many implementations of electric motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The most commonly used for EVs are AC induction due to their lower cost implementation thanks to the absence of permanent magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These components characterize an EV and differentiate it from other types of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, the conventional motor is replaced with an electric one, and a battery pack re- places the fuel tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Notice that all the components mentioned above need to share messages inside and outside the vehicle to guarantee the correct functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker might exploit some of these messages to create inconsistencies on the EV status or to cause damages to both the vehicle and driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provide a detailed security analysis based on these components in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Electric Vehicles Charging Infrastructure The EV needs to charge its battery periodically to provide power to its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To this aim, the EV shall be connected to a charging infrastructure with whom it negotiates a charging session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' According to the negotiated session, the infrastructure then delivers the needed energy to the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Charging may happen either in public areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', shopping malls or offices) or at a private site (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', home).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To prevent possible malfunctioning, the charging infrastructure must be carefully managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This is particularly true when considering a scenario where handling a massive number of EVs may lead to blackout and other grid malfunctions [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Charging an EV differs from other devices, such as smart- phones or laptops, as it requires dedicated hardware and a drastically larger energy supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Indeed, if many EVs are concurrently charging, there can be grid overloading, leading to malfunctions and local blackouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To avoid these issues, the grid must employ a communication channel with the EV to negotiate to charge parameters that respect the vehicle’s battery requirements without overloading the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' V2G refers to the technology enabling this communication type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' There are two solutions to manage a charging session: wired and wireless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' While the former is more diffused and widely implemented nowadays, the latter is still in the initial stage and under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Unfortunately, there is no unique world standard to regulate this communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Instead, different manufacturers implement different standards based on the technologies used for the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, CHAdeMO [33] (Japan) or GB/T [34] (China) can be used only with wired charging, while ISO 15118 (Europe, North America) also supports WPT [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 1) Wired Charging: With this setting, the EV is connected to an EVSE through a cable that transmits both the control signals ad the charging current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In turn, EVSEs negotiate with power grids for the energy needed to charge the vehicle, based on both EV and grid requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, these basic functions are integrated by every charging standard, which employs different communication methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Low current charging levels, such as AC Level 1 or AC Level 2, require a simple control channel which is generally provided by a Pulse- Width Modulation (PWM) communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' More advanced charging, such as DC charging, needs better management of the energy provided by a High-Level Communication (HLC) provided by protocols such as CAN or Power Line Commu- nication (PLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These technologies enable the development of additional services, such as the automatizing of the billing process [37], [38], or the download of firmware updates [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In case of a lack of automated authentication solutions, EVSE 5 may be equipped with RFID readers through which users can authenticate and pay for the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' EVSEs can be deployed at private or public premises: private charging columns are generally less advanced and support less charging level with respect to public EVSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' There are mainly two protocols supporting the HLCs be- tween EV and EVSE during DC charging sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The first one, employed by Combined Charging System (CCS), is the ISO 15118 [38], which modulates data over the control pilot pin using PLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The second one, CHAdeMO [33] employs a CAN channel for the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The physical connection between EV and EVSE may be implemented with different plugs according to different stan- dards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, we can classify EVSEs according to different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Figure 2 shows the different charger levels together with their lead characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' (a) Level 2 (b) Level 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Different types of EV chargers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' L1 = AC line 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' N = AC line neutral, P1 and P2 = proximity lines, PE = ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Level 1 and Level 2 EVSEs exploit a five-leads connector implementing the SAE J1772 protocol [40], as shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This connector exploits two leads to deliver the charging current, two leads for pilot signals, and one lead for ground (or protective earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The two current leads plus the ground one are used by the EVSE for metering and computing the session cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The two pilot lines have two different functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The first one, the control pilot, is used to exchange information with the EV during the charging session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The signals exchanged through the control pilot either control the amount of current delivered to the EV [41] or are used to check the connection status and remove power from the adapter in case of disconnection to prevent the user injuries [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The second pilot line is the proximity pilot, used by the EV to check whether a proper physical connection has been established with the EVSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Level 3 EVSEs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', those allowing for fast charging, are based on different implementations and are showed in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The first is the CCS expansion of the SAE J1772, which allows for direct current exchange for fast charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, it implements PLC to exchange information between the EV, the EVSE, and the smart grid [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The second implementation is the Japanese CHAdeMO [42], which implements a fast charging protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Besides delivering power, this implementation allows for data exchange via the CAN bus protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Thanks to this type of connection, it is possible to avoid applying power to the connector in case of a non-safe connection or to exchange information related to the battery SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, CHAdeMO allows V2G communication, where the EV battery is later used as energy storage to provide service to the grid [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Other protocols exist, such as the proprietary protocol employed by Tesla vehicles and the Chinese GB/T, which will probably be replaced by Chaoji, an evolution of CHAdeMO [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The main differences between Level 3 and Level 2 chargers lie in the higher number of leads in Level 3, and in the implemented circuitry which converts AC to DC, which is inside the charging columns for Level 3, while it is onboard in the EV for Level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, Level 3 charging includes richer communication capabilities thanks to the support of HLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 2) Wireless Power Transfer: Charging via WPT allows charging an EV’s battery without physically connecting the ve- hicle to the charging infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In WPT, a source (powered by the grid) generates a time-varying electromagnetic field that triggers the generation of a current at the receiver’s (EV’s) side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This current is generated thanks to a coil mounted on the EV’s side that receives the transmitted electromagnetic field and, due to Faraday’s law of induction, generates an AC [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Via WPT, it is possible to create multiple charging scenarios depending on the mobility of the EV [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, thanks to the absence of a physical connection, EVs can be either charged while parked or while driving in a dynamic scenario [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The static scenario is similar to the one previously described in Section III-C1, where a user books a charging session and receives the power from the grid while parked at a charging facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Instead, the dynamic case requires a suitably designed infrastructure composed of multiple sequential WPT transmitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Figure 3 shows a pictorial representation of a dynamic WPT system for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The street is equipped with multiple WPT transmitters deployed underneath the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These transmitters are connected to the grid that provides the power needed to charge the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Representation of a WPT system for EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Dynamic WPT can be further divided into two categories: quasi-dynamic and fully-dynamic [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In the former case, L1 N P1 P2 PEPE L1 N P1 P2 PE DC+ DC- X DC+ DC-6 charging is limited to the cases where the EV is not moving, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', while waiting at stops or traffic lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fully-dynamic WPT, charging is continuously delivered to the EV as long as it drives near transmitting coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In both dynamic scenarios, the challenge is to guarantee that transmitters are activated only when needed to avoid energy waste and that only legitimate users access the emitted power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, due to the absence of a medium, users could steal power by driving close to an EV that paid for the charging session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We discuss all the security problems related to WPT in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As specified in the ISO 15118 standard [35], the connection between the vehicle and the charging column during a static WPT scenario uses WiFi (IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The vehicle can connect before being correctly parked or when it is already over the coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If needed, the EVSE provides the EV with fine positioning messages to help the driver correctly place the vehicle to reduce energy dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' After establishing the connection, the two entities communicate similarly to wired cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some modifications are introduced to adapt to the wireless scenario, including the WPT charging mode and the fine positioning messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Due to their novelty, dynamic and quasi-dynamic charging are not yet covered by approved and widely adopted standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some research works [46] adopt Dedicated Short-Range Com- munications (DSRC) to create a channel between vehicles and the Road Side Units (RSUs), which are in charge of controlling a portion of the road coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Another possible solution can be to extend WPT to deliver also information along with power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, Wireless Infor- mation and Power Transfer (WIPT) represents a technology that might be exploited for electric vehicle [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' WIPT can be adopted to implement a system similar to that exploited in wired EV charging, where control signals are exchanged through the pilot line and the charging current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In WIPT, control signals can be coded into the time-varying electromag- netic field to deliver power and check the connection’s status simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, this solution can be exploited to authenticate EVs and solve part of the security challenges in WPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Although not yet discussed in the literature, we believe that WIPT represents a suitable line of research for EV charging technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' IN-VEHICLE SECURITY AND PRIVACY CHALLENGES This section discusses the security and privacy challenges related to the components and protocols used inside EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' First discuss in Section IV-A the challenges related to the battery and the BMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Then, we discuss the challenges related to the controller and charger in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The security of CAN bus has been extensively studied in the literature, as it does not envision secure by design solutions [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, how these attacks may impact EVs has never been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Since all in-vehicle messages are exchanged through CAN and LIN buses, we discuss how their vulnerabilities can be exploited to impact those components specific to the EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We summarize in Table IV the in-vehicle security and privacy challenges together with their effects, impact severeness, and possible countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Battery and BMS The battery pack is a sensitive component of an EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In case of malfunctions, it may catch fire and even explode [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Such situations can severely harm the passengers and create financial damage to the owner and a reputation loss to the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Less severe cyberattacks can, however, create financial damage, for instance, by reducing the battery’s lifespan, forcing the owner to a premature battery replace- ment [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The battery pack is managed by the BMS, which handles communication with the other ECUs via the vehicle bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Again, this channel has been proven to be vulnerable to many cyberattacks [48], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In the following, we discuss how cyberattacks impact EVs, extend their effects to the CPS domain, and highlight their effect on the battery and BMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' a) Denial of Service: The BMS is responsible for re- porting information on the battery status and managing the energy delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker might flood the BMS controller by forging and sending a vast number of requests, in similar ways to what may happen with Denial of Service (DoS) attacks against websites [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An overload of the BMS may slow responses to legitimate requests or even prevent the BMS from sending response messages completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This may lead to multiple effects depending on the information requested to the BMS and how the requester device reacts to the absence of a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, this might cause damage to the battery if power is not properly removed in case of abnormal behavior or physical tampering by the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A DoS may target sensor measurements, such as temperature, and it may prevent the activation of cooling mechanisms, forcing the battery into critical temperatures, which may be irreversible [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, this attack may also prevent the user from obtaining information on the amount of charge left, causing range anxiety and possibly jeopardizing the drivers’ safety in case of a sudden EV stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Flow control might prevent the BMS from handling many fake requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this context, source authentication may pro- vide information regarding the legitimacy of the sender [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A solution for flow control may be given by an adapted version of time-lock puzzles [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, rate limiting can help mitigate against DoS attacks [55], while intrusion detection strategies can help in identifying the attack before it creates damage [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Redundancy on the controllers can also help in mitigating severe DoS attacks against the BMS [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' b) Tampering: An attacker might physically tamper with the battery and the BMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Depending on the specific tampered component, an attacker may be able to cause a short circuit that may lead to catastrophic events such as the start of a fire that might harm both the vehicle and the passenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This consideration holds for battery and BMS, as they both manage high voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Tampering may also lead to less severe consequences, such as the BMS being unable to communicate with the battery or to deliver the full power to the battery during charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These attacks may also include detaching or cutting cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As a possible countermeasure, the battery and BMS shall include an anomaly detection system to prevent applying a 7 TABLE I SUMMARY OF IN-VEHICLE CHALLENGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Attack Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Effect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Impact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Possible Solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='DoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent energy delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent information reception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Increase energy consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Physically damage the battery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Flow control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Time-lock puzzles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Rate limiting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tampering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Short circuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent energy delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Anomaly detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tamper-proof hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Malicious Code Injection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Modify BMS response to command ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Collect sensitive information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Remote attestation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Battery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='BMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Spoofing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Replaying,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and MitM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Report false information to the driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Report false information to the other ECUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Physically damage the battery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Disrupt charging process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Excessive discharging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Overcharging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Identity management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Redundancy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Timestamps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Integrity protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='MitM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Report false information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Isolate charger components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Modify control signals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Increase energy consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Anomaly detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion prevention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Integrity Protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='DoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent the exchange of energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Cookies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Time-lock puzzles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Rate limiting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Spoofing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Replaying Report false information Physical damage Increase energy consumption High Intrusion detection Identity management Timestamps Malicious Code Injection Modify EV response to commands Collect sensitive information Remote control/hijack High Authentication Remote attestation Intrusion detection Tampering Impair the charging process Power loss and overvoltage High Anomaly detection Tamper-proof hardware Controller and Charger Eavesdropping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Side Channels Track the user Profile users’ preferences Low Differential privacy Encryption voltage to tampered components and causing the aforemen- tioned damages [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Another solution may be the physical protection of these components with tamper-proof hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, in case of physical tampering, the battery should be designed so that it cannot receive or deliver power [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The SAE J2464 standard contains safety measures that can also be effective against tampering [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' c) Malicious Code Injection: The battery pack is man- aged by the BMS, which is a piece of hardware with firmware onboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Attackers may try to reverse engineer the software to discover vulnerabilities and build exploit against them [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To patch bugs, EVs’ software may be updated over the air or via the charging cable [39], thus easing the update process for manufacturers and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, this represents a security challenge, as software updates need to access the overall EV network [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A malicious user may inject malware via software update to gain control of the BMS [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' By having partial or complete control over it, the attacker may thus impact the normal functioning of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, the malware may prevent the BMS from requesting energy from the battery, causing a blackout in the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Contrarily, the BMS may be forced into requesting more energy than needed to speed up the discharging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, thanks to the malware, the attacker may measure other sensitive information of the driver, which may lead to privacy leakages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To prevent code injection and its effects, access to the EV’s internal network shall be strictly regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible solutions include the use of external source authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In case of a successful injection, it is fundamental to iden- tify and mitigate its effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To this aim, remote attestation and its collective extension may be used to validate the in- vehicle components [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, anomaly and intrusion detection techniques may help identify attacks to the in- vehicle network [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Injection of malicious updates can be detected by integrity verification on the new software, possibly employing a blockchain [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' d) Spoofing, Replaying, and Man-in-the-Middle: An at- tacker may spoof or modify messages to report to the driver 8 false information on the battery SoC, thus impairing a safe drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The attacker may also report incorrect information to the charging infrastructure by impersonating the BMS or modifying information in the middle of the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This may cause the charging process to provoke damage to the battery or the EV circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, an attacker may report false information to prevent the correct exchange of energy from the battery to the BMS, for instance, by lowering the current demand and preventing the exchange of a sufficient amount of power from the battery to the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' By requiring excessive power, an attacker can discharge the battery faster than expected in a battery exhaustion attack [64] or may force the battery to overcharge, leading to a massive shortening of the battery lifetime [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Finally, through Man- in-the-Middle (MitM), an attacker may modify the voltage values of the battery pack, leading to over-discharging and consequent battery degradation [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To prevent these attacks, the battery and BMS should be given an identity, and all messages shall provide source authen- tication and integrity protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The cryptographic material shall be embedded in these devices, with examples in trusted platform modules [66] or physical unclonable functions [67], and shall not be disclosed during communications to prevent MitM attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An intrusion detection system can help in identifying ongoing attack [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Redundant controllers can be employed to enhance the resilience of the BMS against adver- sarial attacks during charging [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [68] proposed to employ blockchain to provide authentication and access control in the communication between the BMS and the other devices inside the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Strategies to mitigate the effect of an attacker who has gained direct access to the vehicle’s bus have been proposed [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some works have considered peculiar features of EV to detect spoofing attacks: Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [69] proposed a physically-guided machine learning method to detect replay and false data injection attacks on the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Their system, tested in a Hardware-In-the-Loop (HIL) simulation testbed, could identify the attacks with an accuracy of more than 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Finally, to prevent replaying attacks, designers can consider the addition of timestamps to packets and signals transmitted [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Controller, Charger, and Electric Motors The controller and charger are fundamental elements that communicate with the BMS to exchange power to recharge the battery and to feed the EV components with energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The charger communicates with the EVSE to negotiate the parameters of the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Moreover, it manages the energy received and forward it to the battery pack according to the BMS requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If bidirectional charging is available to EV, the charger may also deliver energy from the EV battery to the charging column upon request [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The controller manages the energy delivered from the battery to the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some of these components are powered by the battery also in petrol-based vehicles, such as the infotainment system or the lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Others, such as the electric motors, are instead specific to EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The controller sends energy to them following the driver’s input, such as the torque pedal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This section discusses how an attacker may impair their correct functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' a) Man-in-the-Middle: Modifying the data in the bus may disrupt the regular operation of the charger since the control signals are usually transmitted through this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may isolate certain charger components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', the load relay), leading to a surge in the DC voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These attacks can damage the battery causing degradation in the performance and shortening the lifetime of the battery [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may also modify the signals managing the electric motors by adding noise or other mutations to the original signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This attack can damage the correct functionality of the motors and put the driver in dangerous situations [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Mitigation techniques can be applied using algorithms that can detect the attack in almost real-time by monitoring the physical properties of the vehicle, such as sensor data [13], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To make the receiver aware of possible MitM attacks targeting certain packets, integrity protection mechanisms must be in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, intrusion detection and prevention systems that monitor the data exchanged on the bus can also be implemented to strengthen the defense mechanism [64], [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' b) Denial of Service: The operations handled by the charger and controller heavily rely on the sensors reporting information on the charging status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A malicious user can generate a large number of requests to the sensors reporting data or overload the charger and control modules by flooding them with packets, thus preventing the receipt of legitimate messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If not properly handled, this attack may cause the controller to stop receiving correct state information, impairing the overall state control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Flow control may prevent controllers and motors from handling a large number of fake requests similarly to the BMS, for instance, by employing an adapted version of time- lock puzzles [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Source authentication might be employed to verify the sender’s lawfulness [53], while rate limiting can help mitigate against DoS attacks [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, intrusion detection can be adopted to identify ongoing DoS attacks [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' c) Spoofing, and Replaying: An attacker may spoof sensor identities to create multiple packets with legitimate identifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' By exploiting the same concept, an attacker may also report false information to the charger and con- troller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, the controller may take actions based on false data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This may cause damage to the hardware, possi- bly impairing the whole charging system [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' False data, if correctly crafted, may also impact the electric motors’ functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, they could force a stop of the motors by sending false control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Encryption can be a countermeasure to spoofing, preventing a malicious user from freely creating new packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, replay attacks can be employed to send correct sensor measurements or actuator updates previously recorded from the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may also spoof the information from the infotainment system, acceleration pedal, or other energy-hungry devices in the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The malicious entity may demand a power amount higher than the truly needed one, thus causing higher energy consumption and shortening the battery’s lifespan causing the driver to charge the EV frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The controller also handles the 9 information regarding acceleration and breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may spoof the related sensors to report false state changes to the electricity supplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, an attacker may spoof the gas pedal and prevent the receipt of the amount of power needed by the driver to speed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This may cause safety issues, for instance, when the driver needs to surpass another vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As already depicted, encryption can only prevent certain kinds of spoofing attacks, but it is insufficient to mitigate replay attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To prevent the latter, a combination of unique identifiers and timestamps can be adopted [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Identity man- agement may be another fundamental countermeasure against these threats [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, the controller and charger need to have, by design, access to the identities of all legitimate components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The identification of attacks is possible using intrusion and anomaly detection techniques [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' d) Malicious Code Injection: Similarly to the BMS case, controllers, chargers, and motors also contain software, which may often require updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, software updates repre- sent a security challenge since it needs access to the overall EV network [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A malicious user may force the installation of a malicious software update to gain control of some components of the EV’s internal network [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The attacker may thus impact the safety of the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, the malware may cause the EV to respond to the driver commands oppositely (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', decelerating while pushing on the gas pedal) and may also propagate to all the EV’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, thanks to the malware, the attacker may measure sensitive information on the driver (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', location) or profile the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A further threat is due to the implementation of controllers and ECUs via Field Programmable Gate Array (FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this case, an attacker may be able to inject malicious software into the central management system via communication lines by manipulating the FPGA controller [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Countermeasures are similar to the BMS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The access to the EV’s internal network shall be strictly regulated, for instance, using strong authentication mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Remote attestation and its collective extension may be used to validate the in-vehicle components [62], while blockchain can have a role in the verification of new software [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Finally, intrusion detection techniques may help the identification of ongoing attacks [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' e) Tampering: An attacker might physically tamper one of the controllers, charger, or motor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this case, for instance, the attacker may prevent the charger from cor- rectly detecting the presence of a power source (either wired or wireless), thus impairing the possibility of charging the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This is the case for proximity sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, the attacker may attach a shield to the pin on the EV side such that it cannot correctly communicate with the proximity pilot line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, an attacker may tamper with the power converter to degrade its quality, causing power losses or overvoltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To prevent physical tampering, controllers, chargers, and motors may implement anomaly detection frameworks to detect the application of a voltage in non-safe situations and react to the attack [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, these devices can be designed as tamper-proof, so they will stop functioning in case of tampering [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Although this may impair the vehicle’s functioning, it allows for safeguarding the user’s safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' f) Eavesdropping, and Side Channels: The current ex- changed during the charging process leaks features that can be exploited for user tracking and profiling [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may attach a module to the charger and controller to collect the current exchanged during the charging process and extract those features, thanks to the absence of encryption methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For the same reason, an attacker may also eavesdrop on the information exchanged between the controller, the charger, the motors, and the BMS to launch the attacks mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An adversary may also analyze the power exchanged between the controller and the infotainment system to obtain users’ sensitive information, such as preferences, habits, and passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This attack has been shown in other scenarios [77], such as smartphone charging, where users’ activities can also be detected in case of encrypted traffic [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, it is fundamental to include methods to prevent malicious users from accessing the communication among these entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To guarantee the user’s privacy, the current exchanged may be altered via a noisy signal that hides the original signal’s features similarly to differential privacy in other contexts [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' On the receiver side, the components shall be able to guarantee that the input current does not cause any damage to the circuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These solutions may hold for all the involved sources of current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Cryptographic methods may not always represent a viable solution, as they would add computational overhead to a possibly safety-critical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, they do not represent a solution to side channels, which are challenging to mitigate [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' EV CHARGING SECURITY AND PRIVACY CHALLENGES This section discusses the security and privacy issues related to the EV charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, we discuss the chal- lenges associated with wired charging in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Then, we discuss the challenges associated with WPT and WIPT in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For both technologies, we also discuss possible solutions and countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In Table V, we summarize all the security and privacy challenges associated with the charging process, their effects, impact severeness, and possible countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Wired Charging Challenges In the following, we focus on attacks targeting a wired charging scenario, which is the most common way at the moment of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some of the attacks are specific to cases where HLC is available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', MitM, spoofing), while others are suitable for every type of wired charging, such as tampering or side channel analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' a) Tampering Attacks: In this attack, a malicious user physically tampers with the devices involved in the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, an attacker might manipulate the pilot lines and tamper with the proximity sensor to prevent an EV from deeming a secure connection and hence prevent charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, this can also impact users’ safety, as it might be possible to detach the cable before removing the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' By observing electromagnetic leaks or operations in the chip components both in the EVSE and EV, an attacker might infer 10 TABLE II SUMMARY OF EV CHARGING CHALLENGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Effect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Impact ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Possible Solutions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tampering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent charging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Cause a shock to the driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Get sensitive information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tamper-proof hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Inconsistencies handler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Energy repudiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Cheat on billing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Steal energy from the system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Aggregate signature schemes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Blockchain for energy transactions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='DoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent EV charging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Disruption of the charging service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Medium ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Identity verification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='MitM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Prevent proper charging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Modify charging parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Integrity protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Spoofing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Replaying Create charging state inconsistencies Steal energy from another EV Medium Identity management Authentication Encryption Timestamps Relaying Steal energy from another EV Medium Distance bounding Fingerprinting Eavesdropping Steal sensitive EV information Medium Encryption Wired Charging Side Channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Information Leaking Track user Profile user’s preferences Low Differential privacy Secondary batteries Overpower Damage to EV battery High Energy-efficient overvoltage protection Anomaly detection Jamming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and DoS Prevent EV charging Medium Channel hopping Identity verification Authentication Intrusion detection Freeride attack Steal energy from the system Low Authentication Blockchain Energy repudiation Cheat on billing Steal energy from the system Low Aggregate signature schemes Blockchain for energy transactions Spoofing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Replaying Create charging state inconsistencies Steal energy from another EV Medium Authentication Encryption Physical layer authentication Timestamps Relaying Steal energy from another EV High Distance bounding MitM Prevent proper charging Modify charging parameters Medium Integrity protection Encryption Physical layer authentication Eavesdropping Steal sensitive EV information Medium Encryption WPT Side Channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Information Leaking Track user Profile users’ preferences Medium Differential privacy Secondary batteries sensitive information on the user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' such as private keys used for billing purposes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, by tampering with the charging cable, an attacker might prevent the proper charging of the victim EV or steal energy from an EV in charge by connecting additional cables [81], [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As possible countermeasures, tamper-proof hardware may represent a viable solution [59], [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Thanks to these devices, the attack may be limited to the car functioning without im- pacting the users’ safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Lastly, proper inconsistency handling mechanisms may be implemented to check that all involved components report the same physical status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' b) Energy Charging Repudiation: A malicious user may report to the EVSE that the EV’s battery did not receive any power by exploiting the behavior of the pilot line and the feedback associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this situation, the attacker may be able to charge a smaller amount compared to the amount of energy effectively used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If bidirectional charging is available, an attacker may pretend to have sold more energy than it has actually sold, thus stealing money from the energy provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A possible countermeasure to energy repudiation is the use blockchain technology to handle transactions and guarantee traceability and non-repudiation [84], [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Aggregate signa- ture schemes from different physical components can represent another possible mitigation to the problem [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' c) Denial of Charging: A malicious actor may try to prevent a vehicle from charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It may be done at the data level by modifying values on the packets exchanged during the handshake between the EV and EVSE [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In some cases, DoS can also be performed remotely, exploiting unshielded cables, which are often used for the recharge [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' DoS may also be launched against more than one vehicle, trying to compromise a portion of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A greedy attacker may falsify the information on the battery’s SoC, such that s/he can demand an energy amount higher than needed, thus preventing 11 other users from benefiting from the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The number of users that can simultaneously charge their EVs and the energy effectively delivered each moment depends on the grid’s capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If the grid capacity is limited, the attacker can successfully launch this attack and prevent other users from charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible countermeasures to DoS attacks include low- complexity authentication services in all the packets ex- changed such that the EVSE can rapidly decide whether to accept or discard a request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Identity-based traffic filtering may be combined with a physical state update related to the charge level of a certain user to prevent multiple malicious requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Intrusion detection can be employed to detect ongo- ing DoS attacks which may generate strange communication patterns [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, enforcing physical security by adopting shielded cables can prevent some kinds of DoS and eavesdropping attacks [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' d) Man-in-the-Middle: When operating charging modes employing HLCs, such as CHAdeMo or ISO 15118, the EV and EVSE exchange data through network packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A MitM attack can be employed to modify the content of this communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It may be a consequence of tampering if the malicious actor can insert a device on the pilot line between the vehicle and the charging column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In some cases, MitM can be performed from other charging columns attacking the SECC Discovery Protocol [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A malicious actor may exploit this channel to manipulate the exchanged information and create inconsistencies in the recharging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, an attacker who can modify packets on the fly may prevent proper charging by modifying request and response parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Further attacks can be launched starting from MitM, such as malware injections or DoS [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To identify modified data, integrity protection can be added to packets [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Another possible countermeasure is encryp- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Novel versions or ISO 15118 mandates the usage of Transport Layer Security (TLS) for all the communications between the vehicle and charging column, even if in real life, data are often exchanged in plaintext [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' e) Spoofing, and Replaying: An attacker might interact in the communication link between the vehicle and the charging column by injecting packets spoofing other devices’ identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, a malicious user can spoof the identity of an ECU and report false information on the battery SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, an attacker may inject false information by spoofing the identity of an EVSE and stealing sensitive information from an EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For example, in the case of automatic billing based on the EV features, a malicious user can extrapolate those features from an EV and store them for later use to bill the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The same concept can also be applied to other types of connectors, as long as billing is based on automatic feature recognition [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible countermeasures to these attacks include using a proper identity management scheme, authentication, and data encryption [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Authentication systems shall include infor- mation related to the charging status of the EV or the energy delivered by the EVSE to help guarantee the consistency between the reported information and the actual physical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It is important to consider that encryption cannot prevent the replaying of packets, which may instead be enforced with unique identifiers and timestamps [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' f) Relaying: A relay attack is possible if an attacker has access to the network traffic and can relay it to a nearby charging column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' By relaying information, a malicious user can manipulate the billing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, a malicious user can relay the data between two neighboring EVSEs to bill a closely-located victim user for a charging session [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If bidirectional charging is available, a malicious user can sell the energy of a victim’s vehicle and get paid for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The location information of EV and EVSE may be exploited to prevent relay attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', employing distance bounding protocols [92], [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, the physical features of the EV may be exploited to design dedicated authentication protocols [93], [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' g) Eavesdropping: An attacker may be able to read the information exchanged between the vehicle and charging columns in different ways, similarly to what was presented before for MitM attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' With access to all the network traffic, a malicious entity can steal sensitive information from the user, from simple charging parameters to credit card numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To protect against eavesdropping, encryption can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' As already explained, novel versions of ISO 15118 mandate the usage of TLS for all the communications between the vehicle and the charging column, even if real-life data are still often exchanged in plaintext [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It is important to recall that even if the exchanged data are encrypted, side channel analysis is possible to extract some users’ preferences, as presented in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' h) Side Channels, and Information Leakages: An at- tacker in control of an EVSE may be able to track and profile users who authenticate to the EVSE even if data are encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It may rely on different information, such as the MAC address of the EV or the certificate employed by Plug and Charge [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, in Level 1 and Level 2 charging, these kinds of data are unavailable since no HLC is generated between the two entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In that case, an attacker may rely on other features, such as the exact voltage of the control pilot pin or the duration of the handshake at the beginning of the charging process [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Another side channel that may transfer information is the effective current exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This does not convey information in a network sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', it does not involve the creation of packets with the sender’s and receiver’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, no encryption method is applied to this signal, which is transmitted in plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, it has been shown that it is possible to profile users by extracting features from the charging current [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In particular, the charging current contains features peculiar to each EV, allowing for EV tracking and user profiling based on the current demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, it is fundamental to manipulate the current signal to prevent these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Countermeasures to privacy threats shall not undermine the efficiency of the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, possible solutions must allow the involved parties to retrieve sufficient information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', to the SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Differential privacy methods may represent a viable solution [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An alternative is represented by the use of secondary batteries to create a connection between the EV and EVSE, similarly to what was discussed 12 in [75], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' When HLC is available, MAC address random- ization may represent a good mitigation technique to reduce the profiling power of an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' WPT Challenges Due to the exposure of the wireless medium, WPT incurs in a large number of safety, security, and privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, it is likely that WPT signals to impact more vehicles and that an attacker gets access to the signals or information wirelessly exchanged [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this section, we review and extend the taxonomy of the possible attacks to WPT presented in [96] and adapted it to the EV case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' a) Overpower attack: The wireless medium’s intrinsic vulnerability makes it possible that a single EV receives both its signal and the signal intended for another vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, if two cars are closely located, and both are charging their batteries via WPT, they will receive more power than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This is even more likely when considering fully-dynamic WPT, where vehicles move and cannot hence guarantee that a reasonable safety space is kept between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The excessive received power might harm some components of the BMS or the battery if a proper overvoltage regulator is not deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, an attacker might exploit this concept to launch an overvoltage attack to damage the EV’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible countermeasures include implementing overvoltage protection mechanisms at the EV’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Such mechanisms shall, however, guarantee the efficiency of the charging process to avoid requiring excessive charging times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Anomaly detec- tion methods can also be applied to detect the reception of abnormal power values or other anomalies in the charging pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Design choices can help mitigate overpower attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, the distance between coils must be designed to make overpower attack unfeasible or, at least, more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' b) Jamming, and Denial of Service: In the case of WIPT, the reception of multiple signals might cause exces- sive interference at the receiver’s side, thus preventing the correct reception of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Due to the openness of the WPT medium, an attacker might be able to simultaneously jam multiple EVs by sending random WIPT messages and degrading the channel quality up to the point where messages are not correctly received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, this concept can be exploited to prevent a successful charging negotiation phase, thus preventing a connection between the EV and the charging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This represents a DoS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Similarly, an attacker may launch a jamming attack against the charging column’s WiFi access point, preventing legitimate users from connecting and using the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' An attacker may also target a portion of the energy grid by continuously sending charging requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If many users engage in this session, they might prevent other users from benefiting from the service availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Although feedback mechanisms to report on the SoC of the receiver might be implemented to automatically detach an EV when fully charged, an expert attacker might be able to craft feedback packets to avoid showing full battery’s SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible solutions include frequency hopping mechanisms, where channels are selected according to different strate- gies to avoid using a channel under jamming attack [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Low-complexity authentication services in all the packets exchanged such that the EVSE can rapidly decide whether to accept or discard a request can help in preventing DoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To detect a DoS attack, intrusion detection systems can be deployed [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Identity-based traffic filtering may be combined with a physical state update related to the charge level of a specific user to prevent multiple malicious requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' c) Freeriding attack: As previously mentioned, a user might connect to public infrastructure and pay for charging via WPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Due to the openness of the WPT medium, a malicious user could exploit the proximity to a vehicle in charge to steal energy and charge his/her EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A similar scenario envisions the collusion of multiple EV owners when a single one registers for the service and multiple users share the bill and benefit from the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These attacks are feasible in all types of dynamic WPT models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' the only requirement is a short inter-EV distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This attack is challenging to detect, as it does not impact the legitimate channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, although a second EV might be connected to the charging channel, the main channel will not face any performance degradation, thus making it unfeasible to detect the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' WPT sessions need to be authenticated to prevent other users from benefiting from a charging session they are not pay- ing for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, authentication procedures might include the physical features of the involved devices and the amount of power transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The blockchain solutions proposed by Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [98] may be adapted to the EV case to guarantee security against this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' d) Energy repudiation: WPT is less efficient compared to its wired counterpart, as the wireless medium is characterized by losses due to both attenuation and the relative position of the transmitter and receiver devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, part of the transmitted energy may be lost during the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A fair system requires that users pay for the actually received energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore the billing system needs to compare the transmitted power with the received one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, this might create security issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, a malicious user might continu- ously report a received power value smaller than the true one or report zero received energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This is commonly known as a repudiation attack, where the user denies benefiting from a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To guarantee the correctness of the reported power us- age information, possible solutions might include the use of aggregate signature schemes from different physical compo- nents [86] or the blockchain technology [85], [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' e) Spoofing, and Replaying: A malicious user who knows the standard employed or which is able to eavesdrop on the communication can easily craft malicious packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Based on the crafted information, this class of attacks may have different impacts on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, an attacker may use the identifier of another vehicle to negotiate a charging session that the victim will pay for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, a malicious actor can craft packets declaring weird SoC and spoof other vehicles’ identifiers to create inconsistencies in the charging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The use of authentication and integrity protection mecha- nisms can be effective countermeasures against spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this context, using physical layer authentication may help in 13 designing suitable protocols [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In the context of WPT, the transmission frequency can be regulated to encrypt information and guarantee that only the legitimate party can receive power [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This also represents a possible solution to the attacks aforementioned in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Finally, timestamps can be added to identify multiple sending of the same packet in a replaying attack [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' f) Eavesdropping: Due to the exposure of the wireless medium, an attacker may easily intercept WPT packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' These packets may contain different types of information, such as the vehicle identifier, SoC information, or billing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible solutions include the use of cryptographic tech- niques to hide information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The newest release of ISO- 15118 [36] mandates TLS on every communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' g) Relaying: An attacker may relay information from a victim vehicle to the access point of the attacker’s charging column to steal energy [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This kind of attack work even if the traffic is encrypted since the data is only relayed and not modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' With respect to the wired counterpart, where the attacker has to tamper in some way with the charging column, a wireless relay attack does not need any hardware modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To protect against relay attacks, a distance bounding proto- col can be employed to assess if a malicious entity is relaying the network flow [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' h) Man-in-the-Middle attack: With respect to wireless charging, when dealing with WPT, the interception and for- warding of communication flow are easier due to the openness of the medium [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' At the same time, directional jamming can be employed in some cases to prevent the receiver from getting both the original and the modified data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' If the communication flow is unencrypted or the cryptography is weak, an attacker can launch a MitM attack to modify on- the-fly packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, a malicious user might modify the information sent by the victim (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', report full SoC) after establishing a connection with the service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To perform such an attack, a malicious entity may set up a fake access point and use it to relay the communication to the legitimate one, gaining the ability to modify packets at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Partial solutions include the previously mentioned solutions, such as encryption, authentication, and integrity protection mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In the context of WPT, the transmission fre- quency can be regulated to encrypt information and guarantee that only the legitimate party can receive power [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Further- more, physical layer authentication may enhance the security of the authentication process [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' i) Side Channels, and Information Leaking: Although WPT signals might be encrypted or avoid sensitive reporting information on the user, the power signal can be exploited for profiling purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This attack has been proven feasible for smartphones, where the WPT signal analysis reveals informa- tion on the user’s activity [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This might also be the case for EVs, where an attacker can infer different types of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This attack is similar to the profiling performed in the wired case, where it might be possible to track a user and obtain information on her habits and power demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Preventing this attack represents a challenging task, as it cannot be detected, and data encryption is not sufficient [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A possible solution is represented by differential privacy, where data is corrupted with a noisy pattern that might prevent inferring sensitive users’ data [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, as for the wired counterpart, secondary batteries may prevent the attacker from inferring sensitive users’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' FUTURE DIRECTIONS Looking at the impact of the different attacks in Tables IV and V, we can conclude that many security issues related to the cyber-physical nature of EVs may impact the safety of the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We notice that some of the attacks and countermeasures discussed can also be applied to other EV assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, due to space limitations and to avoid being repetitive, we only discussed those we considered to be the most interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Nevertheless, we summarize all the attacks and countermea- sures in the comprehensive Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Although many threats concern the charging infrastructure, the most severe in terms of safety are related to the in-vehicle network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, the electric component of EVs may be tampered with or impaired to electroshock the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Furthermore, the increasing cyber nature of the EVs’ components leads to challenges regarding the coherency of the information coming from the cyber and the physical worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Lastly, the increasing interest in the application of the WPT technology to EVs impose significant challenges that still need to be properly addressed from a CPS point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Based on our analysis, we foresee the following future directions and needs in the field of EV CPS security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Denial of Service (DoS) represents one of the most chal- lenging threats in EVs security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' It is known as difficult to prevent, and almost every component of the EV can suffer from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Compromised internal components can attack other ECU to compromise the in-vehicle network, but DoS attacks can be launched from charging columns to EV during charging, or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In certain cases, DoS can have an impact not only on a single vehicle but can compromise EVSEs in a certain geographical area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To mitigate this risk, not only do all the vehicle entities need to be associated with an identity, but their allowed flow of information (and hence generated traffic) should depend on the vehicle’s physical situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, it might be that a specific ECU needs to send messages at a higher rate when the vehicle is experiencing certain physical stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' At the same time, all ECUs shall be guaranteed a sufficient amount of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Therefore, future protections against DoS for in-vehicle networks should account for the physical factors and the possible impact on the whole electric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' The potential tampering with the EV components might represent a significant threat to the user’s safety and may also have repercussions on other elements of the vehicle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' All vehicle components shall be equipped with anomaly detection capabilities or should prevent the application of a voltage or current flow in case of tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Possible future solutions might include the collective verification of multiple components to make tampering with a single unit ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 14 TABLE III SUMMARY TABLE WITH ATTACKS AND COUNTERMEASURES FOR EACH ASSET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' THE FIRST ROW INDICATE THE ASSETS INTERESTED BY EACH ATTACK, WHILE THE FOLLOWING ROWS POINT OUT WHICH COUNTERMEASURE IS EFFECTIVE AGAINST EACH ATTACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' � : BMS AND BATTERY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' � : CONTROLLERS, CHARGER AND MOTORS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' � : WIRED CHARGING;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' AND �: WPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='DoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tampering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='MitM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Replaying ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Spoofing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Malware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Overpower ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Freeride ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Jamming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Repudiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Eavesdropping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Side-Channels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Relaying ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Affected assets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Aggregate signature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Anomaly detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Authentication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Blockchain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Channel hopping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Cookies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Differential privacy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Distance bounding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Fingerprinting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Flow control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Intrusion detection/prevention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Identity management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Identity verification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Inconsistencies handler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Integrity protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Overvoltage protection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Physical layer security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Rate limiting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Redundancy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Remote Attestation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Secondary battery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Tamper-proof hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Time-lock puzzles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Timestamps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='� � � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Countermeasure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='The increasing attackers’ capabilities impose additional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='challenges in guaranteeing the cyber-security of EVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In fact, an attacker might be able to combine multiple attacks to impair the EV functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To strengthen the defense mechanisms, it is essential to implement in EVs frameworks collecting information from multiple sources, combining the cyber and the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, verifying the message integrity might employ data from different sensors and actuators to increase the difficulty of information manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Similarly, intrusion detec- tion techniques might combine network data exchanged through the bus with physical signals from sensors to model better the state of the EV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' This will also help prevent attacks related to malicious ECUs controlling ac- tuators for mechanical operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', steering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Future work should consider the EV-specific components such as the battery or the charger as data sources regarding the vehicle’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' For instance, the charge and discharge curves of batteries can be modeled by computers with discrete confidence [105], [106], [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' A simple applica- tion of these simulations is a reference to identify packets declaring modified SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' One of the strengths of EVs compared to previous genera- tions of vehicles relies on the software managing all their operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, this implies that vehicles are more subject to cybersecurity attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some of these attacks may include malware injection into some of the EV’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' To this aim, collective remote attestation can be used to verify the integrity of all the EV’s components and prevent possible safety threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Remote attestation measures should, however, account for the resource- limited nature of EVs’ components and the time-critical nature of the exchanged information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' WPT is one of the promising technological solutions to alleviate the range anxiety of drivers fearing not reaching their destination with the available charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Thanks to the charging while driving paradigm, EVs can be charged during their operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' However, deploying the required public infrastructure poses many security challenges both to the operators and the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Some examples include the billing process and the openness of the wireless medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' WPT related challenges heavily rely on the cyber-physical nature of the overall infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' There- fore, security solutions in this area should account for the coherency of information from the cyber and the physical domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' CONCLUSION The increasing market for EVs demands an in-depth anal- ysis of EV technology’s security and privacy challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this paper, we provided an overview of the components of an EV, focusing on their characteristic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provided the basic information needed to understand how in-vehicle communication networks work and which devices need to communicate with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We then discussed how an EV battery could be charged via wire and WPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We provided the information needed to understand both technologies and discussed the different implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We also provided the 15 security and privacy issues of in-vehicle communications and those related to the charging infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Focusing on a CPS perspective, we discussed how different attacks might impact both the user and the system’s security and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We then discussed possible countermeasures and proposed some future direction to improve the overall EV ecosystem security and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' We conclude that the EV technology currently presents a large attack surface that 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' San Mart´ın, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Zamora, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Asensio, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' O˜nederra, “Analysis of the current electric battery models for electric vehicle simulation,” Energies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 2750, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' [107] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Chen and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Rincon-Mora, “Accurate electrical battery model capable of predicting runtime and i-v performance,” IEEE Transactions on Energy Conversion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 504–511, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Alessandro Brighente is assistant professor at the University of Padova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He was visiting researcher at Nokia Bell Labs, Stuttgart, Germany in 2019 and University of Washington, Seattle, USA, in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He served as TPC for several conferences, including Globecom, VTC, and WWW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is guest editor for IEEE Transactions on Industrial Informatics and program chair of DevSecOpsRA, co-located with EuroS&P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His current research interests include se- curity and privacy in cyber-physical systems, ve- hicular networks, blockchain, and communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Mauro Conti is Full Professor at the University of Padua, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is also affiliated with TU Delft and University of Washington, Seattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He obtained his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' from Sapienza University of Rome, Italy, in 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' After his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=', he was a Post-Doc Researcher at Vrije Universiteit Amsterdam, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In 2011 he joined as Assistant Professor the Uni- versity of Padua, where he became Associate Pro- fessor in 2015, and Full Professor in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has been Visiting Researcher at GMU, UCLA, UCI, TU Darmstadt, UF, and FIU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has been awarded with a Marie Curie Fellowship (2012) by the European Commission, and with a Fellowship by the German DAAD (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His research is also funded by companies, including Cisco, Intel, and Huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His main research interest is in the area of Security and Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In this area, he published more than 400 papers in topmost international peer-reviewed journals and conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is Area Editor-in-Chief for IEEE Communications Surveys & Tutorials, and has been Associate Editor for several journals, including IEEE Communications Surveys & Tutorials, IEEE Transactions on Dependable and Secure Computing, IEEE Transactions on Information Forensics and Security, and IEEE Transactions on Network and Service Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He was Program Chair for TRUST 2015, ICISS 2016, WiSec 2017, ACNS 2020, and General Chair for SecureComm 2012, SACMAT 2013, CANS 2021, and ACNS 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is Senior Member of the IEEE and ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is a member of the Blockchain Expert Panel of the Italian Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is Fellow of the Young Academy of Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Denis Donadel received his MSc in Telecommu- nication Engineering from the University of Padua, Italy, in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is now a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Student in Brain, Mind and Computer Science (BMCS) at the Univer- sity of Padua where he joined the SPRITZ Security and Privacy Research Group under the supervision of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Mauro Conti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Together with his academic course, Denis is also working with Omitech SRL as part of his high apprenticeship program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' During the 2021 Summer, he was granted the New Generation Internet (NGI) Explorers grant to support a collabo- ration with the University of Washington (Seattle, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His research interests lie primarily in Cyber-Physical Systems security, focusing particularly on Vehicles Security and Critical Infrastructures Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' 18 Radha Poovendran is Professor of the Department of Electrical & Computer Engineering at the Univer- sity of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is the founding director of the Network Security Lab and is a founding member and associate director of research for the UW’s Center for Excellence in Information Assurance Research and Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has also been a member of the advisory boards for Information Security Educa- tion and Networking Education Outreach at UW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' In collaboration with NSF, he served as the chair and principal investigator for a Visioning Workshop on Smart and Connected Communities Research and Education in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Poovendran’s research focuses on wireless and sensor network security, adversarial modeling, privacy and anonymity in public wireless networks and cyber-physical systems security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He co-authored a book titled Submodularity in Dynamics and Control of Networked Systems and co-edited a book titled Secure Localization and Time Synchronization in Wireless Ad Hoc and Sensor Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Poovendran is a Fellow of IEEE and has received various awards including Distinguished Alumni Award, ECE Department, University of Maryland, College Park, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' NSA LUCITE Rising Star 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' NSF CAREER 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ARO YIP 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' ONR YIP 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' PECASE 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' and Kavli Fellow of the National Academy of Sciences 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Federico Turrin received the Master’s Degree in Computer Engineering from the University of Padova, Italy, in 2019, where he is currently pur- suing the interdisciplinary Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' in Brain, Mind, and Computer science, since October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has been visiting researcher at SUTD Singapore in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His research interests lie primarily in Cyber-Physical System Security with a particular focus on Industrial Control Systems Security, Vehicles Security, and Anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' Jianying Zhou is a professor and co-center director for iTrust at Singapore University of Technology and Design (SUTD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He received PhD in Infor- mation Security from Royal Holloway, University of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' His research interests are in applied cryptography and network security, cyber-physical system security, mobile and wireless security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has published 300 referred papers at international conferences and journals with 13,000 citations, and received ESORICS’15 best paper award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has 2 technologies being standardized in ISO/IEC 29192-4 and ISO/IEC 20009-4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is a co-founder & steering committee co-chair of ACNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is also steering committee chair of ACM AsiaCCS, and steering committee member of Asiacrypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has served 200 times in international cyber security conference committees (ACM CCS & AsiaCCS, IEEE CSF, ESORICS, RAID, ACNS, Asiacrypt, FC, PKC etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=') as general chair, program chair, and PC member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He has also been in the editorial board of top cyber security journals including IEEE Security & Privacy, IEEE TDSC, IEEE TIFS, Computers & Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He is an ACM Distinguished Member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} +page_content=' He received the ESORICS Outstanding Contribution Award in 2020, in recognition of contributions to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE3T4oBgHgl3EQfjQqW/content/2301.04587v1.pdf'} diff --git a/otE1T4oBgHgl3EQfiAQh/content/tmp_files/2301.03246v1.pdf.txt b/otE1T4oBgHgl3EQfiAQh/content/tmp_files/2301.03246v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc92299bfe92907c3f979f38a153dc4bd84df030 --- /dev/null +++ b/otE1T4oBgHgl3EQfiAQh/content/tmp_files/2301.03246v1.pdf.txt @@ -0,0 +1,3029 @@ +An instrumental variable method for point processes: +generalised Wald estimation based on deconvolution +Zhichao Jiang∗§ +Shizhe Chen†§ +Peng Ding‡ +January 10, 2023 +Abstract +Point processes are probabilistic tools for modeling event data. While there exists a fast- +growing literature studying the relationships between point processes, it remains unexplored how +such relationships connect to causal effects. In the presence of unmeasured confounders, param- +eters from point process models do not necessarily have causal interpretations. We propose an +instrumental variable method for causal inference with point process treatment and outcome. We +define causal quantities based on potential outcomes and establish nonparametric identification +results with a binary instrumental variable. We extend the traditional Wald estimation to deal +with point process treatment and outcome, showing that it should be performed after a Fourier +transform of the intention-to-treat effects on the treatment and outcome and thus takes the form +of deconvolution. We term this as the generalised Wald estimation and propose an estimation +strategy based on well-established deconvolution methods. +Keywords: Causal inference, Identification, Intensity, Principal stratification, Unmeasured con- +founding +∗School +of +Mathematics, +Sun +Yat-sen +University, +Guangzhou, +Guangdong +510275, +China. +Email: +jiangzhch7@mail.sysu.edu.cn +†Department of Statistics, University of California, Davis, California 95616, U.S.A. Email: szdchen@ucdavis.edu +‡Department of Statistics, University of California, Berkeley, California 94720, U.S.A. Email: +pengding- +pku@berkeley.edu +§Equal contribution +arXiv:2301.03246v1 [stat.ME] 9 Jan 2023 + +1 +Introduction +Point processes have long been used for modeling event data. The past decade has witnessed a surge +of interest in point process models in many fields including neuroscience, finance, and social sciences. +In this paper, we consider the analysis of neural data as a concrete motivation. Modern technologies +allow neuroscientists to simultaneously record neural spike trains, i.e., arrays of timestamps when +neurons fire, across the brain. With these data, one can hope to peek into the mechanisms of neural +computing. The nature of these scientific questions is the inference of causal effects. +Current technologies, however, present a major challenge for causal inference with neural data. +Except for experiments on very simple animals, even state-of-the-art technologies can record only +a very small fraction of neurons in chosen regions in the nerve systems, leaving the vast majority +unobserved. The unmeasured neural activities inevitably lead to the issue of unmeasured confound- +ing; that is, unmeasured activities might be the common causes of observed neural activities. As a +result, any relationship inferred based on the partially observed system might not reflect the true +causal relationship, but rather a spurious association. +Fortunately, advances in optogenetics create new opportunities to address unmeasured confound- +ing effects. Neuroscientists are able to instigate neural activities in a living brain via optical stim- +ulation, which alters the activity of any chosen neuron with high spatial and temporal precision +(Mardinly et al., 2018; Carrillo-Reid et al., 2019). From a causal inference perspective, such inter- +ventions can serve as instrumental variables for inferring the causal relationship between neurons, +as they affect the outcome neuron only through the treatment neuron while introducing exogenous +variation in the treatment neuron. +Instrumental variable methods are powerful tools for inferring causal effects in the presence of +unmeasured confounding between the treatment and the outcome. +In a seminal paper, Angrist +et al. (1996) clarify the role of a binary instrumental variable in identifying the causal effect of a +binary treatment for an unmeasured subgroup, known as the complier average causal effect. They +propose two crucial identification assumptions, monotonicity and exclusion restriction. Under these +assumptions, they show that the complier average causal effect is identified by the Wald estimator +(Wald, 1940; Ridder and Moffitt, 2007) that equals the ratio of the differences in means of the +outcome and the treatment when the instrumental variable changes from 0 to 1. +With an instrumental variable, most existing work considers non-dynamic settings and the instru- +mental variable methods in survival analysis mainly focus on a scalar treatment and a non-recurrent +outcome (e.g., Li et al., 2015; Martinussen et al., 2017; Richardson et al., 2017; Jiang et al., 2018). +1 + +To the best of our knowledge, there is no formal instrumental variable framework for point processes +that addresses nonparametric identification. +We propose an instrumental variable method for causal inference when both the treatment and +the outcome take the form of point processes. We define several causal quantities for the effect of the +treatment on the outcome over time. Using a binary instrumental variable, we establish the nonpara- +metric identification of causal effects allowing for the unmeasured treatment-outcome confounding. +The identification assumptions hold as long as the impact of the unmeasured confounders on the +outcome is additive. Our identification result implies that the causal effects can be obtained by solv- +ing a convolution equation. This extends the Wald estimation in traditional instrumental variable +method to take the form of deconvolution, leading to the proposed generalised Wald estimation. +We also examine several commonly-used models under our framework, studying the identification +of the causal effects and the causal interpretation of the model parameters with a binary instru- +mental variable. When the unmeasured confounders are additive on the outcome, the causal effects +are identifiable without any distributional assumptions on the confounders based on the proposed +generalised Wald estimation. Our finding justifies the identifiability of many commonly-used models +such as the Hawkes process, broadening their applicability with fewer assumptions. +We use the following notation. Let A +B | C denote the conditional independence of A and B +given C. Let R denote the set of real numbers and B(R) denote the Borel σ-algebra of the whole +real line. Let L1(R) denote the set of functions f(x) such that +� ∞ +−∞ |f(x)|dx < ∞. Unless specified +otherwise, we assume all functions used in this paper belong to L1(R). Let Ψ denote the Fourier +transform, i.e., for any f(x) ∈ L1(R) and ν ∈ R, define +(Ψf)(ν) = +� ∞ +−∞ +f(x)e−i2πνxdx, +where i = √−1. Let Ψ−1 denote the inverse Fourier transform. +2 +An instrumental variable framework for point processes +2.1 +A brief review of the binary instrumental variable model +We begin by reviewing the binary instrumental variable model in the context of noncompliance +(Angrist et al., 1996). For unit i, let Zi be the binary treatment assigned, Ni the actual treatment +received, and Yi the outcome of interest. Let Niz be the potential value of the treatment receipt +if the assigned treatment condition is z, Yizn the potential value of the outcome if the assigned +treatment is z and the actually received treatment is n. The joint values of Ni1 and Ni0 define the +2 + +unmeasured compliance type Ui = (Ni1, Ni0). Units with (Ni1 = 1, Ni0 = 0) are compliers who take +the treatment assigned, units with (Ni1 = 1, Ni0 = 1) are always-takers who always take treatment +1, units with (Ni1 = 0, Ni0 = 0) are never-takers who always take treatment 0, and units with +(Ni1 = 0, Ni0 = 1) are defiers who take the treatment opposite to the assigned. +Angrist et al. (1996) invoke three assumptions: (1) exclusion restriction that the treatment +assigned affects the outcome only through the treatment received, i.e., Yizn = Yiz′n for all z, z′, n; +(2) randomization that Zi is independent of Niz and Yizn for z, n = 0, 1; (3) monotonicity that the +assigned treatment does not negatively affect the treatment receipt for all units, i.e., Ni1 ≥ Ni0. +Exclusion restriction simplifies Yizn to Yin. Randomization rules out the confounding between the +treatment assignment and the treatment receipt as well as the confounding between the treatment +assignment and the outcome. Monotonicity rules out defiers. Under these assumptions, Angrist et al. +(1996) introduce the complier average causal effect as the average effect of the treatment receipt on +the outcome for compliers, CACE = E(Yi1 − Yi0 | Ni1 = 1, Ni0 = 0), and show that it is identified +by +CACE = +E(Yi | Zi = 1) − E(Yi | Zi = 0) +E(Ni | Zi = 1) − E(Ni | Zi = 0). +(1) +In this model, the treatment assignment Zi is the instrumental variable. +The expression in (1) +suggests the Wald estimator (Wald, 1940) for the CACE, i.e., the ratio of the differences in means +of the outcome and the treatment receipt when the treatment assigned changes from 0 to 1. +Angrist et al. (1996) identify only the treatment effect in the complier subpopulation. +For +extrapolation to the whole population, we can invoke the homogeneity assumption (cf., Heckman, +1996; Chen et al., 2009) that the treatment effect is the same across compliance groups: +E(Yi1 − Yi0 | Ni1, Ni0) = E(Yi1 − Yi0). +(2) +Under the assumption in (2), the treatment effect in the whole population equals the CACE. +2.2 +Notation and basic assumptions with a point process treatment +We now consider the setting when both the treatment Ni and the outcome Yi are point processes. +We will establish a similar ratio relationship as in (1) for the point process treatment and outcome, +but in the frequency domain. As a concrete example, we consider the neuroscience application from +Bolding and Franks (2018a). In this application, the treatment and the outcome are the neural +activities of the mouse olfactory bulb and piriform cortex within each brain region, respectively. In +experiments at the single-cell resolution, one can also model single-neuron activities as the treatment +3 + +and outcome. As shown in Figure 1(a) and (b), these data take the form of spike trains that are +commonly modeled as point processes. Bolding and Franks (2018a) apply light pulses to randomly +selected trials to stimulate the olfactory bulb without affecting other brain regions. +Therefore, +the light pulse serves as an instrumental variable. To formally discuss causal inference, we need to +generalise the model in Angrist et al. (1996) to account for the point process treatment and outcome. +Ni +Zi +Ui +Yi +(d) +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +N (Z=1) +Y (Z=1) +N (Z=0) +Y (Z=0) +(c) +Time (s) +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(a) +Time (s) +Trials +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(b) +Time (s) +Trials +Figure 1: Neural data from Bolding and Franks (2018a) and the causal diagram depicting the +relationships among the variables in the instrumental variable framework. Panel (a) shows the spike +trains collected in the mouse olfactory bulb in the stimulated (in blue) and unstimulated (in red) +trials. Each row represents a spike train in the olfactory bulb in one trial (Ni). The shaded area +depicts the duration of the light pulse. Panel (b) shows the spike trains collected in the mouse +piriform cortex in the stimulated (in blue) and unstimulated (in red) trials. Each row represents a +spike train in the piriform cortex in one trial (Yi). Panel (c) zooms in on two randomly selected +trials, where the solid curves are the smoothed intensities. Panel (d) shows the causal diagram for +the relationship among the variables. Variables in the solid square and circles are observed, and the +variable in the dashed circle is unobserved. The subscript i represents the ith unit. +Let i = 1, . . . , m index the units. We use point processes to describe the neuron activities (see, +e.g., Chapter 2 in Cox and Isham, 1980 or Chapter 3 in Daley and Vere-Jones, 2003). Define the +treatment point process Ni(·) as a family of random non-negative integers {Ni(A)}A∈B(R) counting +4 + +the number of events in each set A. +Let dNi(t) ≡ Ni([t, t + dt)). +Throughout this paper, we +consider point processes that are simple, Pr{dNi(t) = 0 or 1 for all t} = 1, and with bounded +intensity Pr{dNi(t) = 1}/dt < ∞. In a similar manner, we introduce the outcome point process +Yi(·). We focus on a binary instrumental variable, Zi ∈ {0, 1}, and present the results on a discrete +instrumental variable in §S1 in the Supplementary Material. Without loss of generality, we assume +the instrumental variable onsets at time 0, and Ni(·) and Yi(·) are observed from 0 to T. To avoid +cumbersome bookkeeping, we constrain the processes to the observed period and ignore the history +before time 0, and we write Ni([0, t]) as Ni(t). +For ease of discussion, we first consider the treatment Ni(·) being a point process with at most +one event. We refer to Ni(·) as a single-point process if Ni(T) ≤ 1. We will extend the methodology +to a general point process in §3.2. We can characterize a single-point process Ni(·) using its event +time: define Ti = T + if Ni(T) = 0 and Ti ≡ τ if Ni(t) = 1 for t ≥ τ and Ni(t) = 0 for t < τ. +We adopt the potential outcomes framework under the following stable unit treatment value +assumption (Rubin, 1980). +Assumption 1 There is no interference between units and there are no different versions of the +instrument and the treatment process. +Assumption 1 rules out the spillover effect of other units’ instrumental variable on one’s treatment +process and that of other units’ instrumental variable and treatment process on one’s outcome +process. It also requires that there is only one version of the instrument and the treatment process. +In our motivating example, one unit corresponds to one trial, and trials conducted at different +times might use the same mouse. The no-interference assumption would be violated if the neural +dynamics of a mouse adapt to stimulation over time, causing activities in one trial to depend on +previous trials. This phenomenon is known as neural plasticity. To restrict spillover between trials, +adequate washout periods are incorporated to separate trials sufficiently apart. As a result, we can +reasonably assume away spillover effects. Furthermore, uniform stimulation is employed to ensure +that there is only one version of the instrument. For the treatment process, we follow the common +practice in neural data analysis to focus on the effect of the timings of spikes, ignoring the variation +in the spike intensities (cf., Brillinger, 1988; Yu et al., 2009; Zhao and Park, 2017; Wu et al., 2017). +Assumption 1 allows us to define the potential values as the function of a unit’s own instrument +and treatment process. Let Niz(·) and Yiz(·) be the potential processes of the treatment and outcome, +and Tiz be the potential event time of the treatment process if the instrumental variable were set to +5 + +Zi = z. Also, define Yizτ(·) as the potential process of the outcome if the instrumental variable were +set to Zi = z and the event time were set to Ti = τ. By definition, the two versions of the potential +outcome process satisfy Yiz(·) = Yiz,Tiz(·). The observed treatment process is Ni(·) = ZiNi1(·)+(1− +Zi)Ni0(·), and the observed outcome process can be written as Yi(·) = ZiYi1(·) + (1 − Zi)Yi0(·) or +Yi(·) = ZiYi1τ(·)+(1−Zi)Yi0τ(·) if Ti = τ. We assume {Zi, Niz(·), Yizτ(·) : z = 0, 1; τ ∈ [0, T]∪T +}m +i=1 +are independently and identically distributed, and thus the observables {Zi, Ni(·), Yi(·)}m +i=1 are also +independently and identically distributed. We simplify Ni(·) as Ni and Yi(·) as Yi when no confusion +arises. In our motivating example, the experiment is carefully designed to ensure that the trials +are independent and identically distributed. +For instance, the optical stimulation is targeted at +the same location at the same chosen power to eliminate unintentional variability to ensure the +identical distribution condition; the trials are separated with adequate washout periods to ensure +the independence between units; the power of optical stimulation and the length of the experiment +are limited to avoid physical damage to the neural circuits. +Under Assumption 1, we impose the following three assumptions throughout the paper. First, +we generalise the exclusion restriction assumption in Angrist et al. (1996). +Assumption 2 (Exclusion restriction) Yiz′τ = Yizτ for z, z′ = 0, 1 and all i. +Assumption 2 means that the instrumental variable affects the outcome only through the treatment. +It holds in optogenetic experiments since only the targeted neurons respond to optical stimulation. +Under Assumption 2, we can simplify Yizτ as Yiτ. There are two ways to describe the potential +outcome processes under Assumption 2, i.e., Yiz and Yiτ. We will use Yi1 and Yi0 to represent the +potential processes if the instrumental variable were set to Zi = 1 and Zi = 0, respectively, and Yiτ +to represent the potential process if the event time of Ni were set to Ti = τ. +Second, the following independence assumption holds automatically because trials are randomly +selected for optical stimulation. +Assumption 3 (Randomization) Zi +{Niz(·), Yiτ(·) : z ∈ {0, 1}, τ ∈ [0, T] ∪ T +}. +Assumption 3 implies Zi +{Tiz, Yiz(t) : z ∈ {0, 1}, t ∈ [0, T]} under Assumption 2. It allows for the +identification of the intention-to-treat effects of the instrumental variable on the treatment and the +outcome. However, it is insufficient to identify the effect of the treatment on the outcome due to the +possibility of unmeasured confounders. +Lastly, we invoke the following no anticipation assumption because the event time of Ni at a +later time point cannot reversely affect Yi at a previous time. +6 + +Assumption 4 (No anticipation) Yiτ(t) = Yiτ ′(t) for τ, τ ′ ≥ t and all i. +Assumption 4 is well-known in causal inference with time series data (e.g., Bojinov and Shephard, +2019). The use of a non-strict inequality sign instead of a strict inequality in Assumption 4 indicates +that the effect of Ni on Yi is not instantaneous. Replacing the non-strict inequality with a strict +inequality allows for Yiτ(τ) ̸= Yiτ ′(τ) for τ < τ ′, i.e., the event at time τ has an effect on the outcome +at the same time. The non-strict inequality in Assumption 4 also implies YiT + = YiT because the +event at time T does not have an effect on Yi in [0, T]. +Under Assumptions 1–4, the relationships among Zi, Ni, Yi, and the unmeasured confounder Ui +can be illustrated by the causal diagram in Figure 1(d). The randomized stimulation Zi affects the +treatment Ni, which in turn affects the outcome Yi. Because the treatment Ni is not randomized, +unmeasured confounders Ui may exist between Ni and Yi. +2.3 +Definitions of causal effects with point process treatment and outcome +We are now ready to define the causal quantities of interest. First, we define the average causal +effect (ACE) of the instrumental variable on the treatment and outcome processes at time t as +ACEN(t) += +E{Ni1(t) − Ni0(t)} and ACEY (t) += +E{Yi1(t) − Yi0(t)}, respectively. Although +ACEN(t) and ACEY (t) are possible quantities of interest in the experiment, they do not directly +answer how the treatment Ni affects the outcome Yi. Therefore, we define the ACE of the treatment +process on the outcome process as +ACE(t; τ1, τ2) = E{Yiτ1(t) − Yiτ2(t)}, +τ1 ≥ τ2 and τ1, τ2 ∈ [0, T] ∪ T +, +(3) +which characterizes how the change in the event time of Ni from τ2 to τ1 affects Yi at time t. A +positive ACE(t; τ1, τ2) with τ1 ≥ τ2 implies that a later event in the treatment process increases the +expected outcome process at time t. This effect varies over time t and depends on the two event +times τ1 and τ2. Define the average causal effect rate (ACER) of Ni on Yi as +ACER(t; τ) = +lim +∆τ→0+ +ACE(t; τ + ∆τ, τ) +∆τ += ∂E{Yiτ(t)} +∂τ +. +(4) +The ACER measures how fast E{Yiτ(t)} changes given an infinitesimal change in the event time τ. +This concept is similar to the infinitesimal shift function defined in Lok (2008). Under Assumption 4, +we have +ACE(t; τ1, τ2) = +� +� +� +� +� +� +� +� +� +� +� +� +� +ACE(t; τ1, τ2), +if τ2 < τ1 < t +ACE(t; t, τ2), +if τ2 < t ≤ τ1 +0, +if t ≤ τ2 ≤ τ1 +, +7 + +and thus ACER(t; τ) = 0 if t ≤ τ. When the treatment is a single-point process, we have the +following relationship between the ACE and ACER of the treatment, +ACE(t; τ1, τ2) += +� τ1 +τ2 +ACER(t; τ)dτ. +(5) +Therefore, we can focus on the ACER because it determines the ACE. +Under Assumption 3, the ACEs of the instrumental variable on the treatment and outcome +processes can be identified by the observed differences between the stimulated and unstimulated +groups, +ACEN(t) = f(t) +with +f(t) = E{Ni(t) | Zi = 1} − E{Ni(t) | Zi = 0}, +(6) +ACEY (t) = h(t) +with +h(t) = E{Yi(t) | Zi = 1} − E{Yi(t) | Zi = 0}. +(7) +However, Assumption 3 is insufficient for the identification of the ACE and the ACER of the treat- +ment process on the outcome process, because the treatment process is not randomized. +3 +Nonparametric identification and estimation +3.1 +Nonparametric identification with a single-point process treatment +We begin by generalising the monotonicity assumption in Angrist et al. (1996). +Assumption 5 (Monotonicity) For each i, the potential event times of Ni satisfy Ti1 ≤ Ti0. +Assumption 5 requires that the potential event time of Ni under stimulation will be no later than that +without stimulation. Under Assumption 5, the ACE of the instrumental variable on the treatment +process at time τ equals the proportion of a subpopulation defined by the joint potential event times +of Ni, i.e., +ACEN(τ) = Pr(Ti1 ≤ τ < Ti0), +τ ∈ [0, T]. +Units in this subpopulation would have the event time of treatment process before or equal to +τ with stimulation and after τ without stimulation. Thus, these can be viewed as the compliers +whose treatment is positively affected by the stimulation. +With a point process treatment, the +definition of compliers is time-dependent. Similarly, the other three subpopulations, Ti0 ≤ τ < Ti1, +max(Ti1, Ti0) ≤ τ, and τ < min(Ti1, Ti0), generalise the defiers, always-takers, and never-takers in +the binary instrumental variable model, respectively. +We cannot validate Assumption 5 since it depends on unit-level potential outcomes. However, +Assumption 5 implies a testable condition that can be checked using the observed data. +8 + +Proposition 1 Under Assumption 3, Assumption 5 implies, for all τ ∈ [0, T], +Pr(Ti > τ | Zi = 1) ≤ Pr(Ti > τ | Zi = 0). +Proposition 1 states the stochastic dominance of the survival function of Ti under stimulation over +that without stimulation. We can assess Assumption 5 by comparing the empirical survival functions +of Ti in the stimulated and unstimulated groups. If the two curves cross, then the testable condition +in Proposition 1 is violated, which in turn falsifies Assumption 5. Therefore, our identification results +will consider scenarios both with and without Assumption 5. +Angrist et al. (1996) show that the effect of the instrumental variable on the outcome equals the +product of the effect of the instrumental variable on the treatment and the effect of the treatment +on the outcome. The following theorem generalises their result to our setting. +Theorem 1 Suppose that Ni is a single-point process and Assumptions 1–4 hold. For any t ∈ [0, T], +we have +ACEY (t) += +� T +0 +E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1}Pr(Ti0 ≤ τ < Ti1)dτ +− +� T +0 +E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0}Pr(Ti1 ≤ τ < Ti0)dτ. +(8) +If Assumption 5 holds in addition, then for any t ∈ [0, T], we have +ACEY (t) += +− +� T +0 +E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} · ACEN(τ)dτ. +(9) +In Theorem 1, ∂Yiτ(t)/∂τ is a generalised derivative that may consists of Dirac δ functions (Lax, +2002, Appendix B). Since the conditional set Ti1 ≤ τ < Ti0 depends on τ, it is important to note +that E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = ∂E{Yiτ ′(t) | Ti1 ≤ τ < Ti0}/∂τ ′ |τ ′=τ, which is generally not +equal to ∂E{Yiτ(t) | Ti1 ≤ τ < Ti0}/∂τ that takes into account the change of the conditional set. +By rewriting ACER(t; τ) as E{∂Yiτ(t)/∂τ}, we can view E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} as +the ACER in the subpopulation Ti1 ≤ τ < Ti0. In a sense, E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} gen- +eralises the complier average causal effect in the binary instrumental variable model. +Similarly, +E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1} generalises the average causal effect for the defiers. These conditional +expectations might not equal the ACER because Yiτ(t) and (Ti1, Ti0) might not be independent due +to the unmeasured confounding between Ni and Yi. +The formula in (8) shows that the average causal effect of the instrumental variable on the out- +come process, ACEY (t), equals the difference between the weighted averages of the two subpopula- +9 + +tion ACERs over the timeline. The weights rely on the joint distribution of (Ti1, Ti0). When Assump- +tion 5 holds, the first term on the right hand side of (8) vanishes and the weight Pr(Ti1 ≤ τ < Ti0) +is equal to ACEN(τ). As a result, (8) reduces to (9) under monotonicity. +Under Assumption 3, ACEY (t) and ACEN(t) are identifiable. Thus, we can view (8) and (9) +as integral equations for the subgroup ACERs (Newey and Powell, 2003). Unfortunately, these sub- +population ACERs are not identifiable without additional assumptions. To provide some intuition, +consider (9) under monotonicity. Based on the observed data, (6) and (7) give the identification +formulas for ACEY (t) and ACEN(τ) for all t, τ ∈ [0, T] under Assumption 3. So (9) is an integral +equation for the unknown quantity defined as γ(τ, t) = E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0}. Consider +a discrete approximation of γ(τ, t) by evaluating its values over a K1 × K2 two-dimensional grid +of (τ, t). Equation (9) generates only K2 equations by considering the K2 grid of t, which cannot +sustain the identification of K1 × K2 unknown values of γ(·, ·). Consequently, the identification of +the ACERs is infeasible without additional assumptions. +To address this problem, we invoke the following identification assumption. +Assumption 6 (Stationarity) ACE(t; τ1, τ2) = ACE(t − τ1; 0, τ2 − τ1) for τ1 ≤ τ2 ≤ t. +Assumption 6 states that the ACE of the treatment on the outcome is invariant to timeline shifts. +The left-hand side is the effect of the treatment when the event time is τ1 versus τ2 on the outcome +at time t. In contrast, the right-hand side represents the same effect, but with the timeline shifted +forward by τ1. Therefore, Assumption 6 means that the ACE of the treatment is invariant regardless +of the absolute time. Under Assumption 6, we have ACER(t; τ) = ACER(t − τ; 0) and thus can +simplify ACER(t; τ) as ACER(t − τ) with ACER(t − τ) = 0 if t ≤ τ. We can show that, together +with Assumption 4, Assumption 6 leads to +ACER(t; 0) = −∂E{Yi0(t)}/∂t. +(10) +The formula in (10) offers a more natural interpretation of ACER, that is, −ACER(t; 0) describes +the expected change rate in the potential outcome at time t when the event in Ni happens at time +0. Theorem 2 below gives sufficient conditions for identifying the ACER. +Theorem 2 Suppose that Ni is a single-point process and Assumptions 1–4 and 6 hold. Further- +more, if either (i) Assumption 5 holds and, for all t, τ ∈ [0, T], +E {∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = ∂E {Yiτ(t)}/∂τ, +(11) +10 + +or (ii) for all t, τ ∈ [0, T], +E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1} = ∂E{Yiτ(t)}/∂τ, +(12) +then ACER satisfies +ACER(t; τ) = +� +� +� +� +� +ACER(t − τ; 0), +if t > τ +0, +if t ≤ τ +, +(13) +and +h(t) = − +� T +0 +ACER(t − τ; 0)f(τ)dτ +(14) +for t ∈ [0, T]. If further (Ψf)(ν) ̸= 0 for all ν ∈ R, then the ACER is identified by ACER(t; τ) = +−Ψ−1� +G +� +(t − τ) for t > τ and ACER(t; τ) = 0 for t ≤ τ, where +G(ν) = (Ψh)(ν) +(Ψf)(ν) +for all +ν ∈ R. +(15) +The condition in (11) means that the ACERs are homogenous across subpopulations defined by +Ti1 ≤ τ < Ti0 with different values of τ, generalising the homogeneity assumption in (2). Without +monotonicity, the condition in (12) further requires that the ACERs are homogenous across sub- +populations defined by Ti0 ≤ τ < Ti1. Similar to the instrumental variable methods in survival +analysis (e.g. Li et al., 2015; Tchetgen Tchetgen et al., 2015), these conditions are satisfied as long +as the impact of the confounders on the outcome is additive. With a binary treatment and a scalar +outcome, Wang and Tchetgen Tchetgen (2018) also use a similar condition assuming no additive +interaction between the treatment and the unmeasured confounders on the outcome. We will study +this condition in detail under several commonly-used outcome models in §4. +The deconvolution problem (14) belongs to the family of Wiener–Hopf equations (see, among +others, Noble, 1959). It is essentially the same as the well-studied deconvolution of densities in +statistics (e.g., Fan, 1991; Diggle and Hall, 1993; Pensky and Vidakovic, 1999; Johannes, 2009; +Dattner et al., 2011, 2016). From the Paley–Wiener–Schwartz theorem, we know that (Ψf)(ν) ̸= 0 +for all ν ∈ R if f(t) = E{Ni(t) | Zi = 1} − E{Ni(t) | Zi = 0} is a non-zero function with bounded +support. This holds as long as the effect of the instrument Zi on the treatment process Ni vanishes in +finite time. The non-zero condition of (Ψf)(ν) is also employed in the nonparametric deconvolution +problem (see, e.g., Fan, 1991). +In the binary instrumental variable model with the homogeneity assumption, the effect of the +treatment on the outcome equals the ratio of the effects of the instrumental variable on the treat- +ment and the outcome. Theorem 2 shows that this ratio relationship also holds with point process +11 + +treatment and outcome, but in the frequency domain. The well-known convolution theorem ensures +that the Fourier transform of a convolution of two functions is equal to the product of their Fourier +transforms. Therefore, by applying the Fourier transform on each term of the convolution equation +in (14), we can obtain the generalised Wald estimation formula (15) in Theorem 2. +3.2 +Treatment with multiple events +We generalise the identification result in §3.1 to a treatment process with possibly multiple events. +We begin by generalising the definition of potential values and causal effects. Let Yi,n(·)(·) be the +potential process of the outcome if the treatment were set to a fixed process n(·). The observed +outcome process is Yi(·) = Yi,n(·)(·) if Ni(·) = n(·). Then, we can define the ACE of the treatment +n(·) versus n′(·) on the outcome as +ACE{t; n(·), n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)}. +(16) +For single-point process treatments, +(16) reduces to the definition in (3). Using the linearity of +expectation, we can write +ACE{t; n(·), n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)} += E{Yi,n(·)(t) − YiT+(t)} − E{Yi,n′(·)(t) − YiT+(t)}, +where E{Yi,n(·)(t) − YiT+(t)} and E{Yi,n′(·)(t) − YiT+(t)} are the effects of n(·) and n′(·) versus a +null process with no events in [0, T], respectively. Similar to §3.1, we can characterize the treatment +process using event times. Suppose that n(·) has l events at times τ1, . . . , τl. Then Yi,n(·)(t) can be +written as Yi,τ1,...,τl(t), so its expectation decomposes as +E{Yi,τ1,...,τl(t)} = E{YiT+(t)} + +l +� +s=1 +E +� +Yi,τ1,...,τs(t) − Yi,τ1,...,τs−1(t) +� +(17) +with Yi,τ1,...,τs−1(t) = YiT+(t) for s = 1. The following assumption simplifies the decomposition by +assuming away the interactive effects of the event times in the potential outcome process. +Assumption 7 (Additivity) E +� +Yi,τ1,...,τs(·) − Yi,τ1,...,τs−1(·) +� += E{Yi,τs(·)−YiT+(·)} for any s ≥ 1 +and any event times (τ1, . . . , τs) satisfying τ1 < τ2 < · · · < τs. +Point processes with event times (τ1, . . . , τs) and (τ1, . . . , τs−1) have the same trajectory up to time +τs−1, where the former has an additional event at τs. Assumption 7 means that the effect of the +process with event times (τ1, . . . , τs) versus that with event times (τ1, . . . , τs−1) does not depend +on their common trajectory up to time τs−1. +Hence, the causal effect remains the same when +12 + +the first s − 1 events are removed from both processes. Under Assumption 7, (17) simplifies to +E{Yi,τ1,...,τl(t) − YiT+(t)} = �l +s=1 E {Yi,τs(t) − YiT+(t)} , which means that the effect of each event +time on the outcome process is additive. In §S2.5 in the supplementary material, we show that +Assumption 7 holds under the Hawkes process (Hawkes, 1971) or Aalen’s additive hazard model +(Aalen, 1980) for the potential outcome process. Assumption 7 may be violated due to the interactive +effect of the event times in the treatment process. For instance, neural ensembles are famous for their +neural plasticity in the long term — the ability to reorganize themselves in response to stimulation, +which clearly violates Assumption 7. Such violations of Assumption 7 are sometimes of scientific +interest. We leave the investigation of such effects for future research. +Under Assumption 7, we can separately study the effect of each event in Ni. Proposition 2 below +generalises (5) to treatment processes with multiple events. +Proposition 2 Under Assumptions 1, 2, 4, and 7, we have +ACE{t; n(·), n′(·)} = − +� t +0 +ACER(t; τ){n(τ) − n′(τ)}dτ. +Based on Proposition 2, we can focus on the identification of ACER. Theorem 3 below generalises +Theorem 2 to treatment processes with multiple events. +Theorem 3 Suppose that Assumptions 1–4, 6, and 7 hold. If for all t ∈ [0, T] and any fixed processes +n(·) and n′(·), +E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1(·) = n(·), Ni0(·) = n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)}, +(18) +then the ACER satisfies (13) and (14). If (Ψf)(ν) ̸= 0 for all ν ∈ R, then the ACER is identified +by ACER(t; τ) = −Ψ−1� +G +� +(t − τ) for t > τ and ACER(t; τ) = 0 for t ≤ τ, with G(ν) defined in +(15). +When Ni is a single-point process, Theorem 3 does not require Assumption 7, and the condition +in (18) reduces to (12) in Theorem 2. As a result, Theorem 3 reduces to Theorem 2 when Ni has at +most one event. Similar to Theorem 2, the condition in (18) means that the ACERs are homogenous +across subpopulations defined by Ni1 and Ni0. +3.3 +Estimation +We consider the estimation of the ACER based on identification results from Theorems 2 and 3. This +is essentially the deconvolution problem commonly studied in the literature (see Diggle and Hall, +13 + +1993; Pensky and Vidakovic, 1999; Johannes, 2009; Dattner et al., 2011, 2016, for more discussion). +Since an optimal estimation procedure is not the focus of this paper, we only provide a simple +regression-based procedure to estimate ACER. To be specific, we use a two-step procedure by first +obtaining the estimates of f and h and then solving the ACER from the empirical version of the +convolution equation in (14). +Let ˆf and ˆh denote the estimators of f and h defined in (6) and (7), which equal the empirical +mean differences in the treatment and outcome processes in the stimulated and unstimulated groups. +We approximate the true ACER with truncated basis expansions, for ∆ ∈ [0, T], +ACER(∆; 0) ≈ +J +� +j=1 +ψj(∆)βj, +(19) +where J is a tuning parameter for the number of bases and {ψj(·) : j = 1, 2, . . . , J} is a set of pre- +specified basis functions. Here the support of ACER(·; 0) can be determined by prior knowledge. +Then, we estimate β = (β1, . . . , βJ) by minimizing the following penalized ℓ2-distance based on the +convolution equation (14) +�β = arg min +β∈RJ +������ +ˆh + +J +� +j=1 +(ψj ∗ ˆf)βj +������ +2 +2 ++ η∥β∥2 +2, +(20) +where ∗ denotes the convolution between two functions and introduce the ridge penalty to re- +duce boundary effects. +An analytic solution for �β is available since the objective function in +(20) is quadratic in β. +Recalling that we consider independent trials, we can choose the tun- +ing parameter J and η using cross-validation or based on prior knowledge such as the smooth- +ness of the ACER. +Denoting the selected parameter by �J and ˆη, the final estimator is given +as +� +ACER(·; 0) = � � +J +j=1 ψj(·)ˆβj,ˆη. +We can then construct the confidence band for the function +� +ACER(·; 0) using the bootstrap. The asymptotic properties for � +ACER as the sample size m in- +creases follow from the standard theory assuming independent samples. +We leave the rigorous +discussion for future analysis, as it is not the main focus of this paper. +4 +The role of models: causal interpretability and identifiability +4.1 +Conditional intensity +In this section, we study several commonly-used models for point process outcomes in applied re- +search when an instrumental variable is available, allowing for the presence of unmeasured con- +founders. We do not impose any distributional assumptions on the unmeasured confounders. Con- +sequently, it is difficult to study the identifiability of the model parameters themselves. We take +14 + +an alternative route by connecting the model parameters to the causal effects and considering the +identifiability and estimation of the causal effects directly. With a binary instrumental variable, we +show that the ACER is identifiable and can be estimated using the generalised Wald estimation +under many commonly-used models. This estimation strategy does not rely on the identification +or estimation of the model parameters, as long as the unmeasured confounding is additive in the +underlying outcome model. +We begin by introducing some additional notation to characterize a point process. Let Ui(·) +denote the unmeasured confounding process on R. We use Hit− to represent the σ-algebra induced +by the history up to, but not including, time t. Define the conditional intensity of Yi as +λY (t) = E {dYi(t)/dt | Hit−} . +(21) +The conditional intensity, or intensity, is the conditional mean of the event rate of Yi in an infinitesi- +mal time interval [t, t+dt), which is analogous to the conditional mean of the outcome in the binary +instrumental variable model. It fully characterizes the probabilistic structure of a point process and +is closely related to the hazard function in survival analysis. See Chapter 7 in Daley and Vere-Jones +(2003) for more discussion of the intensity. +In (21), the conditional intensity could depend on the history of Yi. +When the outcome Yi +describes recurrent events, it is common to allow the conditional intensity to depend on past events +of Yi (e.g., Hawkes, 1971; Brillinger, 1988; Lawrence, 2004; Kulkarni and Paninski, 2007; Yu et al., +2009; Gao et al., 2015; Macke et al., 2015; Gao et al., 2016; Wu et al., 2017; Zhao and Park, 2017; +Pandarinath et al., 2018). As concrete examples, in the context of neural data, the dependence on +past events captures the known phenomenon that a single neuron cannot fire consecutively in a very +short period of time and activities in a region may trigger inhibitory circuits to stabilize the activity +on a longer time scale. +An inherent constraint on the intensity is that it must be non-negative for the probabilistic +model to be well-defined. A similar constraint is well acknowledged in modeling the hazard function +in survival analysis. +This constraint of the intensity creates a schism in the modeling of point +processes — whether to employ a linear working model (Aalen, 1980) or a non-negative generative +model (Cox, 1972). In either model, since Ui is unobserved, existing methods usually impose strong +parametric assumptions on Ui in order to estimate the parameters. A common assumption is that +Ui is a Gaussian process (see, e.g., Yu et al., 2009; Zhao and Park, 2017), primarily due to its +simplicity for the Bayesian computation. However, the analysis can be sensitive to these parametric +assumptions. In §4.2, we will study both types of models. +15 + +Before specifying λY (t), we introduce the following assumption on the relationships among Ni, +Yi, and Ui, which are commonly used in instrumental variable methods when outcome models are +employed (see Tchetgen Tchetgen et al., 2015, for an example in survival analysis). +Assumption 8 (a) Zi +{Niz(t), Yiτ(t), Ui(t) : z ∈ {0, 1}, t ∈ [0, T], τ ∈ [0, T] ∪ T +}; (b) For t ∈ +[0, T], Yi,n(·)(t) +Ni(·) | {H∗\Ni +it− , Ni(s) = n(s), s ∈ [0, t)} and any fixed point process n(·), where +H∗\Ni +it− +denotes the σ-algebra induced by all potential processes including Ui, except for Ni up to time +t. +See Lok (2008) for the measure-theoretic description of the independence given histories of point +processes. Assumption 8(a) is a restatement of Assumption 3 with the additional notation of Ui. It +holds because Zi is randomized. Assumption 8(b) generalises the latent sequential ignorability in +Ricciardi et al. (2020) to a continuous-time setting with point process treatment and outcome. It +assumes that Ui fully characterizes the confounding between the treatment and the outcome so the +treatment is independent of the potential outcome at time t given the histories of Ni, Yi, and Ui. +Under Assumption 8, we have, for any t ∈ [0, T], +E +� +dYi,n(·)(t)/dt | Hit− +� += +E +� +dYi(t)/dt | Ni(·) = n(·), H∗\Ni +it− +� +, +(22) +which links the potential processes to the conditional intensity of the observed outcome. Therefore, +the discussion in §4.2 will focus on the models for the observed outcome. Under each of the models, +we will connect the model parameters with the ACER and study its identification. +4.2 +Identification of causal effects with linear additive unmeasured confounding +We start with linear models for the intensity. This type of models has been widely used in different +contexts because of its mathematical tractability (e.g., Hawkes, 1971; Aalen, 1980; Tchetgen Tchet- +gen et al., 2015; Jiang et al., 2018). In particular, consider Yi(·) to be a linear Hawkes process with +the following intensity, +λY (t) = µY + +� t +0 +g(t − s)dNi(s) + +� t +0 +ω(t − s)dYi(s) + ψUi(t), +(23) +where g(∆) = ω(∆) = 0 for ∆ ≤ 0 and ψUi(t) represents any function of {Ui(s) : s ∈ [0, t)}. +Proposition 3 below connects the ACER with parameters in (23) and shows the identification. +Proposition 3 Suppose that Assumptions 1–3 and 8 hold, and the underlying outcome model sat- +isfies (23). (a) We have +ACER(t; τ) = ACER(t − τ; 0) = − +� +Ψ−1 �G +� +(t − τ), +16 + +where �G(ν) = +� +1 + (Ψω)(ν) +�−1� +Ψg +� +(ν) if 1 + (Ψω)(ν) ̸= 0 for all ν ∈ R. (b) When (Ψf)(ν) ̸= 0 for +all ν ∈ R, ACER is identified by ACER(t; τ) = − +� +Ψ−1G +� +(t − τ) for t > τ and ACER(t; τ) = 0 for +t ≤ τ, with G(ν) defined in (15). +In practice, the function g(·) is often interpreted as the effect of an event in Ni on the outcome +Yi conditional on the history up to time t. Proposition 3(a) expresses the ACER in terms of the +model parameters, showing that g(·) and ω(·) jointly characterize the ACER of Ni on Yi. When +the dependence on past Yi does not exist, i.e., ω(·) ≡ 0, we have ACER(t; τ) = −g(t − τ). From +Proposition 3(a), we can obtain the ACER if we can estimate the model parameters in (23). How- +ever, this requires specifying the distribution of Ui(·). Fortunately, Proposition 3(b) shows that we +can identify the ACER without any distributional assumption on Ui when a binary instrumental +variable is available, and hence estimate it using the method in §3.3. It broadens the applicability +of Model (23) with fewer parametric assumptions. +Proposition 3(b) is an application of Theorem 2 under Model (23). The linearity in Model (23) +plays a key role in the causal interpretation of the model parameters and nonparametric identification +of the ACER. The linear terms of Ni and Yi connect the ACER with g(·) and ω(·), and the linear +term of Ui implies Assumption 6 and the condition in (12). +4.3 +Identification of causal effects with nonlinear additive unmeasured confound- +ing +We now consider the following nonlinear model that is similar to models in survival analysis with +an instrument (e.g., MacKenzie et al., 2014; Li et al., 2015; Tchetgen Tchetgen et al., 2015): +λY (t) = φ +� +µY + +� t +0 +g(t − s)dNi(s) +� ++ ψUi(t). +(24) +Model (24) generalises Model (23) by allowing for a nonlinear relationship between Ni and Yi through +the link function φ while requiring the unmeasured confounding effect to be additive. For Model (24), +the following proposition characterizes the causal effect and its identifiability. +Proposition 4 Suppose that Assumptions 1–3 and 8 hold, Ni is a single-point process, and the +underlying outcome model satisfies (24). (a) We have, for t, τ ∈ [0, T], +ACER(t; τ) = φ(µY ) − φ{µY + g(t − τ)}. +(b) When (Ψf)(ν) ̸= 0 for all ν ∈ R, ACER is identified by ACER(t; τ) = − +� +Ψ−1G +� +(t − τ) for +t > τ and ACER(t; τ) = 0 for t ≤ τ, where G(ν) is defined in (15). +17 + +From Proposition 4(a), the causal interpretation of g(·) depends on µY and the link function φ(·) +under Model (24). As a result, even with the same link function, the interpretation of g(·) differs +in populations with different values of µY . This warns us of interpreting g(·) as some causal effects. +Proposition 4(b) is an application of Theorem 2. Similar to Model (23), we can use the method in +§3.3 to estimate the ACER without the knowledge of φ(·) or Ui. Therefore, Proposition 4 suggests +directly targeting the ACER instead of the model parameter g(·). This circumvents the daunting task +to identify, estimate, and interpret the model parameters in Model (24), broadening its applicability +with fewer parametric assumptions. +Critically, although Model (24) allows for nonlinearity, it restricts the effect of unmeasured con- +founder to be additive. Relaxing this modeling assumption is challenging. In §S2.7 in the sup- +plementary material, we show that when the confounding effect on Yi is non-additive, the ACER +would depend on the distribution of the confounder, making the identification not possible without +a distributional assumption on Ui. +5 +Numerical analysis +5.1 +Simulation +We use simulation to illustrate the numerical performance of the proposed nonparametric estimation +procedure. In this simulation study, we generate the treatment Ni and outcome Yi from the following +model +λN(t) += +µN + φβ0 +� +α(t; aN, bN)Zi + Ui(t) +� +, +(25) +λY (t) += +φβ2 +� +φβ1 +� +µY + +� t +0 +α(∆; aY , bY )dNi(t − ∆) +� ++ φβ1 {Ui(t − dU)} +� +, +(26) +where α(·; a, b) = ba2t exp(−at) is the alpha function (see, among others, Chapter 7 in Ermentrout +and Terman, 2010) and φβ(x) = xβ is the link function. The confounding variable Ui is generated +as a Gaussian process with mean zero and a squared exponential kernel cov{Ui(t), Ui(t + d)} = +σ2 +U exp{−d2/(2l2 +U)}. The parameters in (25) and (26) are set as µN = µY = 0.2, aN = 10, bN = 0.5, +aY = 8, bY = 1, dU = 0.5, σU = 0.2, and lU = 0.1. We consider five scenarios in this simulation. +Scenario 1a (β0 = β1 = β2 = 1): A linear model for Yi with a single-point process Ni, which is +achieved by suppressing the intensity (25) to zero after the first event in Ni is generated. +Scenario 1b (β1 = 2, β0 = β2 = 1): An additive confounding model for Yi with a single-point +process Ni. +Scenario 2a (β0 = β1 = β2 = 1): A linear model for Yi with multiple events in Ni. +18 + +Scenario 2b (β0 = 3, β1 = 2, β2 = 1): A linear model for Yi with multiple events in Ni and +non-additive confounding effects on Ni. +Scenario 3 (β0 = β1 = 1, β2 = 3): A non-additive confounding model for Yi. +For each scenario, we generate m trials with m ranging from 40 to 800. In each simulated dataset, +half of the trials are set to have Zi = 1 and the other half Zi = 0. The processes Ni and Yi are +generated from 0 to T = 3 using thinning process, while the unmeasured confounding process Ui is +generated from −1 to 3 to account for its delayed effect on Yi. +For Scenario 3, the identification of ACER is difficult with nonlinear confounding effects, as +illustrated in the supplementary material. +So we use the Monte Carlo method to calculate the +ACER to show its dependence on the distribution of the unmeasured confounder. For Scenarios +1a to 2b, we apply the proposed generalised Wald estimation procedure in §3.3. We estimate the +function h(t) as the difference between the empirical cumulative intensities of Yi in the treatment +group (Zi = 1) and control group (Zi = 0). The function f(t) is estimated in a similar manner. +We approximate the ACER using a cubic B-splines with 6 knots evenly-spaced in [0, 1] where the +mass of α(·; aY , bY ) resides. The tuning parameter of the ridge penalty η is set to be m−1 to reduce +boundary effects from the nonparametric approximation. To measure the performance, we calculate +the proportion of integrated squared errors with respect to the true ACER, that is +r = +� 1 +0 +� +� +ACER(∆) − ACER(∆; 0) +�2 +d∆ +� 1 +0 ACER2(∆; 0)d∆ +, +(27) +where the true ACER is calculated from (26) based on Propositions 3 and 4. +Figure 2 shows the simulation results averaged over 1000 replicates. Figures 2(a) and (b) show +that the performances of estimators improve as the number of trials increases in Scenarios 1a, 1b, +2a, and 2b. In particular, the proportion of integrated squared error in Scenario 1a is larger than +that in Scenario 2a, despite having the same model for Yi. This reveals a feature for point process +treatments that, given the same amount of trials, more events in Ni contribute more information for +recovering the causal effects of Ni on Yi. The four curves in Figures 2(a) and (b) converge slowly +towards zero due to the existence of approximation error in the basis expansion and the bias from +penalization. Figures 2(c) and (d) show the calculated true ACER in Scenario 3 under two different +distributions of Ui. +Within each of Figures 2(c) and (d), the five curves correspond to τ being +0, 0.25, 0.5, 0.75, and 1. Even with the same t, the shape of ACER(t; τ) varies as τ changes in both +figures, implying that ACER(t; τ) does not equal to ACER(t − τ). Moreover, the contrast between +Figure 2(c) and (d) shows that ACER depends on the distribution of the unmeasured process Ui, +19 + +demonstrating the sensitivity of the ACER to distributional assumptions on Ui. +200 +400 +600 +800 +0.0 +0.4 +0.8 +Scenario 1a +Scenario 1b +Number of trials (m) +MISE (prop.) +(a) +200 +400 +600 +800 +0.0 +0.4 +0.8 +Scenario 2a +Scenario 2b +Number of trials (m) +MISE (prop.) +(b) +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.4 +0.8 +1.2 +Time (t) +−ACER(t; τ) +σU = 0.1 +(c) +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.4 +0.8 +1.2 +Time (t) +−ACER(t; τ) +σU = 0.3 +(d) +Figure 2: Identification of ACER and performance of the generalised Wald estimation averaged +across 1000 replicates. Panels (a) and (b) illustrates the performance of the proposed estimation +procedures in Scenarios 1a, 1b (Panel a), 2a, and 2b (Panel b). The x-axes are the numbers of trials +(m), and the y-axes are the measure defined in (27). The shaded areas are the interquartile bands +from the 1000 replicates. Estimation performances in the four scenarios are not directly comparable +given the huge difference between the data generating mechanisms. Based on the interpretation +in (10), Panels (b) and (c) show the true value of −ACER(t; τ) with non-additive confounding +effects on Y (Scenario 3). The spectrum of gray curves is calculated with σU = 0.1, and the red +curves with σU = 0.3. Within each spectrum, the five curves correspond to τ being 0, 0.25, 0.5, 0.75 +and 1, respectively. Curves in each color spectrum are not shift-invariant. +5.2 +Empirical analysis +We now apply the proposed methodology to the neural data from Bolding and Franks (2018a). +We provide the basic background to help understand the experiment. For more details, see the +supplementary material and Bolding and Franks (2018a). Bolding and Franks (2018a) conduct an +experiment to understand how a mouse brain maintains stationarity in odor detection regardless +of odor concentration. To be specific, it is known that neural activities in the olfactory bulb (OB) +increase in response to a higher concentration of odor particles, and that a spike in OB triggers +neural activities of principal neurons (PN) in the piriform cortex, where the odor is perceived by the +brain. To avoid other neural processes that normalize odor responses, Bolding and Franks (2018a) +20 + +use optogenetics to stimulate neurons in OB with 1-s light pulses, which meet the requirements as an +instrumental variable in our methodology. Bolding and Franks (2018a) also take an optogenetic to +circumventing the contribution of centrifugal inputs and other intrabulbar processes, which effectively +cuts of the feedback from PN to OB. Figure 3(d) shows a causal diagram for the relationship among +the stimulation, OB, and PN. +The dataset contains spike trains recorded in OB and PN during the experiment. A total of +160 trials are conducted on 8 mice, where each mouse has 10 trials without stimulation (Zi = 0) +and 10 trials with a one-second light pulse at 20 mW/mm2 (Zi = 1). The light pulse, if present, +onsets at time 0 and ends at 1s. In our analysis, we consider the first 3.5 seconds of a trial, from +−0.5 to 3, as there are hardly any residual effects afterward. We consider the treatment Ni as the +process of events in OB, and the outcome Yi as the process of events in PN. Each recorded event +in Ni is a spike of one neuron in OB that may instigate a distinct group of PN in the piriform +cortex. Given the vast amount of PN in the piriform cortex, the instigated groups may share few +or no overlaps, limiting the interactive effect of the treatment process. Therefore, the additivity in +Assumption 7 is a plausible approximation to the true underlying mechanism. Figures 3(a) and (b) +show the smoothed intensities of neural activities in the stimulated (blue) and unstimulated (red) +groups. We can see that the stimulation triggers increased activities in OB in all trials. However, +there are large variations in the neural activities across trials. +We first conduct a preliminary analysis assuming no unmeasured confounders between the treat- +ment and outcome processes. In this case, the identification of the causal effects does not require the +instrumental variable. We fit a model of Yi on Ni and directly interpret the coefficient function of Ni +as the causal effect. However, the conclusion is inconsistent with the findings in Bolding and Franks +(2018a), implying possible unmeasured confounders or model misspecification. See more details in +§S3.3 in the supplementary material. +We then apply the generalised Wald estimation procedure to estimate the causal effects of neural +activities in OB on the neural activities of PN. From the observed data, we estimate the functions +h and f using differences between empirical cumulative intensities in the stimulated (Zi = 1) and +unstimulated (Zi = 0) groups. We approximate the unknown ACER using cubic B-splines with +evenly-spaced knots in [0, 1], where two knots are selected by a 5-fold cross-validation. We set the +tuning parameter for the ridge penalty to be η = 0.01 to handle boundary effects. A 90% confidence +band is constructed using the bootstrap with details in the supplementary material. We use the +bootstrap confidence band to approximate the uncertainty in the estimates. +21 + +OB +Light +U +PN +ACER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−30 +0 +30 +60 +90 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +500 +1000 +1500 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +500 +1000 +1500 +(a) +Trials +(b) +Time (in seconds) +Trials +∆ (in seconds) +−ACER(∆; 0) +(c) +Time (in seconds) +(d) +Figure 3: Empirical intensities and fitted ACER on data from Bolding and Franks (2018b). Panels +(a) and (b) show the empirical intensities of the neural activities of OB (Panel a) and PN (Panel +b) in the stimulated (blue) and unstimulated (red) groups. The solid curves represent the average +intensity over 80 trials, and the dashed curves demonstrate the empirical intensity from 20 randomly +selected trials in each group. The shaded area in Panel (a) represents the duration of the light +pulse. Based on the interpretation in (10), Panel (c) shows the estimated −ACER(∆; 0) from the +full data set. The shaded area represents a 90% confidence band for visualizing the uncertainty of +the estimates from 5000 bootstrap samples. Panel (d) shows the causal diagram for the relationship +among the variables. +Figure 3(c) shows the estimated ACER. The curve shows that an event in OB elicits high activities +in PN immediately after the event (< 0.1 seconds), but the effect quickly turns negative for an +extended duration (between 0.1 to 0.4 seconds) before it dies down. This is consistent with the +findings in Bolding and Franks (2018a) that a temporal mechanism is in place to stabilize the neural +activities of PN after the initial detection of odors. Additional analysis of the neural dataset can +be found in §S3 in the supplementary material. +The confidence band shows that the proposed +generalised Wald estimation procedure yields high uncertainty near the boundaries, despite a large +number of events and the ridge penalty. In this particular case, it appears that the ACER vanishes +after 0.5 seconds, but the boundary effect causes spurious estimates in the bootstrap samples. In +practice, we recommend practitioners applying the generalised Wald estimation procedure, and then +22 + +applying a suitable parametric form or shape constraint on the ACER. +References +Aalen, O. (1980). A model for nonparametric regression analysis of counting processes. In Mathe- +matical statistics and probability theory, pp. 1–25. Springer. +Angrist, J. D., G. W. Imbens, and D. B. Rubin (1996). Identification of causal effects using instru- +mental variables. Journal of the American Statistical Association 91(434), 444–455. +Bojinov, I. and N. Shephard (2019). Time series experiments and causal estimands: exact random- +ization tests and trading. Journal of the American Statistical Association 114(528), 1665–1682. +Bolding, K. A. and K. M. Franks (2018a). 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Neural Computation 29(5), 1293–1316. +26 + +Supplementary Material +§S1 generalises the framework to discrete instrumental variables. +§S2 provides the proofs of the theorems, propositions, and claims in the main text. +§S3 provides more details about the empirical analysis. +Let 1(·) denote the indicator function. Recall i = √−1. +S1 +Generalisation to discrete instrumental variables +In the main text, we discuss the case when the instrumental variable takes binary values. Here we +briefly outline the generalisation to the case with discrete instrumental variables where there could +be multiple levels of Z. For a discrete instrumental variable, we can apply the proposed methodology +by comparing two levels z and z′ (instead of 0 and 1 in the binary instrumental variable case) under +the monotonicity and homogeneity assumptions with respect to these two levels. For simplicity, we +only give the result with a single-point process Ni. Define +ACEY (t; z, z′) += +E{Yiz(t) − Yiz′(t)}, +ACEN(t; z, z′) = E{Niz(t) − Niz′(t)}. +We can obtain a similar result as Theorem 1 with respect to z and z′: +ACEY (t; z, z′) += +� T +0 +E{∂Yiτ(t)/∂τ | Tiz′ ≤ τ < Tiz}Pr(Tiz′ ≤ τ < Tiz)dτ +− +� T +0 +E{∂Yiτ(t)/∂τ | Tiz ≤ τ < Tiz′}Pr(Tiz ≤ τ < Tiz′)dτ, +which, under monotonicity Tiz ≤ Tiz′, simplifies as +ACEY (t; z, z′) += +− +� T +0 +E{∂Yiτ(t)/∂τ | Tiz ≤ τ < Tiz′} · ACEN(τ; z, z′)dτ. +Define +hz,z′(t) += +E{Yi(t) | Zi = z} − E[Yi(t) | Zi = z′}, +fz,z′(t) += +E{Ni(t) | Zi = z} − E{Ni(t) | Zi = z′}. +Under randomization (Assumption 3) and stationarity (Assumption 6), if +E +�∂Yiτ(t) +∂τ +���� Tiz ≤ τ < Tiz′ +� += E +�∂Yiτ(t) +∂τ +���� Tiz′ ≤ τ < Tiz +� += ∂E {Yiτ(t)} +∂τ +, +then +hz,z′(t) = − +� T +0 +ACER(t − τ; 0)fz,z′(τ)dτ. +27 + +This is similar to Theorem 2 with respect to z and z′. Therefore, if (Ψfz,z′)(ν) ̸= 0 for all ν ∈ R, +then the ACER is identified by +ACER(t; τ) = −Ψ−1(G)(t − τ) with G(ν) = +� +Ψhz,z′� +(ν) +� +Ψfz,z′� +(ν). +For estimation, we can use the data with Zi = z, z′ and apply the same method as in the binary +instrumental variable case. +S2 +Proofs +S2.1 +Proof of Theorem 1 +First, we have +ACEY (t) = E{Yi1,Ti1(t)} − E{Yi0,Ti0(t)} = E{Yi,Ti1(t)} − E{Yi,Ti0(t)}, +(S1) +where the second equality follows from Assumption 2. +Using the Stieltjes integral, we can write +Yi,Ti1(t) += +� +[0,T] +Yiτ(t)dNi1(τ) + YiT+(t)1{Ni1(T) = 0} += +� +[0,T] +Yiτ(t)dNi1(τ) + YiT (t)1{Ni1(T) = 0}, +(S2) +where the second equality follows from Assumption 4. Similarly, we have +Yi,Ti0(t) += +� +[0,T] +Yiτ(t)dNi0(τ) + YiT (t)1{Ni0(T) = 0}. +(S3) +Plugging (S2) and (S3) into (S1) yields +ACEY (t) += +E +�� +[0,T] +Yiτ(t)dNi1(τ) + YiT (t)1{Ni1(T) = 0} +� +−E +�� +[0,T] +Yiτ(t)dNi0(τ) + YiT (t)1{Ni0(T) = 0} +� += +E +�� +[0,T] +Yiτ(t){dNi1(τ) − dNi0(τ)} +� ++ E [YiT (t)1{Ni1(T) = 0} − 1{Ni0(T) = 0}] += +E +�� +[0,T] +Yiτ(t){dNi1(τ) − dNi0(τ)} +� ++ E [YiT (t){Ni0(T) − Ni1(T)}] , +(S4) +where the last equality holds because Ni0 and Ni1 are single-point processes. +28 + +We then focus on the first term of (S4). Let HN +izt = {Niz(s) : s ∈ [0, t]} denote the history of Niz +up to time t for z = 0, 1. By switching the order of the expectation and integral, we have +E +�� +[0,T] +Yiτ(t){dNi1(τ) − dNi0(τ)} +� += +� +[0,T] +E [Yiτ(t){dNi1(τ) − dNi0(τ)}] += +� +[0,T] +E +� +E +� +Yiτ(t) | HN +i1T , HN +i0T +� +d{Ni1(τ) − Ni0(τ)} +� += +E +�� +[0,T] +E +� +Yiτ(t) | HN +i1T , HN +i0T +� +d{Ni1(τ) − Ni0(τ)} +� += +E +� +E +� +Yiτ(t) | HN +i1T , HN +i0T +� +{Ni1(τ) − Ni0(τ)} +��� +τ=T +τ=0 +� +−E +�� +[0,T] +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +{Ni1(τ) − Ni0(τ)}dτ +� +, +(S5) +where the second equality follows from the law of total expectation and the last equality follows from +integration by parts. We can write the first term of (S5) as +E +� +E +� +Yiτ(t) | HN +i1T , HN +i0T +� +{Ni1(τ) − Ni0(τ)} +��� +τ=T +τ=0 +� += +E +� +E +� +YiT (t) | HN +i1T , HN +i0T +� +{Ni1(T) − Ni0(T)} − 0 +� += +E [YiT (t){Ni1(T) − Ni0(T)}] , +where the first equality follows from Ni1(0) − Ni0(0) = 0 and the second equality follows from the +law of total expectation. Therefore, the first term of (S4) becomes +E +�� +[0,T] +Yiτ(t){dNi1(τ) − dNi0(τ)} +� += +E [YiT (t){Ni1(T) − Ni0(T)}] +−E +�� +[0,T] +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +{Ni1(τ) − Ni0(τ)}dτ +� +. +(S6) +Plugging (S6) into (S4), we obtain +ACEY (t) = −E +�� +[0,T] +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +{Ni1(τ) − Ni0(τ)}dτ +� +. +(S7) +Because Ni(·) is a single-point process, Ni1(τ) − Ni0(τ) takes values only in {−1, 0, 1}. In addition, +Ni1(τ) − Ni0(τ) = 1 is equivalent to Ti1 ≤ τ < Ti0 and Ni1(τ) − Ni0(τ) = −1 is equivalent to +Ti0 ≤ τ < Ti1. Therefore, by switching the order of the expectation, integral, and derivative in (S7), +we have +ACEY (t) += +− +� +[0,T] +E +� +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +����� Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ +29 + ++ +� +[0,T] +E +� +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +����� Ti0 ≤ τ < Ti1 +� +Pr(Ti0 ≤ τ < Ti1)dτ += +− +� +[0,T] +E +� +E +�∂Yiτ(t) +∂τ +���� HN +i1T , HN +i0T +� ����� Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ ++ +� +[0,T] +E +� +E +�∂Yiτ(t) +∂τ +| HN +i1T , HN +i0T +� ����� Ti0 ≤ τ < Ti1 +� +Pr(Ti0 ≤ τ < Ti1)dτ += +− +� +[0,T] +E +�∂Yiτ(t) +∂τ +���� Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ ++ +� +[0,T] +E +�∂Yiτ(t) +∂τ +���� Ti0 ≤ τ < Ti1 +� +Pr(Ti0 ≤ τ < Ti1)dτ, +where the first equality follows from the law of total probability and the third equality follows from +the law of total expectation. This proves (8). +When Assumption 5 holds, Ni1(τ)−Ni0(τ) takes values only in {0, 1} and ACEN(τ) = Pr(Ti1 ≤ +τ < Ti0), Therefore, by switching the order of the expectation, integral, and derivative in (S7), we +have +ACEY (t) += +− +� +[0,T] +E +� +∂E +� +Yiτ(t) | HN +i1T , HN +i0T +� +∂τ +����� Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ += +− +� +[0,T] +E +�∂Yiτ(t) +∂τ +���� Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ += +− +� +[0,T] +E +�∂Yiτ(t) +∂τ +���� Ti1 ≤ τ < Ti0 +� +ACEN(τ)dτ. +This proves (9). +□ +S2.2 +Proof of Theorem 2 +First, from Assumption 6 and the definition of ACER, we have +ACER(t; τ) = +lim +∆τ→0+ +E{Yi,τ+∆τ(t) − Yiτ(t)} +∆τ += +lim +∆τ→0+ +E{Yi,∆τ(t − τ) − Yi0(t − τ)} +∆τ +, +which depends only on t − τ. Therefore, we can write ACER(t; τ) = ACER(t − τ). +Second, under Assumption 3, we have +ACEY (t) = E{Yi(t) | Zi = 1} − E{Yi(t) | Zi = 0} = h(t) +and +Pr(Ti1 ≤ τ < Ti0) − Pr(Ti0 ≤ τ < Ti1) = ACEN(τ) = f(τ). +Finally, we can write (8) in Theorem 1 as +h(t) += +− +� +[0,T] +E +�∂Yiτ(t) +∂τ +����Ti1 ≤ τ < Ti0 +� +Pr(Ti1 ≤ τ < Ti0)dτ +30 + ++ +� +[0,T] +E +�∂Yiτ(t) +∂τ +����Ti0 ≤ τ < Ti1 +� +Pr(Ti0 ≤ τ < Ti1)dτ += +� +[0,T] +∂E {Yiτ(t)} +∂τ +{Pr(Ti0 ≤ τ < Ti1) − Pr(Ti1 ≤ τ < Ti0)} dτ += +− +� +[0,T] +ACER(t − τ)f(τ)dτ, +(S8) +where the second equality follows from the condition in (12). +Taking the Fourier transform on both sides of (S8) yields, for all ν, +(Ψh)(ν) = −(ΨACER)(ν)(Ψf)(ν). +Therefore, if (Ψf)(ν) ̸= 0 for all ν ∈ R, then ACER(t; τ) = −Ψ−1� +G +� +(t − τ) for t > τ and 0 +otherwise, where G(ν) = (Ψh)(ν)/(Ψf)(ν) for all ν ∈ R. +□ +S2.3 +Proof of Proposition 2 +From Assumption 7, we have +E{Yi,n(·)(t) − YiT+(t)} = +l +� +j=1 +E +� +Yi,τj(t) − YiT+(t) +� +. +Under Assumption 4, we have Yi,τj(t) = YiT+(t) for j > n(t) and YiT+(t) = Yit(t). Therefore, for +j > n(t), we have +E +� +Yi,τj(t) − YiT+(t) +� += 0. +(S9) +Therefore, +E{Yi,n(·)(t) − YiT+(t)} = +n(t) +� +j=1 +E +� +Yi,τj(t) − YiT+(t) +� += +n(t) +� +j=1 +ACE(t; τj, T +) = +n(t) +� +j=1 +ACE(t; τj, t). +As a result, +E{Yi,n(·)(t) − YiT+(t)} += +− +n(t) +� +j=1 +� t +τj +ACER(t; τ)dτ += +− +n(t) +� +j=1 +�� t +τn(t) +ACER(t; τ)dτ + +� τn(t) +τj +ACER(t; τ)dτ +� += +−n(t) · +� t +τn(t) +ACER(t; τ)dτ − +n(t)−1 +� +j=1 +� +� +� +n(t)−1 +� +k=j +� τk+1 +τk +ACER(t; τ)dτ +� +� +� += +−n(t) · +� t +τn(t) +ACER(t; τ)dτ − +n(t)−1 +� +k=1 +k +� +j=1 +� τk+1 +τk +ACER(t; τ)dτ +31 + += +−n(t) · +� t +τn(t) +ACER(t; τ)dτ − +n(t)−1 +� +k=1 +k · +� τk+1 +τk +ACER(t; τ)dτ. +Because +n(τ) = +� +� +� +� +� +� +� +� +� +� +� +� +� +k, +if τ ∈ [τk, τk+1) +0, +if τ ∈ [0, τ1) +n(t), +if τ ∈ [τn(t), t] +, +we have +E{Yi,n(·)(t) − YiT+(t)} += +− +� t +τn(t) +ACER(t; τ)n(τ)dτ − +n(t)−1 +� +k=1 +� τk+1 +τk +ACER(t; τ)n(τ)dτ += +− +� t +0 +ACER(t; τ)n(τ)dτ. +As a result, we have +E{Yi,n(·)(t) − Yi,n′(·)(t)} = − +� t +0 +ACER(t; τ) +� +n(τ) − n′(τ) +� +dτ. +□ +S2.4 +Proof of Theorem 3 +Under Assumptions 2 and 3, we can write, for any t ∈ [0, T], +h(t) += +E{Yi,Ni1(t)} − E{Yi,Ni0(t)} += +� +n(·),n′(·)∈N +E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n(·), Ni0 = n′(·)}Pr{Ni1 = n(·), Ni0 = n′(·)}, +where N is the sample space of simple point process on [0, T]. Here we abuse the notation Pr(·) to +represent the density and take Σ as the summation over all possible pairs of n(·) and n′(·) in N. +From the condition in (18), we know that +h(t) = +� +n(·),n′(·)∈N +E{Yi,n(·)(t) − Yi,n′(·)(t)}Pr{Ni1 = n(·), Ni0 = n′(·)}. +It follows from Proposition 2 that +h(t) = − +� +n(·),n′(·)∈N +Pr{Ni1 = n(·), Ni0 = n′(·)} +� t +0 +ACER(t; τ) +� +n(τ) − n′(τ) +� +dτ. +As a result, we have +h(t) = − +� t +0 +ACER(t; τ)E{Ni1(τ) − Ni0(τ)}dτ. +32 + +From Assumption 3, we know that +E{Ni1(τ) − Ni0(τ)} = E{Ni(τ) | Zi = 1} − E{Ni(τ) | Zi = 0} = f(τ). +Finally, using that ACER(t; τ) = ACER(t − τ) from Assumption 6, we have +h(t) = − +� t +0 +ACER(t − τ)f(τ)dτ. +The rest follows from the same argument as in §S2.2. +□ +S2.5 +Proof of Proposition 3 +We first prove Proposition 3(a). Consider a single-point treatment process Ni. +From Assumption 8, we know (22) holds that, for any t ∈ [0, T], +E +�dYiτ(t) +dt +���� Hit− +� += +E +�dYi(t) +dt +���� Ti = τ, H∗\Ni +it− +� += λY (t), +where the last equality follows from the definition of λY (t) in (21). +Plugging in Model (23), we know that +E +�dYiτ(t) +dt +���� Hit− +� += µY + g(t − τ) + +� t +0 +ω(t − s)dYiτ(s) + ψUi(t). +(S10) +Taking the expectation on both sides of (S10), we obtain +E +�dYiτ(t) +dt +� += +µY + g(t − τ) + +� t +0 +ω(t − s)E{dYiτ(s)} + E{ψUi(t)}. +(S11) +Integrating both sides of (S11) from 0 to t yields +E {Yiτ(t)} = µY t + +� t +0 +g(s − τ)ds + +� t +0 +� s +0 +ω(s − ∆)E {dYiτ(∆)} ds + +� t +0 +E{ψUi(s)}ds. +(S12) +For the third term in (S12), we have +� t +0 +� s +0 +ω(s − ∆)E {dYiτ(∆)} ds += +� t +0 +� s +0 +ω(s − ∆)E +�dYiτ(∆) +d∆ +� +d∆ds += +� t +0 +� s +0 +ω(∆′)E +�dYiτ(s − ∆′) +d(s − ∆′) +� +d(s − ∆′)ds +(∆′ = s − ∆) += +� t +0 +� 0 +s +ω(∆′)E +� +−dYiτ(s − ∆′) +d(s − ∆′) +� +d∆′ds +(d(s − ∆) = −d∆′) += +� t +0 +� s +0 +ω(∆′)E +�dYiτ(s − ∆′) +d(s − ∆′) +� +d∆′ds +33 + += +� t +0 +� s +0 +ω(∆′)E +�dYiτ(s − ∆′) +ds +� +d∆′ds += +� t +0 +� t +∆′ ω(∆′)E +�dYiτ(s − ∆′) +ds +� +dsd∆′ +(Fubini’s Theorem) += +� t +0 +ω(∆′) +�� t +∆′ E +�dYiτ(s − ∆′) +ds +� +ds +� +d∆′ += +� t +0 +ω(∆)E{Yiτ(t − ∆)}d∆ − +�� t +0 +ω(∆)d∆ +� +E{Yiτ(0)}. +Therefore, +E {Yiτ(t)} += +µY t + +� t +0 +g(s − τ)ds + +� t +0 +ω(∆)E{Yiτ(t − ∆)}d∆ +− +�� t +0 +ω(∆)d∆ +� +E{Yiτ(0)} + +� t +0 +E{ψUi(s)}ds. +Taking the derivative with respect to τ on both sides of the above equation yields +ACER(t; τ) += +∂E{Yiτ(t)} +∂τ += +− +� t +0 +∂g(s − τ) +∂s +ds − +� t +0 +ω(∆) ∂ +∂τ E{Yiτ(t − ∆)}d∆ += +−g(t − τ) − +� t +0 +ω(∆)ACER(t − ∆; τ)d∆, +where ∂E{Yiτ(0)}/∂τ = 0 follows from (S12). Using the property of the Dirac δ function, we obtain, +� t +0 +{ω(∆) + δ(∆)}ACER(t − ∆; τ)d∆ += +−g(t − τ). +(S13) +Taking the Fourier transform on both sides of (S13) yields, for all ν, +{(Ψω)(ν) + 1}(ΨACERτ)(ν) += +− exp(−iντ)(Ψg)(ν), +where ΨACERτ is a short-hand notation for the Fourier transform of ACER(·; τ) and the right- +hand side follows from the Fourier shift theorem that Ψg(t − τ) = exp(−iντ)Ψg(ν). Therefore, we +have +(ΨACERτ)(ν) = − exp(−iντ) +(Ψg)(ν) +(Ψω)(ν) + 1 = − exp(−iντ) �G(ν), +where �G(ν) = +� +1 + (Ψω)(ν) +�−1� +Ψg +� +(ν). By setting τ = 0, we obtain ACER(t; 0) = − +� +Ψ−1 �G +� +(t). +Therefore, we have +(ΨACERτ)(ν) = exp(−iντ)(ΨACER(t; 0))(ν), +where ΨACER(t; 0) is the Fourier transform of ACER(t; 0). Using the Fourier shift theorem, we +obtain ACER(t; τ) = ACER(t − τ; 0). +34 + +We now prove Proposition 3(b). The result is an application of Theorem 3. In particular, we +know that Assumption 6 holds from Part (a), we need to verify the condition in (18), Assumption 7, +and Assumption 4. +Verifying the condition in (18): Define +x(t; n(·), n′(·), n1(·), n0(·)) ≡ E +� +Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) +� +. +By the law of total expectation, we have, for any fixed point processes n1(·) and n0(·), +E +� +Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) +� += +E +� +E +� +Yi,n(·)(t) − Yi,n′(·)(t) | Hit−, Ni1 = n1(·), Ni0 = n0(·) +� ��Ni1 = n1(·), Ni0 = n0(·) +� += +E +� +E +� +Yi,n(·)(t) | Hit− +� +− E +� +Yi,n′(·)(t) | Hit− +� ��Ni1 = n1(·), Ni0 = n0(·) +� +, +where the last equality holds since the Ni1 and Ni0 are fixed up to time t conditioning on the history +Hit. Similar to the derivation of (S12), we can obtain +E +� +Yi,n(·)(t) | Hit− +� +− E +� +Yi,n′(·)(t) | Hit− +� += +µY t + +� t +0 +� s +0 +g(s − l)dn(l)ds + +� t +0 +ψUi(s)ds + +� t +0 +� s +0 +ω(s − l)dYi,n(·)(l)ds +−µY t − +� t +0 +� s +0 +g(s − l)dn′(l)ds − +� t +0 +ψUi(s)ds − +� t +0 +� s +0 +ω(s − l)dYi,n′(·)(l)ds += +� t +0 +� s +0 +g(s − l)d{n(l) − n′(l)}ds + +� t +0 +� s +0 +ω(s − l)d{Yi,n(·)(l) − Yi,n′(·)(l)}ds. +Therefore, we have +E +� +Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) +� += +E +� +E +� +Yi,n(·)(t) | Hit− +� +− E +� +Yi,n′(·)(t) | Hit− +� +| Ni1 = n1(·), Ni0 = n0(·) +� += +� t +0 +� s +0 +g(s − l)d{n(l) − n′(l)}ds + +� t +0 +� s +0 +ω(s − l)dx(l; n(·), n′(·), n1(·), n0(·))ds, +where +� t +0 +� s +0 +ω(s − l)dx(l; n(·), n′(·), n1(·), n0(·))ds += +� t +0 +� s +0 +ω(s − l)dx(l; n(·), n′(·), n1(·), n0(·)) +dl +dlds += +� t +0 +� s +0 +ω(l′)dx(s − l′; n(·), n′(·), n1(·), n0(·)) +d(s − l′) +d(s − l′)ds +(l′ = s − l) += +� t +0 +� 0 +s +ω(l′)−dx(s − l′; n(·), n′(·), n1(·), n0(·)) +d(s − l′) +dl′ds += +� t +0 +� s +0 +ω(l′)dx(s − l′; n(·), n′(·), n1(·), n0(·)) +ds +dl′ds +35 + += +� t +0 +� t +l′ ω(l′)dx(s − l′; n(·), n′(·), n1(·), n0(·)) +ds +dsdl′ +(Fubini’s Theorem) += +� t +0 +ω(l′) +�� t +l′ +dx(s − l′; n(·), n′(·), n1(·), n0(·)) +ds +ds +� +dl′ += +� t +0 +ω(l′)x(t − l′; n(·), n′(·), n1(·), n0(·))dl′ += +� t +0 +ω(s)x(t − s; n(·), n′(·), n1(·), n0(·))ds. +As a result, for t ∈ [0, T], +x(t; n(·), n′(·), n1(·), n0(·)) += +� t +0 +� s +0 +g(s − l)d{n(l) − n′(l)}ds + +� t +0 +ω(s)x(t − s; n(·), n′(·), n1(·), n0(·))ds. +(S14) +Taking the derivative with respect to t on both sides of (S14) leads to +˙x(t; n(·), n′(·), n1(·), n0(·)) += +� t +0 +g(t − l)d{n(l) − n′(l)} + ω(t) ˙x(0; n(·), n′(·), n1(·), n0(·)) ++ +� t +0 +ω(s) ˙x(t − s; n(·), n′(·), n1(·), n0(·))ds, +where ˙x(t; n(·), n′(·), n1(·), n0(·)) denotes the derivative of x(t; n(·), n′(·), n1(·), n0(·)) with respect to +t. We know that ˙x(0; n(·), n′(·), n1(·), n0(·)) = 0 by definition. Therefore, using the Fourier trans- +form, we know that ˙x(t; n(·), n′(·), n1(·), n0(·)) = − +� +Ψ−1 ˜G +� +(t), where ˜G(ν) = +� +1+(Ψω)(ν) +�−1� +Ψ˜g +� +(ν) +and ˜g(t) ≡ +� t +0 g(t − l)d{n(l) − n′(l)}. As a result, ˙x(t; n(·), n′(·), n1(·), n0(·)) does not depend on +n1(·) and n0(·). Combining this with the fact that x(0; n(·), n′(·), n1(·), n0(·)) = 0, we know that +x(t; n(·), n′(·), n1(·), n0(·)) does not depend on n1(·) and n0(·) and can write x(t; n(·), n′(·), n1(·), n0(·)) = +x(t; n(·), n′(·)). +Taking n1(·) = n(·) and n0(·) = n′(·), we have, +E +� +Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n(·), Ni0 = n′(·) +� += x(t; n(·), n′(·)). +From the law of total probability, we obtain +E +� +Yi,n(·)(t) − Yi,n′(·)(t) +� += +� +n1(·),n0(·)∈N +E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·)}Pr{Ni1 = n1(·), Ni0 = n0(·)} += +� +n1(·),n0(·)∈N +x(t; n(·), n′(·))Pr{Ni1 = n1(·), Ni0 = n0(·)} += +x(t; n(·), n′(·)). +36 + +As a result, the condition in (18) holds. +Verifying Assumption 7 (Additivity): We have shown that, for any n(·) and n′(·), +E +� +Yi,n(·)(t) − Yi,n′(·)(t) +� += x(t; n(·), n′(·)), +where x(t; n(·), n′(·)) is the solution of (S14). To verify additivity, we only need to show that, for all +t, +x(t; n1(·), n′ +1(·)) = x(t; n2(·), n′ +2(·)), +where, without loss of generality, n1(·) has events {τ1, τ2, . . . , τl}, n′ +1(·) has events {τ1, τ2, . . . , τl−1}, +n2(·) has a single event {τl}, and n′ +2(·) has no events. +From (S14), we have +x(t; n(·), n′(·)) += +� t +0 +� s +0 +g(s − l)d{n(l) − n′(l)}ds + +� t +0 +ω(s)x(t − s; n(·), n′(·))ds += +n(t) +� +j=1 +� t +τj +g(s − τj)ds − +n′(t) +� +j=1 +� t +τj +g(s − τ ′ +j)ds + +� t +0 +ω(s)x(t − s; n(·), n′(·))ds. +Therefore, x(t; n1(·), n′ +1(·)) is the solution of +x(t; n1(·), n′ +1(·)) = +� +� +� +� +� +� t +τl g(s − τj)ds + +� t +0 ω(s)x(t − s; n1(·), n′ +1(·))ds +t > τl +0 +t ≤ τl +. +Similarly, x(t; n2(·), n′ +2(·)) is the solution of +x(t; n2(·), n′ +2(·)) = +� +� +� +� +� +� t +τl g(s − τj)ds + +� t +0 ω(s)x(t − s; n2(·), n′ +2(·))ds +t > τl +0 +t ≤ τl +. +Therefore, we have x(t; n1(·), n′ +1(·)) = x(t; n2(·), n′ +2(·)). +Verifying Assumption 4: We only need to show that, for any t ≤ τ,∂E{Yiτ(t)}/∂τ = 0. From (S13), +we have +� t +0 +{ω(∆) + δ(∆)}ACER(t − ∆; τ)d∆ += +0 +(S15) +for t ≤ τ, where we use g(t − τ) = 0 for t ≤ τ. For fixed τ, denote m(t) = ACER(t; τ) for t ≤ τ and +x(t) = 0 for t > τ. Then, for any t +� t +0 +{ω(∆) + δ(∆)}x(t − ∆)d∆ += +0. +(S16) +37 + +Taking the Fourier transform on both sides yields, for all ν, +{(Ψω)(ν) + 1}(Ψm)(ν) += +0. +Recalling that {(Ψω)(ν) + 1} ̸= 0, we have Ψm = 0, and thus m(t) = 0 for all t. +Therefore, +∂E{Yiτ(t)}/∂τ = ACER(t; τ) = 0 for t ≤ τ. +□ +S2.6 +Proof of Proposition 4 +Using a similar argument as in (S10), we have +E +�dYiτ(t) +dt +���� Hit− +� += φ {µY + g(t − τ)} + ψUi(t). +Taking expectation and then integrating from 0 to t on both sides, we obtain +E {Yiτ(t)} = +� t +0 +φ {µY + g(s − τ)} ds + +� t +0 +E {ψUi(s)} ds. +(S17) +Taking the derivative with respect to τ yields Part (a) of the proposition that +ACER(t; τ) = − +� t +0 +φ′ {µY + g(s − τ)} g′(s − τ)ds = φ(µY ) − φ{µY + g(t − τ)}. +(S18) +Similar to the proof of Proposition 3, it suffices to verify Assumption 6, the condition in (12), +and Assumption 4. +From (S22), we have +E{Yiτ1(t) − Yiτ2(t)} += +� t +τ1 +φ {µY + g(s − τ1)} ds − +� t +τ2 +φ {µY + g(s − τ2)} ds += +� t−τ1 +0 +φ {µY + g(s)} ds − +� t−τ1 +τ2−τ1 +φ {µY + g(s − τ2 + τ1)} ds += +E{Y0(t − τ1) − Yτ2−τ1(t − τ1)}. +for τ1 ≤ τ2 ≤ t. Thus, Assumption 6 holds. +We then verify the condition in (12). Similar to the proof of Proposition 3, we can obtain +E +�∂Yiτ(t) +∂τ +· 1(Ti1 ≤ τ) +� += +E +�∂E{Yiτ(t) | Hit−} +∂τ +· 1(Ti1 ≤ τ) +� += +E +� +∂E{Yi(t) | Ti = τ, H∗\Ni +it +} +∂τ +· 1(Ti1 ≤ τ) +� +, +where, from Model (24), +∂E{Yi(t) | Ti = τ, H∗\Ni +it +} +∂τ += +∂ +∂τ +� +φ(µY )τ + +� t +τ +φ{µY + g(s − τ)}ds + +� t +0 +ψUi(s)ds +� +38 + += +φ(µY ) − φ(µY ) − +� t +τ +g′(s − τ)φ′{µY + g(s − τ)}ds += +φ(µY ) − φ(µY ) − [φ{µY + g(t − τ)} − φ(µY )] += +φ(µY ) − φ{µY + g(t − τ)}. +Therefore, we obtain +E +�∂Yiτ(t) +∂τ +· 1(Ti1 ≤ τ) +� += [φ(µY ) − φ{µY + g(t − τ)}] Pr(Ti1 ≤ τ). +(S19) +Similarly, we obtain +E +�∂Yiτ(t) +∂τ +· 1(Ti0 ≤ τ) +� += [φ(µY ) − φ{µY + g(t − τ)}] Pr(Ti0 ≤ τ). +(S20) +Combining (S19) and (S20) yields the condition in (12). +From (S18), we have +ACER(t; τ) = φ(µY ) − φ{µY + g(t − τ)} = φ(µY ) − φ(µY ) = 0 +for t ≤ τ, where we use g(∆) = 0 for ∆ ≤ 0. Therefore, Assumption 4 holds. +□ +S2.7 +Comment on the difficulties of the nonlinear non-additive model +We now comment on the difficulties of the following nonlinear non-additive model +λY (t) = φ +� +µY + +� t +0 +g(t − s)dNi(s) + +� t +0 +ω(t − s)dYi(s) + ψUi(t) +� +, +(S21) +where φ(·) is a non-negative link function and the rest are the same as in (23). Model (S21) is widely +used in the analysis of neural data with different forms of φ (see among others Lawrence, 2004; +Kulkarni and Paninski, 2007; Yu et al., 2009; Gao et al., 2015; Macke et al., 2015; Gao et al., 2016; +Sussillo et al., 2016; Wu et al., 2017; Zhao and Park, 2017; Pandarinath et al., 2018). Compared with +Models (23) and (24), it guarantees a non-negative intensity without additional restrictions on the +unmeasured confounders. However, the impact of the unmeasured confounding is no longer additive +under Model (S21). Even when the treatment is a single-point process, the following proposition +shows that the causal effect depends on the choices of the link function and the distribution of Ui in +Model (S21). +Proposition 5 Suppose that Assumptions 1, 2, and 8 hold and the underlying outcome satis- +fies (S21). When Ni is a single-point process, we have, for t, τ ∈ [0, T], +ACER(t; τ) = +� t +0 +E +� ∂ +∂τ φ +� +µY + g(s − τ) + +� s +0 +ω(t − ∆)dYiτ(∆) + ψUi(s) +�� +ds. +39 + +In Proposition 5, the ACER under Model (S21) does not satisfy the identification assumptions +in Theorems 2 or 3. Thus, we cannot use the identification result in §3. Under Model (S21), the +identification of ACER is difficult. Although it is strenuous to formally study its identifiability due to +the complexity of Model (S21), econometricians have obtained negative results for the identification +of non-separable models in the cross-sectional setting when both the treatment and the outcome are +scalars. In particular, Chesher (2003) gives sufficient conditions for nonparametric identification, +which generally requires the instrumental variable to be continuous. Moreover, when the outcome is +discrete, Chesher (2010) shows that the identification is typically not achieved even under parametric +models. Therefore, the identification of Model (S21) is gloomy in a more complex context with point +processes. Due to the dependence of the ACER on the model parameters and the distribution of Ui, +its identification is also unpromising. +We end this subsection with the proof of Proposition 5. +Proof of Proposition 5. Using a similar argument as in (S10), we have +E +�dYiτ(t) +dt +���� Hit +� += φ +� +µY + g(t − τ) + +� t +0 +ω(t − s)dYi(s) + ψUi(t) +� +. +Taking expectation and then integrating from 0 to t on both sides, we obtain +E {Yiτ(t)} = +� t +0 +E +� +φ +� +µY + g(s − τ) + +� s +0 +ω(t − l)dYiτ(l) + ψUi(s) +�� +ds. +(S22) +Taking the derivative with respect to τ yields +ACER(t; τ) = +� t +0 +E +� ∂ +∂τ φ +� +µY + g(s − τ) + +� s +0 +ω(t − l)dYiτ(l) + ψUi(s) +�� +ds. +(S23) +S3 +Supplement of the numerical analysis +S3.1 +Derivation of ACER in simulation +We know from §4.2 that the ACER in the presence of non-additive confounding takes the form +in (S23). For Scenario 3 in §5.1, we can derive that +∂ +∂τ φβ2 {µY + g(t − τ) + Ui(t − dU)} += +� ∂ +∂τ g(t − τ) +� +β2{µY + g(t − τ) + Ui(t − dU)}β2−1 +=β2bY a2 +Y +� +a2 +Y (t − τ) − 1 +� +exp{−aY (t − τ)}{µY + α(t − τ; aY , bY ) + Ui(t − dU)}β2−1, +where the last equality follows from +∂ +∂τ g(t − τ) = ∂ +∂τ α(t − τ; aY , bY ) +40 + += ∂ +∂τ bY a2 +Y (t − τ) exp{−aY (t − τ)} += bY a2 +Y [− exp{−aY (t − τ)} + aY (t − τ) exp{−aY (t − τ)}]. +We can calculate the true value of ACER(t; τ) using the Monte Carlo method by simulating the +unmeasured confounding process Ui. +S3.2 +Additional information on the real data analysis +In §5.2, we apply our methodology on the neural data to estimate the causal effect of neural activities +in the olfactory bulbs on those in the piriform cortex. In this section, we discuss more scientific +backgrounds and conduct additional analysis. +In addition to the olfactory bulb (OB) mitral cells and the principal neurons (PN) in the piriform +cortex (PCx), layer 1 feedforward interneurons (FFI) and layer 2/3 feedback interneurons (FBI) play +important roles in the neural circuits for odor perception (Bolding and Franks, 2018a). They are +inhibitory neurons that suppress the activities of PN. In particular, OB mitral cells excite both the +PN and FFI in PCx, which may result in an immediate excitation and a slightly delayed inhibition +in the PN. In addition, PN excites the FBI, which will in turn suppress future activities in PN. The +causal pathways among the aforementioned neurons are illustrated in Figure 4. +OB +Odor +FFI +PN +FBI +Figure 4: Causal diagram depicting the relationships among the olfactory bulb mitral cells (OB), +principal neurons (PN), feedforward interneurons (FFI), and feedback interneurons (FBI) in the +piriform cortex. A red arrow represents an excitatory pathway while a blue arrow represents an +inhibitory pathway. The dotted arrow from PN to FBI is disconnected when TeLC is expressed. In +the experiment analyzed here and in §5.2, the odor is replaced by light pulses, and only the neural +activities of OB and PN are recorded. +The fitted ACER in Figure 3(c) shows that, overall, a spike in OB causes an immediate excitation +in PN in PCx, and causes a relatively long-term suppression till the effect vanishes. This finding +corroborates the causal pathways in Figure 4 that the immediate excitation may be due to the +direct effects of OB cells on the PN, while the long-term suppression results from the induced +activities from FFI and FBI. As a result, the stationarity of odor detection is maintained, as shown +41 + +in Figures 3(a) and (b), that the PN return to normal activity level quickly after a sharp increase +in activities. +To further investigate the roles of inhibitory pathways, Bolding and Franks (2018a) selectively +expressed tetanus toxin light chain (TeLC) in PN. The expression of TeLC prevents a principal +neuron to excite any other neurons including the FBIs, while retaining PN’s excitability. In other +words, in a TeLC-expressed mouse, the causal pathway between PN to FBI no longer exists in +Figure 4. Figure 5 shows the resulting intensities. We can see the consequence of the absence of the +FBI, where the activities of PN remain at a higher-than-normal level till the end of the stimulation. +Applying the same estimation procedure to the data from the TeLC-expressed mice reveals a very +different ACER. In this case, there are 15 TeLC-expressed mice. On each mouse, there are 10 trials +in the treatment group and 10 in the control group. The confidence band is constructed as follows. +Let {ˆgb : b = 1, . . . , B} be the estimates from the bootstrap samples, where trials are sampled with +replacement in each treatment group. For each t ∈ [0, T], we estimate the bootstrap mean ¯g(t) +and the bootstrap standard deviation �σg(t) from the bootstrap samples. We can then construct a +(1 − α)% confidence band for ˆg as {(ˆg(t) − qαm−1/2ˆσg(t), ˆg(t) + qαm−1/2ˆσg(t)) : t ∈ [0, T]}, where +qα is the (1 − α)% bootstrap quantile of Qb = maxt |ˆgb(t) − ¯g(t)|/ˆσg(t). The number of trials m is +300 in the TeLC experiment and 160 in the analysis in the main text. +We can see that the causal effect lasts for a much shorter period, in the absence of self-excitation +among PN and the inhibitions from FBI. Note that an inhibition period still exists, which is likely +the effect through the FFI. Furthermore, the 5-fold cross-validation selected 18 knots in the TeLC- +expressed mice, in sharp contrast to the 2 knots selected in Figure 3. +By comparing the fitted +ACER in Figure 5 and Figure 3 in the main text, we can see that the causal pathway between PN +and FFI is a key mechanism in maintaining stationarity in odor perception. We also notice that the +nonparametric estimation procedure suffers from boundary effects, where a spurious inhibitory effect +is estimated towards the right boundary of the support. The estimated effect from the observational +analysis does not reflect the sharp inhibition following an event in OB. +S3.3 +Analysis assuming no unmeasured confounding +In this section, we conduct an observational analysis of the real data assuming there is no unmeasured +confounding between the treatment Ni and the outcome Yi. In particular, we assume the outcome +Yi follows the linear model +λY (t) = E +�dYi(t) +dt +���� Hit− +� += µ + +� t +0 +g(t − s)dNi(s), +(S24) +42 + +OB +Light +U +PN +ACER +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +−30 +0 +30 +60 +90 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +500 +1500 +2500 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +500 +1500 +2500 +(a) +Trials +(b) +Time (in seconds) +Trials +∆ (in seconds) +−ACER(∆; 0) +(c) +Time (in seconds) +(d) +Figure 5: +Empirical intensities and fitted ACER on data from Bolding and Franks (2018b) on +TeLC-expressed mice. Panels (a) and (b) show the empirical intensities of the neural activities of +OB (Panel a) and PN (Panel b) in the stimulated (blue) and unstimulated (blue) groups. The solid +curves represent the average intensity over 150 trials, and the shaped dashed curves demonstrate +the empirical intensity from 20 randomly selected trials in each group. The shaded area in Panel (a) +represents the duration of the light pulse. Panel (c) shows the estimated −ACER(∆; 0) from the +full data set. The shaded area represents a 90% confidence band for visualizing the uncertainty of +the estimates from 5000 bootstrap samples. Panel (d) shows the causal diagram for the relationship +among the variables. +43 + +where g(·) is commonly interpreted as the effect of Ni on Yi. +Here we do not include the self- +dependence of Yi because we have shown in Proposition 3 that the function g is not comparable to +ACER when the self-dependence is included. +Taking the expectation on both sides of (S24), we have +E +�dYi(t) +dt +� += µ + +� t +0 +g(t − s)E{dNi(s)}. +(S25) +Integrating both sides of (S25) from 0 to t yields +E{Yi(t)} += +� t +0 +E +�dYi(l) +dl +� +dl += +µt + +� t +0 +� l +0 +g(l − s)E{dNi(s)}dl += +µt + +� t +0 +� l +0 +g(l − s)E +�dNi(s) +ds +� +dsdl += +µt + +� t +0 +� l +0 +g(∆)E +�dNi(l − ∆) +d(l − ∆) +� +d(t − ∆)dl +(s = l − ∆) += +µt + +� t +0 +� 0 +l +g(∆)E +� +−dNi(l − ∆) +d(l − ∆) +� +d∆dl += +µt + +� t +0 +� l +0 +g(∆)E +�dNi(l − ∆) +dl +� +d∆dl += +µt + +� t +0 +� t +∆ +g(∆)E +�dNi(l − ∆) +dl +� +dld∆ +(Fubini’s Theorem) += +µt + +� t +0 +g(∆) +�� t +∆ +E +�dNi(l − ∆) +dl +� +dl +� +d∆ += +µt + +� t +0 +g(s)E{Ni(t − s)}ds += +µt + +� t +0 +g(t − s)E{Ni(s)}ds. +Define h′(t) = E{Yi(t)} and f′(t) = E{Ni(t)}. We have +h′(t) = µt + +� t +0 +g(t − s)f′(s)ds. +(S26) +We take a similar estimation procedure as in §3.3. First, we estimate f′(t) and h′(t) using empirical +cumulative intensities from all trials, denoted as ˆf′ and ˆh′(t). +Second, we approximate g with +truncated bases {ψj : j = 1, . . . , J}. Finally, we obtain the estimator from +ˆβ = arg min +β∈RJ+1 +������ +ˆh′ − βJ+1t − +J +� +j=1 +(ψj ∗ ˆf′)βj +������ +2 +2 ++ η∥β∥2 +2. +(S27) +As with the proposed procedure, we set η = 0.01 and use cubic B-splines with evenly-spaced knots. +The number of knots is chosen using 5-fold cross-validation. We construct 90% confidence bands +44 + +using the bootstrap to approximate the uncertainty as in the main text. +Results are shown in +Figure 6. We can see that not using the instrumental variable method yields estimates inconsistent +with the findings in Bolding and Franks (2018a). In particular, in Figure 6(a), the observational +analysis fails to capture the long-term inhibition between 0.1 to 0.3 seconds that contributes to +the stable response in PN, while displaying a false excitatory effect between 0.2 to 0.3 seconds. In +Figure 6(b), the observational analysis estimates a close-to-zero effect from OB to PN, contradicting +the findings in Bolding and Franks (2018a) on TeLC-expressed mice. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +−30 +0 +30 +60 +90 +∆ (in seconds) +−ACER(∆; 0) +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +−30 +0 +30 +60 +90 +∆ (in seconds) +−ACER(∆; 0) +(a) +Figure 6: Estimated −ACER(∆; 0) using the proposed instrumental variable method (black) and +the estimated effect from the observational analysis (blue) using data from Bolding and Franks +(2018b) on normal mice (Panel a) and TeLC-expressed mice (Panel b). The shaded area represents +a 90% confidence band for visualizing the uncertainty of the estimates from 5000 bootstrap samples. +45 + diff --git a/otE1T4oBgHgl3EQfiAQh/content/tmp_files/load_file.txt b/otE1T4oBgHgl3EQfiAQh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e74b452351694448f5e4ef4015e86df3025683d3 --- /dev/null +++ b/otE1T4oBgHgl3EQfiAQh/content/tmp_files/load_file.txt @@ -0,0 +1,1688 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf,len=1687 +page_content='An instrumental variable method for point processes: generalised Wald estimation based on deconvolution Zhichao Jiang∗§ Shizhe Chen†§ Peng Ding‡ January 10, 2023 Abstract Point processes are probabilistic tools for modeling event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' While there exists a fast- growing literature studying the relationships between point processes, it remains unexplored how such relationships connect to causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In the presence of unmeasured confounders, param- eters from point process models do not necessarily have causal interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We propose an instrumental variable method for causal inference with point process treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We define causal quantities based on potential outcomes and establish nonparametric identification results with a binary instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We extend the traditional Wald estimation to deal with point process treatment and outcome, showing that it should be performed after a Fourier transform of the intention-to-treat effects on the treatment and outcome and thus takes the form of deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We term this as the generalised Wald estimation and propose an estimation strategy based on well-established deconvolution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Keywords: Causal inference, Identification, Intensity, Principal stratification, Unmeasured con- founding ∗School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Email: jiangzhch7@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='cn †Department of Statistics, University of California, Davis, California 95616, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Email: szdchen@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='edu ‡Department of Statistics, University of California, Berkeley, California 94720, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Email: pengding- pku@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='edu §Equal contribution arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='03246v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ME] 9 Jan 2023 1 Introduction Point processes have long been used for modeling event data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The past decade has witnessed a surge of interest in point process models in many fields including neuroscience, finance, and social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this paper, we consider the analysis of neural data as a concrete motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Modern technologies allow neuroscientists to simultaneously record neural spike trains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', arrays of timestamps when neurons fire, across the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' With these data, one can hope to peek into the mechanisms of neural computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The nature of these scientific questions is the inference of causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Current technologies, however, present a major challenge for causal inference with neural data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Except for experiments on very simple animals, even state-of-the-art technologies can record only a very small fraction of neurons in chosen regions in the nerve systems, leaving the vast majority unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The unmeasured neural activities inevitably lead to the issue of unmeasured confound- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' that is, unmeasured activities might be the common causes of observed neural activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, any relationship inferred based on the partially observed system might not reflect the true causal relationship, but rather a spurious association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Fortunately, advances in optogenetics create new opportunities to address unmeasured confound- ing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Neuroscientists are able to instigate neural activities in a living brain via optical stim- ulation, which alters the activity of any chosen neuron with high spatial and temporal precision (Mardinly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Carrillo-Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From a causal inference perspective, such inter- ventions can serve as instrumental variables for inferring the causal relationship between neurons, as they affect the outcome neuron only through the treatment neuron while introducing exogenous variation in the treatment neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Instrumental variable methods are powerful tools for inferring causal effects in the presence of unmeasured confounding between the treatment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In a seminal paper, Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) clarify the role of a binary instrumental variable in identifying the causal effect of a binary treatment for an unmeasured subgroup, known as the complier average causal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' They propose two crucial identification assumptions, monotonicity and exclusion restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under these assumptions, they show that the complier average causal effect is identified by the Wald estimator (Wald, 1940;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Ridder and Moffitt, 2007) that equals the ratio of the differences in means of the outcome and the treatment when the instrumental variable changes from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' With an instrumental variable, most existing work considers non-dynamic settings and the instru- mental variable methods in survival analysis mainly focus on a scalar treatment and a non-recurrent outcome (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Martinussen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 1 To the best of our knowledge, there is no formal instrumental variable framework for point processes that addresses nonparametric identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We propose an instrumental variable method for causal inference when both the treatment and the outcome take the form of point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We define several causal quantities for the effect of the treatment on the outcome over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using a binary instrumental variable, we establish the nonpara- metric identification of causal effects allowing for the unmeasured treatment-outcome confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The identification assumptions hold as long as the impact of the unmeasured confounders on the outcome is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Our identification result implies that the causal effects can be obtained by solv- ing a convolution equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This extends the Wald estimation in traditional instrumental variable method to take the form of deconvolution, leading to the proposed generalised Wald estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We also examine several commonly-used models under our framework, studying the identification of the causal effects and the causal interpretation of the model parameters with a binary instru- mental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When the unmeasured confounders are additive on the outcome, the causal effects are identifiable without any distributional assumptions on the confounders based on the proposed generalised Wald estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Our finding justifies the identifiability of many commonly-used models such as the Hawkes process, broadening their applicability with fewer assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let A B | C denote the conditional independence of A and B given C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let R denote the set of real numbers and B(R) denote the Borel σ-algebra of the whole real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let L1(R) denote the set of functions f(x) such that � ∞ −∞ |f(x)|dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Unless specified otherwise, we assume all functions used in this paper belong to L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Ψ denote the Fourier transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', for any f(x) ∈ L1(R) and ν ∈ R, define (Ψf)(ν) = � ∞ −∞ f(x)e−i2πνxdx, where i = √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Ψ−1 denote the inverse Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 2 An instrumental variable framework for point processes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 A brief review of the binary instrumental variable model We begin by reviewing the binary instrumental variable model in the context of noncompliance (Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For unit i, let Zi be the binary treatment assigned, Ni the actual treatment received, and Yi the outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Niz be the potential value of the treatment receipt if the assigned treatment condition is z, Yizn the potential value of the outcome if the assigned treatment is z and the actually received treatment is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The joint values of Ni1 and Ni0 define the 2 unmeasured compliance type Ui = (Ni1, Ni0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Units with (Ni1 = 1, Ni0 = 0) are compliers who take the treatment assigned, units with (Ni1 = 1, Ni0 = 1) are always-takers who always take treatment 1, units with (Ni1 = 0, Ni0 = 0) are never-takers who always take treatment 0, and units with (Ni1 = 0, Ni0 = 1) are defiers who take the treatment opposite to the assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) invoke three assumptions: (1) exclusion restriction that the treatment assigned affects the outcome only through the treatment received, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Yizn = Yiz′n for all z, z′, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (2) randomization that Zi is independent of Niz and Yizn for z, n = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (3) monotonicity that the assigned treatment does not negatively affect the treatment receipt for all units, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Ni1 ≥ Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Exclusion restriction simplifies Yizn to Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Randomization rules out the confounding between the treatment assignment and the treatment receipt as well as the confounding between the treatment assignment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Monotonicity rules out defiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under these assumptions, Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) introduce the complier average causal effect as the average effect of the treatment receipt on the outcome for compliers, CACE = E(Yi1 − Yi0 | Ni1 = 1, Ni0 = 0), and show that it is identified by CACE = E(Yi | Zi = 1) − E(Yi | Zi = 0) E(Ni | Zi = 1) − E(Ni | Zi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1) In this model, the treatment assignment Zi is the instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The expression in (1) suggests the Wald estimator (Wald, 1940) for the CACE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', the ratio of the differences in means of the outcome and the treatment receipt when the treatment assigned changes from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) identify only the treatment effect in the complier subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For extrapolation to the whole population, we can invoke the homogeneity assumption (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Heckman, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2009) that the treatment effect is the same across compliance groups: E(Yi1 − Yi0 | Ni1, Ni0) = E(Yi1 − Yi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (2) Under the assumption in (2), the treatment effect in the whole population equals the CACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Notation and basic assumptions with a point process treatment We now consider the setting when both the treatment Ni and the outcome Yi are point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We will establish a similar ratio relationship as in (1) for the point process treatment and outcome, but in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a concrete example, we consider the neuroscience application from Bolding and Franks (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this application, the treatment and the outcome are the neural activities of the mouse olfactory bulb and piriform cortex within each brain region, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In experiments at the single-cell resolution, one can also model single-neuron activities as the treatment 3 and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As shown in Figure 1(a) and (b), these data take the form of spike trains that are commonly modeled as point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Bolding and Franks (2018a) apply light pulses to randomly selected trials to stimulate the olfactory bulb without affecting other brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, the light pulse serves as an instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To formally discuss causal inference, we need to generalise the model in Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) to account for the point process treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Ni Zi Ui Yi (d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 N (Z=1) Y (Z=1) N (Z=0) Y (Z=0) (c) Time (s) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 (a) Time (s) Trials −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 (b) Time (s) Trials Figure 1: Neural data from Bolding and Franks (2018a) and the causal diagram depicting the relationships among the variables in the instrumental variable framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (a) shows the spike trains collected in the mouse olfactory bulb in the stimulated (in blue) and unstimulated (in red) trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Each row represents a spike train in the olfactory bulb in one trial (Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area depicts the duration of the light pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (b) shows the spike trains collected in the mouse piriform cortex in the stimulated (in blue) and unstimulated (in red) trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Each row represents a spike train in the piriform cortex in one trial (Yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (c) zooms in on two randomly selected trials, where the solid curves are the smoothed intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (d) shows the causal diagram for the relationship among the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Variables in the solid square and circles are observed, and the variable in the dashed circle is unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The subscript i represents the ith unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , m index the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We use point processes to describe the neuron activities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Chapter 2 in Cox and Isham, 1980 or Chapter 3 in Daley and Vere-Jones, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define the treatment point process Ni(·) as a family of random non-negative integers {Ni(A)}A∈B(R) counting 4 the number of events in each set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let dNi(t) ≡ Ni([t, t + dt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Throughout this paper, we consider point processes that are simple, Pr{dNi(t) = 0 or 1 for all t} = 1, and with bounded intensity Pr{dNi(t) = 1}/dt < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In a similar manner, we introduce the outcome point process Yi(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We focus on a binary instrumental variable, Zi ∈ {0, 1}, and present the results on a discrete instrumental variable in §S1 in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Without loss of generality, we assume the instrumental variable onsets at time 0, and Ni(·) and Yi(·) are observed from 0 to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To avoid cumbersome bookkeeping, we constrain the processes to the observed period and ignore the history before time 0, and we write Ni([0, t]) as Ni(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For ease of discussion, we first consider the treatment Ni(·) being a point process with at most one event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We refer to Ni(·) as a single-point process if Ni(T) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We will extend the methodology to a general point process in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can characterize a single-point process Ni(·) using its event time: define Ti = T + if Ni(T) = 0 and Ti ≡ τ if Ni(t) = 1 for t ≥ τ and Ni(t) = 0 for t < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We adopt the potential outcomes framework under the following stable unit treatment value assumption (Rubin, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 1 There is no interference between units and there are no different versions of the instrument and the treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 1 rules out the spillover effect of other units’ instrumental variable on one’s treatment process and that of other units’ instrumental variable and treatment process on one’s outcome process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It also requires that there is only one version of the instrument and the treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In our motivating example, one unit corresponds to one trial, and trials conducted at different times might use the same mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The no-interference assumption would be violated if the neural dynamics of a mouse adapt to stimulation over time, causing activities in one trial to depend on previous trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This phenomenon is known as neural plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To restrict spillover between trials, adequate washout periods are incorporated to separate trials sufficiently apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, we can reasonably assume away spillover effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Furthermore, uniform stimulation is employed to ensure that there is only one version of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For the treatment process, we follow the common practice in neural data analysis to focus on the effect of the timings of spikes, ignoring the variation in the spike intensities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Brillinger, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Zhao and Park, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 1 allows us to define the potential values as the function of a unit’s own instrument and treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Niz(·) and Yiz(·) be the potential processes of the treatment and outcome, and Tiz be the potential event time of the treatment process if the instrumental variable were set to 5 Zi = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Also, define Yizτ(·) as the potential process of the outcome if the instrumental variable were set to Zi = z and the event time were set to Ti = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By definition, the two versions of the potential outcome process satisfy Yiz(·) = Yiz,Tiz(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The observed treatment process is Ni(·) = ZiNi1(·)+(1− Zi)Ni0(·), and the observed outcome process can be written as Yi(·) = ZiYi1(·) + (1 − Zi)Yi0(·) or Yi(·) = ZiYi1τ(·)+(1−Zi)Yi0τ(·) if Ti = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We assume {Zi, Niz(·), Yizτ(·) : z = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ ∈ [0, T]∪T +}m i=1 are independently and identically distributed, and thus the observables {Zi, Ni(·), Yi(·)}m i=1 are also independently and identically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We simplify Ni(·) as Ni and Yi(·) as Yi when no confusion arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In our motivating example, the experiment is carefully designed to ensure that the trials are independent and identically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For instance, the optical stimulation is targeted at the same location at the same chosen power to eliminate unintentional variability to ensure the identical distribution condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' the trials are separated with adequate washout periods to ensure the independence between units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' the power of optical stimulation and the length of the experiment are limited to avoid physical damage to the neural circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 1, we impose the following three assumptions throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' First, we generalise the exclusion restriction assumption in Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 2 (Exclusion restriction) Yiz′τ = Yizτ for z, z′ = 0, 1 and all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 2 means that the instrumental variable affects the outcome only through the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It holds in optogenetic experiments since only the targeted neurons respond to optical stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 2, we can simplify Yizτ as Yiτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' There are two ways to describe the potential outcome processes under Assumption 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Yiz and Yiτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We will use Yi1 and Yi0 to represent the potential processes if the instrumental variable were set to Zi = 1 and Zi = 0, respectively, and Yiτ to represent the potential process if the event time of Ni were set to Ti = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Second, the following independence assumption holds automatically because trials are randomly selected for optical stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 3 (Randomization) Zi {Niz(·), Yiτ(·) : z ∈ {0, 1}, τ ∈ [0, T] ∪ T +}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 3 implies Zi {Tiz, Yiz(t) : z ∈ {0, 1}, t ∈ [0, T]} under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It allows for the identification of the intention-to-treat effects of the instrumental variable on the treatment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, it is insufficient to identify the effect of the treatment on the outcome due to the possibility of unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Lastly, we invoke the following no anticipation assumption because the event time of Ni at a later time point cannot reversely affect Yi at a previous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 6 Assumption 4 (No anticipation) Yiτ(t) = Yiτ ′(t) for τ, τ ′ ≥ t and all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 4 is well-known in causal inference with time series data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Bojinov and Shephard, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The use of a non-strict inequality sign instead of a strict inequality in Assumption 4 indicates that the effect of Ni on Yi is not instantaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Replacing the non-strict inequality with a strict inequality allows for Yiτ(τ) ̸= Yiτ ′(τ) for τ < τ ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', the event at time τ has an effect on the outcome at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The non-strict inequality in Assumption 4 also implies YiT + = YiT because the event at time T does not have an effect on Yi in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumptions 1–4, the relationships among Zi, Ni, Yi, and the unmeasured confounder Ui can be illustrated by the causal diagram in Figure 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The randomized stimulation Zi affects the treatment Ni, which in turn affects the outcome Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Because the treatment Ni is not randomized, unmeasured confounders Ui may exist between Ni and Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 Definitions of causal effects with point process treatment and outcome We are now ready to define the causal quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' First, we define the average causal effect (ACE) of the instrumental variable on the treatment and outcome processes at time t as ACEN(t) = E{Ni1(t) − Ni0(t)} and ACEY (t) = E{Yi1(t) − Yi0(t)}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Although ACEN(t) and ACEY (t) are possible quantities of interest in the experiment, they do not directly answer how the treatment Ni affects the outcome Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we define the ACE of the treatment process on the outcome process as ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2) = E{Yiτ1(t) − Yiτ2(t)}, τ1 ≥ τ2 and τ1, τ2 ∈ [0, T] ∪ T +, (3) which characterizes how the change in the event time of Ni from τ2 to τ1 affects Yi at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A positive ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2) with τ1 ≥ τ2 implies that a later event in the treatment process increases the expected outcome process at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This effect varies over time t and depends on the two event times τ1 and τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define the average causal effect rate (ACER) of Ni on Yi as ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = lim ∆τ→0+ ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ + ∆τ, τ) ∆τ = ∂E{Yiτ(t)} ∂τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (4) The ACER measures how fast E{Yiτ(t)} changes given an infinitesimal change in the event time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This concept is similar to the infinitesimal shift function defined in Lok (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 4, we have ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2) = � � � � � � � � � � � � � ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2), if τ2 < τ1 < t ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' t, τ2), if τ2 < t ≤ τ1 0, if t ≤ τ2 ≤ τ1 , 7 and thus ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 if t ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When the treatment is a single-point process, we have the following relationship between the ACE and ACER of the treatment, ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2) = � τ1 τ2 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (5) Therefore, we can focus on the ACER because it determines the ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 3, the ACEs of the instrumental variable on the treatment and outcome processes can be identified by the observed differences between the stimulated and unstimulated groups, ACEN(t) = f(t) with f(t) = E{Ni(t) | Zi = 1} − E{Ni(t) | Zi = 0}, (6) ACEY (t) = h(t) with h(t) = E{Yi(t) | Zi = 1} − E{Yi(t) | Zi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (7) However, Assumption 3 is insufficient for the identification of the ACE and the ACER of the treat- ment process on the outcome process, because the treatment process is not randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 3 Nonparametric identification and estimation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 Nonparametric identification with a single-point process treatment We begin by generalising the monotonicity assumption in Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 5 (Monotonicity) For each i, the potential event times of Ni satisfy Ti1 ≤ Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 5 requires that the potential event time of Ni under stimulation will be no later than that without stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 5, the ACE of the instrumental variable on the treatment process at time τ equals the proportion of a subpopulation defined by the joint potential event times of Ni, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', ACEN(τ) = Pr(Ti1 ≤ τ < Ti0), τ ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Units in this subpopulation would have the event time of treatment process before or equal to τ with stimulation and after τ without stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Thus, these can be viewed as the compliers whose treatment is positively affected by the stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' With a point process treatment, the definition of compliers is time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similarly, the other three subpopulations, Ti0 ≤ τ < Ti1, max(Ti1, Ti0) ≤ τ, and τ < min(Ti1, Ti0), generalise the defiers, always-takers, and never-takers in the binary instrumental variable model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We cannot validate Assumption 5 since it depends on unit-level potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, Assumption 5 implies a testable condition that can be checked using the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 8 Proposition 1 Under Assumption 3, Assumption 5 implies, for all τ ∈ [0, T], Pr(Ti > τ | Zi = 1) ≤ Pr(Ti > τ | Zi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 1 states the stochastic dominance of the survival function of Ti under stimulation over that without stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can assess Assumption 5 by comparing the empirical survival functions of Ti in the stimulated and unstimulated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' If the two curves cross, then the testable condition in Proposition 1 is violated, which in turn falsifies Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, our identification results will consider scenarios both with and without Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Angrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (1996) show that the effect of the instrumental variable on the outcome equals the product of the effect of the instrumental variable on the treatment and the effect of the treatment on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The following theorem generalises their result to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 1 Suppose that Ni is a single-point process and Assumptions 1–4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For any t ∈ [0, T], we have ACEY (t) = � T 0 E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1}Pr(Ti0 ≤ τ < Ti1)dτ − � T 0 E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0}Pr(Ti1 ≤ τ < Ti0)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (8) If Assumption 5 holds in addition, then for any t ∈ [0, T], we have ACEY (t) = − � T 0 E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} · ACEN(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (9) In Theorem 1, ∂Yiτ(t)/∂τ is a generalised derivative that may consists of Dirac δ functions (Lax, 2002, Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Since the conditional set Ti1 ≤ τ < Ti0 depends on τ, it is important to note that E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = ∂E{Yiτ ′(t) | Ti1 ≤ τ < Ti0}/∂τ ′ |τ ′=τ, which is generally not equal to ∂E{Yiτ(t) | Ti1 ≤ τ < Ti0}/∂τ that takes into account the change of the conditional set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By rewriting ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) as E{∂Yiτ(t)/∂τ}, we can view E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} as the ACER in the subpopulation Ti1 ≤ τ < Ti0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In a sense, E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} gen- eralises the complier average causal effect in the binary instrumental variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similarly, E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1} generalises the average causal effect for the defiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' These conditional expectations might not equal the ACER because Yiτ(t) and (Ti1, Ti0) might not be independent due to the unmeasured confounding between Ni and Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The formula in (8) shows that the average causal effect of the instrumental variable on the out- come process, ACEY (t), equals the difference between the weighted averages of the two subpopula- 9 tion ACERs over the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The weights rely on the joint distribution of (Ti1, Ti0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When Assump- tion 5 holds, the first term on the right hand side of (8) vanishes and the weight Pr(Ti1 ≤ τ < Ti0) is equal to ACEN(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, (8) reduces to (9) under monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 3, ACEY (t) and ACEN(t) are identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Thus, we can view (8) and (9) as integral equations for the subgroup ACERs (Newey and Powell, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Unfortunately, these sub- population ACERs are not identifiable without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To provide some intuition, consider (9) under monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Based on the observed data, (6) and (7) give the identification formulas for ACEY (t) and ACEN(τ) for all t, τ ∈ [0, T] under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' So (9) is an integral equation for the unknown quantity defined as γ(τ, t) = E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Consider a discrete approximation of γ(τ, t) by evaluating its values over a K1 × K2 two-dimensional grid of (τ, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Equation (9) generates only K2 equations by considering the K2 grid of t, which cannot sustain the identification of K1 × K2 unknown values of γ(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Consequently, the identification of the ACERs is infeasible without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To address this problem, we invoke the following identification assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 6 (Stationarity) ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ1, τ2) = ACE(t − τ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0, τ2 − τ1) for τ1 ≤ τ2 ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 6 states that the ACE of the treatment on the outcome is invariant to timeline shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The left-hand side is the effect of the treatment when the event time is τ1 versus τ2 on the outcome at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In contrast, the right-hand side represents the same effect, but with the timeline shifted forward by τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, Assumption 6 means that the ACE of the treatment is invariant regardless of the absolute time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 6, we have ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) and thus can simplify ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) as ACER(t − τ) with ACER(t − τ) = 0 if t ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can show that, together with Assumption 4, Assumption 6 leads to ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) = −∂E{Yi0(t)}/∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (10) The formula in (10) offers a more natural interpretation of ACER, that is, −ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) describes the expected change rate in the potential outcome at time t when the event in Ni happens at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 2 below gives sufficient conditions for identifying the ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 2 Suppose that Ni is a single-point process and Assumptions 1–4 and 6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Further- more, if either (i) Assumption 5 holds and, for all t, τ ∈ [0, T], E {∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = ∂E {Yiτ(t)}/∂τ, (11) 10 or (ii) for all t, τ ∈ [0, T], E{∂Yiτ(t)/∂τ | Ti1 ≤ τ < Ti0} = E{∂Yiτ(t)/∂τ | Ti0 ≤ τ < Ti1} = ∂E{Yiτ(t)}/∂τ, (12) then ACER satisfies ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = � � � � � ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0), if t > τ 0, if t ≤ τ , (13) and h(t) = − � T 0 ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0)f(τ)dτ (14) for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' If further (Ψf)(ν) ̸= 0 for all ν ∈ R, then the ACER is identified by ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = −Ψ−1� G � (t − τ) for t > τ and ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 for t ≤ τ, where G(ν) = (Ψh)(ν) (Ψf)(ν) for all ν ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (15) The condition in (11) means that the ACERs are homogenous across subpopulations defined by Ti1 ≤ τ < Ti0 with different values of τ, generalising the homogeneity assumption in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Without monotonicity, the condition in (12) further requires that the ACERs are homogenous across sub- populations defined by Ti0 ≤ τ < Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to the instrumental variable methods in survival analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Tchetgen Tchetgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015), these conditions are satisfied as long as the impact of the confounders on the outcome is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' With a binary treatment and a scalar outcome, Wang and Tchetgen Tchetgen (2018) also use a similar condition assuming no additive interaction between the treatment and the unmeasured confounders on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We will study this condition in detail under several commonly-used outcome models in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The deconvolution problem (14) belongs to the family of Wiener–Hopf equations (see, among others, Noble, 1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It is essentially the same as the well-studied deconvolution of densities in statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Fan, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Diggle and Hall, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Pensky and Vidakovic, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Johannes, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Dattner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2011, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From the Paley–Wiener–Schwartz theorem, we know that (Ψf)(ν) ̸= 0 for all ν ∈ R if f(t) = E{Ni(t) | Zi = 1} − E{Ni(t) | Zi = 0} is a non-zero function with bounded support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This holds as long as the effect of the instrument Zi on the treatment process Ni vanishes in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The non-zero condition of (Ψf)(ν) is also employed in the nonparametric deconvolution problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Fan, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In the binary instrumental variable model with the homogeneity assumption, the effect of the treatment on the outcome equals the ratio of the effects of the instrumental variable on the treat- ment and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 2 shows that this ratio relationship also holds with point process 11 treatment and outcome, but in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The well-known convolution theorem ensures that the Fourier transform of a convolution of two functions is equal to the product of their Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, by applying the Fourier transform on each term of the convolution equation in (14), we can obtain the generalised Wald estimation formula (15) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Treatment with multiple events We generalise the identification result in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 to a treatment process with possibly multiple events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We begin by generalising the definition of potential values and causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Yi,n(·)(·) be the potential process of the outcome if the treatment were set to a fixed process n(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The observed outcome process is Yi(·) = Yi,n(·)(·) if Ni(·) = n(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Then, we can define the ACE of the treatment n(·) versus n′(·) on the outcome as ACE{t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (16) For single-point process treatments, (16) reduces to the definition in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using the linearity of expectation, we can write ACE{t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)} = E{Yi,n(·)(t) − YiT+(t)} − E{Yi,n′(·)(t) − YiT+(t)}, where E{Yi,n(·)(t) − YiT+(t)} and E{Yi,n′(·)(t) − YiT+(t)} are the effects of n(·) and n′(·) versus a null process with no events in [0, T], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1, we can characterize the treatment process using event times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Suppose that n(·) has l events at times τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Then Yi,n(·)(t) can be written as Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τl(t), so its expectation decomposes as E{Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τl(t)} = E{YiT+(t)} + l � s=1 E � Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τs(t) − Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τs−1(t) � (17) with Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τs−1(t) = YiT+(t) for s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The following assumption simplifies the decomposition by assuming away the interactive effects of the event times in the potential outcome process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 7 (Additivity) E � Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τs(·) − Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τs−1(·) � = E{Yi,τs(·)−YiT+(·)} for any s ≥ 1 and any event times (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τs) satisfying τ1 < τ2 < · · · < τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Point processes with event times (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τs) and (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τs−1) have the same trajectory up to time τs−1, where the former has an additional event at τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 7 means that the effect of the process with event times (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τs) versus that with event times (τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τs−1) does not depend on their common trajectory up to time τs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Hence, the causal effect remains the same when 12 the first s − 1 events are removed from both processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 7, (17) simplifies to E{Yi,τ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=',τl(t) − YiT+(t)} = �l s=1 E {Yi,τs(t) − YiT+(t)} , which means that the effect of each event time on the outcome process is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In §S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 in the supplementary material, we show that Assumption 7 holds under the Hawkes process (Hawkes, 1971) or Aalen’s additive hazard model (Aalen, 1980) for the potential outcome process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 7 may be violated due to the interactive effect of the event times in the treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For instance, neural ensembles are famous for their neural plasticity in the long term — the ability to reorganize themselves in response to stimulation, which clearly violates Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Such violations of Assumption 7 are sometimes of scientific interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We leave the investigation of such effects for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 7, we can separately study the effect of each event in Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 2 below generalises (5) to treatment processes with multiple events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 2 Under Assumptions 1, 2, 4, and 7, we have ACE{t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)} = − � t 0 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ){n(τ) − n′(τ)}dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Based on Proposition 2, we can focus on the identification of ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 3 below generalises Theorem 2 to treatment processes with multiple events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Theorem 3 Suppose that Assumptions 1–4, 6, and 7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' If for all t ∈ [0, T] and any fixed processes n(·) and n′(·), E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1(·) = n(·), Ni0(·) = n′(·)} = E{Yi,n(·)(t) − Yi,n′(·)(t)}, (18) then the ACER satisfies (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' If (Ψf)(ν) ̸= 0 for all ν ∈ R, then the ACER is identified by ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = −Ψ−1� G � (t − τ) for t > τ and ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 for t ≤ τ, with G(ν) defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When Ni is a single-point process, Theorem 3 does not require Assumption 7, and the condition in (18) reduces to (12) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, Theorem 3 reduces to Theorem 2 when Ni has at most one event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to Theorem 2, the condition in (18) means that the ACERs are homogenous across subpopulations defined by Ni1 and Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 Estimation We consider the estimation of the ACER based on identification results from Theorems 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This is essentially the deconvolution problem commonly studied in the literature (see Diggle and Hall, 13 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Pensky and Vidakovic, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Johannes, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Dattner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2011, 2016, for more discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Since an optimal estimation procedure is not the focus of this paper, we only provide a simple regression-based procedure to estimate ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To be specific, we use a two-step procedure by first obtaining the estimates of f and h and then solving the ACER from the empirical version of the convolution equation in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let ˆf and ˆh denote the estimators of f and h defined in (6) and (7), which equal the empirical mean differences in the treatment and outcome processes in the stimulated and unstimulated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We approximate the true ACER with truncated basis expansions, for ∆ ∈ [0, T], ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) ≈ J � j=1 ψj(∆)βj, (19) where J is a tuning parameter for the number of bases and {ψj(·) : j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , J} is a set of pre- specified basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Here the support of ACER(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) can be determined by prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Then, we estimate β = (β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , βJ) by minimizing the following penalized ℓ2-distance based on the convolution equation (14) �β = arg min β∈RJ ������ ˆh + J � j=1 (ψj ∗ ˆf)βj ������ 2 2 + η∥β∥2 2, (20) where ∗ denotes the convolution between two functions and introduce the ridge penalty to re- duce boundary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' An analytic solution for �β is available since the objective function in (20) is quadratic in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Recalling that we consider independent trials, we can choose the tun- ing parameter J and η using cross-validation or based on prior knowledge such as the smooth- ness of the ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Denoting the selected parameter by �J and ˆη, the final estimator is given as � ACER(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) = � � J j=1 ψj(·)ˆβj,ˆη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can then construct the confidence band for the function � ACER(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) using the bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The asymptotic properties for � ACER as the sample size m in- creases follow from the standard theory assuming independent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We leave the rigorous discussion for future analysis, as it is not the main focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 4 The role of models: causal interpretability and identifiability 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 Conditional intensity In this section, we study several commonly-used models for point process outcomes in applied re- search when an instrumental variable is available, allowing for the presence of unmeasured con- founders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We do not impose any distributional assumptions on the unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Con- sequently, it is difficult to study the identifiability of the model parameters themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We take 14 an alternative route by connecting the model parameters to the causal effects and considering the identifiability and estimation of the causal effects directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' With a binary instrumental variable, we show that the ACER is identifiable and can be estimated using the generalised Wald estimation under many commonly-used models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This estimation strategy does not rely on the identification or estimation of the model parameters, as long as the unmeasured confounding is additive in the underlying outcome model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We begin by introducing some additional notation to characterize a point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let Ui(·) denote the unmeasured confounding process on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We use Hit− to represent the σ-algebra induced by the history up to, but not including, time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define the conditional intensity of Yi as λY (t) = E {dYi(t)/dt | Hit−} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (21) The conditional intensity, or intensity, is the conditional mean of the event rate of Yi in an infinitesi- mal time interval [t, t+dt), which is analogous to the conditional mean of the outcome in the binary instrumental variable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It fully characterizes the probabilistic structure of a point process and is closely related to the hazard function in survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' See Chapter 7 in Daley and Vere-Jones (2003) for more discussion of the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In (21), the conditional intensity could depend on the history of Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When the outcome Yi describes recurrent events, it is common to allow the conditional intensity to depend on past events of Yi (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Hawkes, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Brillinger, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Lawrence, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Kulkarni and Paninski, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Macke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Zhao and Park, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Pandarinath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As concrete examples, in the context of neural data, the dependence on past events captures the known phenomenon that a single neuron cannot fire consecutively in a very short period of time and activities in a region may trigger inhibitory circuits to stabilize the activity on a longer time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' An inherent constraint on the intensity is that it must be non-negative for the probabilistic model to be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A similar constraint is well acknowledged in modeling the hazard function in survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This constraint of the intensity creates a schism in the modeling of point processes — whether to employ a linear working model (Aalen, 1980) or a non-negative generative model (Cox, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In either model, since Ui is unobserved, existing methods usually impose strong parametric assumptions on Ui in order to estimate the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A common assumption is that Ui is a Gaussian process (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Zhao and Park, 2017), primarily due to its simplicity for the Bayesian computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, the analysis can be sensitive to these parametric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2, we will study both types of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 15 Before specifying λY (t), we introduce the following assumption on the relationships among Ni, Yi, and Ui, which are commonly used in instrumental variable methods when outcome models are employed (see Tchetgen Tchetgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015, for an example in survival analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 8 (a) Zi {Niz(t), Yiτ(t), Ui(t) : z ∈ {0, 1}, t ∈ [0, T], τ ∈ [0, T] ∪ T +};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (b) For t ∈ [0, T], Yi,n(·)(t) Ni(·) | {H∗\\Ni it− , Ni(s) = n(s), s ∈ [0, t)} and any fixed point process n(·), where H∗\\Ni it− denotes the σ-algebra induced by all potential processes including Ui, except for Ni up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' See Lok (2008) for the measure-theoretic description of the independence given histories of point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 8(a) is a restatement of Assumption 3 with the additional notation of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It holds because Zi is randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Assumption 8(b) generalises the latent sequential ignorability in Ricciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (2020) to a continuous-time setting with point process treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It assumes that Ui fully characterizes the confounding between the treatment and the outcome so the treatment is independent of the potential outcome at time t given the histories of Ni, Yi, and Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 8, we have, for any t ∈ [0, T], E � dYi,n(·)(t)/dt | Hit− � = E � dYi(t)/dt | Ni(·) = n(·), H∗\\Ni it− � , (22) which links the potential processes to the conditional intensity of the observed outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, the discussion in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 will focus on the models for the observed outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under each of the models, we will connect the model parameters with the ACER and study its identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Identification of causal effects with linear additive unmeasured confounding We start with linear models for the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This type of models has been widely used in different contexts because of its mathematical tractability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', Hawkes, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Aalen, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Tchetgen Tchet- gen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, consider Yi(·) to be a linear Hawkes process with the following intensity, λY (t) = µY + � t 0 g(t − s)dNi(s) + � t 0 ω(t − s)dYi(s) + ψUi(t), (23) where g(∆) = ω(∆) = 0 for ∆ ≤ 0 and ψUi(t) represents any function of {Ui(s) : s ∈ [0, t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 3 below connects the ACER with parameters in (23) and shows the identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 3 Suppose that Assumptions 1–3 and 8 hold, and the underlying outcome model sat- isfies (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (a) We have ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) = − � Ψ−1 �G � (t − τ), 16 where �G(ν) = � 1 + (Ψω)(ν) �−1� Ψg � (ν) if 1 + (Ψω)(ν) ̸= 0 for all ν ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (b) When (Ψf)(ν) ̸= 0 for all ν ∈ R, ACER is identified by ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = − � Ψ−1G � (t − τ) for t > τ and ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 for t ≤ τ, with G(ν) defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In practice, the function g(·) is often interpreted as the effect of an event in Ni on the outcome Yi conditional on the history up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 3(a) expresses the ACER in terms of the model parameters, showing that g(·) and ω(·) jointly characterize the ACER of Ni on Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When the dependence on past Yi does not exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', ω(·) ≡ 0, we have ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = −g(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From Proposition 3(a), we can obtain the ACER if we can estimate the model parameters in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' How- ever, this requires specifying the distribution of Ui(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Fortunately, Proposition 3(b) shows that we can identify the ACER without any distributional assumption on Ui when a binary instrumental variable is available, and hence estimate it using the method in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It broadens the applicability of Model (23) with fewer parametric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 3(b) is an application of Theorem 2 under Model (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The linearity in Model (23) plays a key role in the causal interpretation of the model parameters and nonparametric identification of the ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The linear terms of Ni and Yi connect the ACER with g(·) and ω(·), and the linear term of Ui implies Assumption 6 and the condition in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 Identification of causal effects with nonlinear additive unmeasured confound- ing We now consider the following nonlinear model that is similar to models in survival analysis with an instrument (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', MacKenzie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Tchetgen Tchetgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015): λY (t) = φ � µY + � t 0 g(t − s)dNi(s) � + ψUi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (24) Model (24) generalises Model (23) by allowing for a nonlinear relationship between Ni and Yi through the link function φ while requiring the unmeasured confounding effect to be additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For Model (24), the following proposition characterizes the causal effect and its identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 4 Suppose that Assumptions 1–3 and 8 hold, Ni is a single-point process, and the underlying outcome model satisfies (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (a) We have, for t, τ ∈ [0, T], ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = φ(µY ) − φ{µY + g(t − τ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (b) When (Ψf)(ν) ̸= 0 for all ν ∈ R, ACER is identified by ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = − � Ψ−1G � (t − τ) for t > τ and ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 for t ≤ τ, where G(ν) is defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 17 From Proposition 4(a), the causal interpretation of g(·) depends on µY and the link function φ(·) under Model (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, even with the same link function, the interpretation of g(·) differs in populations with different values of µY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This warns us of interpreting g(·) as some causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 4(b) is an application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to Model (23), we can use the method in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 to estimate the ACER without the knowledge of φ(·) or Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, Proposition 4 suggests directly targeting the ACER instead of the model parameter g(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This circumvents the daunting task to identify, estimate, and interpret the model parameters in Model (24), broadening its applicability with fewer parametric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Critically, although Model (24) allows for nonlinearity, it restricts the effect of unmeasured con- founder to be additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Relaxing this modeling assumption is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In §S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='7 in the sup- plementary material, we show that when the confounding effect on Yi is non-additive, the ACER would depend on the distribution of the confounder, making the identification not possible without a distributional assumption on Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 5 Numerical analysis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 Simulation We use simulation to illustrate the numerical performance of the proposed nonparametric estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this simulation study, we generate the treatment Ni and outcome Yi from the following model λN(t) = µN + φβ0 � α(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' aN, bN)Zi + Ui(t) � , (25) λY (t) = φβ2 � φβ1 � µY + � t 0 α(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' aY , bY )dNi(t − ∆) � + φβ1 {Ui(t − dU)} � , (26) where α(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' a, b) = ba2t exp(−at) is the alpha function (see, among others, Chapter 7 in Ermentrout and Terman, 2010) and φβ(x) = xβ is the link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The confounding variable Ui is generated as a Gaussian process with mean zero and a squared exponential kernel cov{Ui(t), Ui(t + d)} = σ2 U exp{−d2/(2l2 U)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The parameters in (25) and (26) are set as µN = µY = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2, aN = 10, bN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5, aY = 8, bY = 1, dU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5, σU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2, and lU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We consider five scenarios in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Scenario 1a (β0 = β1 = β2 = 1): A linear model for Yi with a single-point process Ni, which is achieved by suppressing the intensity (25) to zero after the first event in Ni is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Scenario 1b (β1 = 2, β0 = β2 = 1): An additive confounding model for Yi with a single-point process Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Scenario 2a (β0 = β1 = β2 = 1): A linear model for Yi with multiple events in Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 18 Scenario 2b (β0 = 3, β1 = 2, β2 = 1): A linear model for Yi with multiple events in Ni and non-additive confounding effects on Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Scenario 3 (β0 = β1 = 1, β2 = 3): A non-additive confounding model for Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For each scenario, we generate m trials with m ranging from 40 to 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In each simulated dataset, half of the trials are set to have Zi = 1 and the other half Zi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The processes Ni and Yi are generated from 0 to T = 3 using thinning process, while the unmeasured confounding process Ui is generated from −1 to 3 to account for its delayed effect on Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For Scenario 3, the identification of ACER is difficult with nonlinear confounding effects, as illustrated in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' So we use the Monte Carlo method to calculate the ACER to show its dependence on the distribution of the unmeasured confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For Scenarios 1a to 2b, we apply the proposed generalised Wald estimation procedure in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We estimate the function h(t) as the difference between the empirical cumulative intensities of Yi in the treatment group (Zi = 1) and control group (Zi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The function f(t) is estimated in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We approximate the ACER using a cubic B-splines with 6 knots evenly-spaced in [0, 1] where the mass of α(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' aY , bY ) resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The tuning parameter of the ridge penalty η is set to be m−1 to reduce boundary effects from the nonparametric approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To measure the performance, we calculate the proportion of integrated squared errors with respect to the true ACER, that is r = � 1 0 � � ACER(∆) − ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) �2 d∆ � 1 0 ACER2(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0)d∆ , (27) where the true ACER is calculated from (26) based on Propositions 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figure 2 shows the simulation results averaged over 1000 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figures 2(a) and (b) show that the performances of estimators improve as the number of trials increases in Scenarios 1a, 1b, 2a, and 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, the proportion of integrated squared error in Scenario 1a is larger than that in Scenario 2a, despite having the same model for Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This reveals a feature for point process treatments that, given the same amount of trials, more events in Ni contribute more information for recovering the causal effects of Ni on Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The four curves in Figures 2(a) and (b) converge slowly towards zero due to the existence of approximation error in the basis expansion and the bias from penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figures 2(c) and (d) show the calculated true ACER in Scenario 3 under two different distributions of Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Within each of Figures 2(c) and (d), the five curves correspond to τ being 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='75, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Even with the same t, the shape of ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) varies as τ changes in both figures, implying that ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) does not equal to ACER(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Moreover, the contrast between Figure 2(c) and (d) shows that ACER depends on the distribution of the unmeasured process Ui, 19 demonstrating the sensitivity of the ACER to distributional assumptions on Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 200 400 600 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 Scenario 1a Scenario 1b Number of trials (m) MISE (prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=') (a) 200 400 600 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 Scenario 2a Scenario 2b Number of trials (m) MISE (prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=') (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Time (t) −ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) σU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Time (t) −ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) σU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 (d) Figure 2: Identification of ACER and performance of the generalised Wald estimation averaged across 1000 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panels (a) and (b) illustrates the performance of the proposed estimation procedures in Scenarios 1a, 1b (Panel a), 2a, and 2b (Panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The x-axes are the numbers of trials (m), and the y-axes are the measure defined in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded areas are the interquartile bands from the 1000 replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Estimation performances in the four scenarios are not directly comparable given the huge difference between the data generating mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Based on the interpretation in (10), Panels (b) and (c) show the true value of −ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) with non-additive confounding effects on Y (Scenario 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The spectrum of gray curves is calculated with σU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1, and the red curves with σU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Within each spectrum, the five curves correspond to τ being 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='75 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Curves in each color spectrum are not shift-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Empirical analysis We now apply the proposed methodology to the neural data from Bolding and Franks (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We provide the basic background to help understand the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For more details, see the supplementary material and Bolding and Franks (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Bolding and Franks (2018a) conduct an experiment to understand how a mouse brain maintains stationarity in odor detection regardless of odor concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To be specific, it is known that neural activities in the olfactory bulb (OB) increase in response to a higher concentration of odor particles, and that a spike in OB triggers neural activities of principal neurons (PN) in the piriform cortex, where the odor is perceived by the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To avoid other neural processes that normalize odor responses, Bolding and Franks (2018a) 20 use optogenetics to stimulate neurons in OB with 1-s light pulses, which meet the requirements as an instrumental variable in our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Bolding and Franks (2018a) also take an optogenetic to circumventing the contribution of centrifugal inputs and other intrabulbar processes, which effectively cuts of the feedback from PN to OB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figure 3(d) shows a causal diagram for the relationship among the stimulation, OB, and PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The dataset contains spike trains recorded in OB and PN during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A total of 160 trials are conducted on 8 mice, where each mouse has 10 trials without stimulation (Zi = 0) and 10 trials with a one-second light pulse at 20 mW/mm2 (Zi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The light pulse, if present, onsets at time 0 and ends at 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In our analysis, we consider the first 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 seconds of a trial, from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 to 3, as there are hardly any residual effects afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We consider the treatment Ni as the process of events in OB, and the outcome Yi as the process of events in PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Each recorded event in Ni is a spike of one neuron in OB that may instigate a distinct group of PN in the piriform cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Given the vast amount of PN in the piriform cortex, the instigated groups may share few or no overlaps, limiting the interactive effect of the treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, the additivity in Assumption 7 is a plausible approximation to the true underlying mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figures 3(a) and (b) show the smoothed intensities of neural activities in the stimulated (blue) and unstimulated (red) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can see that the stimulation triggers increased activities in OB in all trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, there are large variations in the neural activities across trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We first conduct a preliminary analysis assuming no unmeasured confounders between the treat- ment and outcome processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this case, the identification of the causal effects does not require the instrumental variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We fit a model of Yi on Ni and directly interpret the coefficient function of Ni as the causal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, the conclusion is inconsistent with the findings in Bolding and Franks (2018a), implying possible unmeasured confounders or model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' See more details in §S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We then apply the generalised Wald estimation procedure to estimate the causal effects of neural activities in OB on the neural activities of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From the observed data, we estimate the functions h and f using differences between empirical cumulative intensities in the stimulated (Zi = 1) and unstimulated (Zi = 0) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We approximate the unknown ACER using cubic B-splines with evenly-spaced knots in [0, 1], where two knots are selected by a 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We set the tuning parameter for the ridge penalty to be η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='01 to handle boundary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A 90% confidence band is constructed using the bootstrap with details in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We use the bootstrap confidence band to approximate the uncertainty in the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 21 OB Light U PN ACER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 −30 0 30 60 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0 500 1000 1500 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0 500 1000 1500 (a) Trials (b) Time (in seconds) Trials ∆ (in seconds) −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) (c) Time (in seconds) (d) Figure 3: Empirical intensities and fitted ACER on data from Bolding and Franks (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panels (a) and (b) show the empirical intensities of the neural activities of OB (Panel a) and PN (Panel b) in the stimulated (blue) and unstimulated (red) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The solid curves represent the average intensity over 80 trials, and the dashed curves demonstrate the empirical intensity from 20 randomly selected trials in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area in Panel (a) represents the duration of the light pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Based on the interpretation in (10), Panel (c) shows the estimated −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) from the full data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area represents a 90% confidence band for visualizing the uncertainty of the estimates from 5000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (d) shows the causal diagram for the relationship among the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figure 3(c) shows the estimated ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The curve shows that an event in OB elicits high activities in PN immediately after the event (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 seconds), but the effect quickly turns negative for an extended duration (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 seconds) before it dies down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This is consistent with the findings in Bolding and Franks (2018a) that a temporal mechanism is in place to stabilize the neural activities of PN after the initial detection of odors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Additional analysis of the neural dataset can be found in §S3 in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The confidence band shows that the proposed generalised Wald estimation procedure yields high uncertainty near the boundaries, despite a large number of events and the ridge penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this particular case, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Gaussian-process factor analysis for low-dimensional single-trial analysis of neural population ac- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In Advances in neural information processing systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 1881–1888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Park (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Variational latent Gaussian process for recovering single-trial dynamics from population spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Neural Computation 29(5), 1293–1316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 26 Supplementary Material §S1 generalises the framework to discrete instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' §S2 provides the proofs of the theorems, propositions, and claims in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' §S3 provides more details about the empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let 1(·) denote the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Recall i = √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' S1 Generalisation to discrete instrumental variables In the main text, we discuss the case when the instrumental variable takes binary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Here we briefly outline the generalisation to the case with discrete instrumental variables where there could be multiple levels of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For a discrete instrumental variable, we can apply the proposed methodology by comparing two levels z and z′ (instead of 0 and 1 in the binary instrumental variable case) under the monotonicity and homogeneity assumptions with respect to these two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For simplicity, we only give the result with a single-point process Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define ACEY (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' z, z′) = E{Yiz(t) − Yiz′(t)}, ACEN(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' z, z′) = E{Niz(t) − Niz′(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can obtain a similar result as Theorem 1 with respect to z and z′: ACEY (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' z, z′) = � T 0 E{∂Yiτ(t)/∂τ | Tiz′ ≤ τ < Tiz}Pr(Tiz′ ≤ τ < Tiz)dτ − � T 0 E{∂Yiτ(t)/∂τ | Tiz ≤ τ < Tiz′}Pr(Tiz ≤ τ < Tiz′)dτ, which, under monotonicity Tiz ≤ Tiz′, simplifies as ACEY (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' z, z′) = − � T 0 E{∂Yiτ(t)/∂τ | Tiz ≤ τ < Tiz′} · ACEN(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' z, z′)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define hz,z′(t) = E{Yi(t) | Zi = z} − E[Yi(t) | Zi = z′}, fz,z′(t) = E{Ni(t) | Zi = z} − E{Ni(t) | Zi = z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under randomization (Assumption 3) and stationarity (Assumption 6), if E �∂Yiτ(t) ∂τ ���� Tiz ≤ τ < Tiz′ � = E �∂Yiτ(t) ∂τ ���� Tiz′ ≤ τ < Tiz � = ∂E {Yiτ(t)} ∂τ , then hz,z′(t) = − � T 0 ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0)fz,z′(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 27 This is similar to Theorem 2 with respect to z and z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, if (Ψfz,z′)(ν) ̸= 0 for all ν ∈ R, then the ACER is identified by ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = −Ψ−1(G)(t − τ) with G(ν) = � Ψhz,z′� (ν) � Ψfz,z′� (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For estimation, we can use the data with Zi = z, z′ and apply the same method as in the binary instrumental variable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' S2 Proofs S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 Proof of Theorem 1 First, we have ACEY (t) = E{Yi1,Ti1(t)} − E{Yi0,Ti0(t)} = E{Yi,Ti1(t)} − E{Yi,Ti0(t)}, (S1) where the second equality follows from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using the Stieltjes integral, we can write Yi,Ti1(t) = � [0,T] Yiτ(t)dNi1(τ) + YiT+(t)1{Ni1(T) = 0} = � [0,T] Yiτ(t)dNi1(τ) + YiT (t)1{Ni1(T) = 0}, (S2) where the second equality follows from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similarly, we have Yi,Ti0(t) = � [0,T] Yiτ(t)dNi0(τ) + YiT (t)1{Ni0(T) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S3) Plugging (S2) and (S3) into (S1) yields ACEY (t) = E �� [0,T] Yiτ(t)dNi1(τ) + YiT (t)1{Ni1(T) = 0} � −E �� [0,T] Yiτ(t)dNi0(τ) + YiT (t)1{Ni0(T) = 0} � = E �� [0,T] Yiτ(t){dNi1(τ) − dNi0(τ)} � + E [YiT (t)1{Ni1(T) = 0} − 1{Ni0(T) = 0}] = E �� [0,T] Yiτ(t){dNi1(τ) − dNi0(τ)} � + E [YiT (t){Ni0(T) − Ni1(T)}] , (S4) where the last equality holds because Ni0 and Ni1 are single-point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 28 We then focus on the first term of (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let HN izt = {Niz(s) : s ∈ [0, t]} denote the history of Niz up to time t for z = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By switching the order of the expectation and integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' we have E �� [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] Yiτ(t){dNi1(τ) − dNi0(τ)} � = � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E [Yiτ(t){dNi1(τ) − dNi0(τ)}] = � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � d{Ni1(τ) − Ni0(τ)} � = E �� [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � d{Ni1(τ) − Ni0(τ)} � = E � E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � {Ni1(τ) − Ni0(τ)} ��� τ=T τ=0 � −E �� [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] ∂E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � ∂τ {Ni1(τ) − Ni0(τ)}dτ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S5) where the second equality follows from the law of total expectation and the last equality follows from integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can write the first term of (S5) as E � E � Yiτ(t) | HN i1T , HN i0T � {Ni1(τ) − Ni0(τ)} ��� τ=T τ=0 � = E � E � YiT (t) | HN i1T , HN i0T � {Ni1(T) − Ni0(T)} − 0 � = E [YiT (t){Ni1(T) − Ni0(T)}] , where the first equality follows from Ni1(0) − Ni0(0) = 0 and the second equality follows from the law of total expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, the first term of (S4) becomes E �� [0,T] Yiτ(t){dNi1(τ) − dNi0(τ)} � = E [YiT (t){Ni1(T) − Ni0(T)}] −E �� [0,T] ∂E � Yiτ(t) | HN i1T , HN i0T � ∂τ {Ni1(τ) − Ni0(τ)}dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S6) Plugging (S6) into (S4), we obtain ACEY (t) = −E �� [0,T] ∂E � Yiτ(t) | HN i1T , HN i0T � ∂τ {Ni1(τ) − Ni0(τ)}dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S7) Because Ni(·) is a single-point process, Ni1(τ) − Ni0(τ) takes values only in {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In addition, Ni1(τ) − Ni0(τ) = 1 is equivalent to Ti1 ≤ τ < Ti0 and Ni1(τ) − Ni0(τ) = −1 is equivalent to Ti0 ≤ τ < Ti1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' by switching the order of the expectation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' and derivative in (S7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' we have ACEY (t) = − � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � ∂E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � ∂τ ����� Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ 29 + � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � ∂E � Yiτ(t) | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � ∂τ ����� Ti0 ≤ τ < Ti1 � Pr(Ti0 ≤ τ < Ti1)dτ = − � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � E �∂Yiτ(t) ∂τ ���� HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � ����� Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ + � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E � E �∂Yiτ(t) ∂τ | HN i1T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' HN i0T � ����� Ti0 ≤ τ < Ti1 � Pr(Ti0 ≤ τ < Ti1)dτ = − � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E �∂Yiτ(t) ∂τ ���� Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ + � [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='T] E �∂Yiτ(t) ∂τ ���� Ti0 ≤ τ < Ti1 � Pr(Ti0 ≤ τ < Ti1)dτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' where the first equality follows from the law of total probability and the third equality follows from the law of total expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This proves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When Assumption 5 holds, Ni1(τ)−Ni0(τ) takes values only in {0, 1} and ACEN(τ) = Pr(Ti1 ≤ τ < Ti0), Therefore, by switching the order of the expectation, integral, and derivative in (S7), we have ACEY (t) = − � [0,T] E � ∂E � Yiτ(t) | HN i1T , HN i0T � ∂τ ����� Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ = − � [0,T] E �∂Yiτ(t) ∂τ ���� Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ = − � [0,T] E �∂Yiτ(t) ∂τ ���� Ti1 ≤ τ < Ti0 � ACEN(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This proves (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Proof of Theorem 2 First, from Assumption 6 and the definition of ACER, we have ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = lim ∆τ→0+ E{Yi,τ+∆τ(t) − Yiτ(t)} ∆τ = lim ∆τ→0+ E{Yi,∆τ(t − τ) − Yi0(t − τ)} ∆τ , which depends only on t − τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we can write ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ACER(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Second, under Assumption 3, we have ACEY (t) = E{Yi(t) | Zi = 1} − E{Yi(t) | Zi = 0} = h(t) and Pr(Ti1 ≤ τ < Ti0) − Pr(Ti0 ≤ τ < Ti1) = ACEN(τ) = f(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Finally, we can write (8) in Theorem 1 as h(t) = − � [0,T] E �∂Yiτ(t) ∂τ ����Ti1 ≤ τ < Ti0 � Pr(Ti1 ≤ τ < Ti0)dτ 30 + � [0,T] E �∂Yiτ(t) ∂τ ����Ti0 ≤ τ < Ti1 � Pr(Ti0 ≤ τ < Ti1)dτ = � [0,T] ∂E {Yiτ(t)} ∂τ {Pr(Ti0 ≤ τ < Ti1) − Pr(Ti1 ≤ τ < Ti0)} dτ = − � [0,T] ACER(t − τ)f(τ)dτ, (S8) where the second equality follows from the condition in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking the Fourier transform on both sides of (S8) yields, for all ν, (Ψh)(ν) = −(ΨACER)(ν)(Ψf)(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, if (Ψf)(ν) ̸= 0 for all ν ∈ R, then ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = −Ψ−1� G � (t − τ) for t > τ and 0 otherwise, where G(ν) = (Ψh)(ν)/(Ψf)(ν) for all ν ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 Proof of Proposition 2 From Assumption 7, we have E{Yi,n(·)(t) − YiT+(t)} = l � j=1 E � Yi,τj(t) − YiT+(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Assumption 4, we have Yi,τj(t) = YiT+(t) for j > n(t) and YiT+(t) = Yit(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, for j > n(t), we have E � Yi,τj(t) − YiT+(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S9) Therefore, E{Yi,n(·)(t) − YiT+(t)} = n(t) � j=1 E � Yi,τj(t) − YiT+(t) � = n(t) � j=1 ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τj, T +) = n(t) � j=1 ACE(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τj, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, E{Yi,n(·)(t) − YiT+(t)} = − n(t) � j=1 � t τj ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ = − n(t) � j=1 �� t τn(t) ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ + � τn(t) τj ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ � = −n(t) · � t τn(t) ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ − n(t)−1 � j=1 � � � n(t)−1 � k=j � τk+1 τk ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ � � � = −n(t) · � t τn(t) ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ − n(t)−1 � k=1 k � j=1 � τk+1 τk ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ 31 = −n(t) · � t τn(t) ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ − n(t)−1 � k=1 k · � τk+1 τk ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Because n(τ) = � � � � � � � � � � � � � k, if τ ∈ [τk, τk+1) 0, if τ ∈ [0, τ1) n(t), if τ ∈ [τn(t), t] , we have E{Yi,n(·)(t) − YiT+(t)} = − � t τn(t) ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)n(τ)dτ − n(t)−1 � k=1 � τk+1 τk ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)n(τ)dτ = − � t 0 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)n(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, we have E{Yi,n(·)(t) − Yi,n′(·)(t)} = − � t 0 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) � n(τ) − n′(τ) � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 Proof of Theorem 3 Under Assumptions 2 and 3, we can write, for any t ∈ [0, T], h(t) = E{Yi,Ni1(t)} − E{Yi,Ni0(t)} = � n(·),n′(·)∈N E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n(·), Ni0 = n′(·)}Pr{Ni1 = n(·), Ni0 = n′(·)}, where N is the sample space of simple point process on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Here we abuse the notation Pr(·) to represent the density and take Σ as the summation over all possible pairs of n(·) and n′(·) in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From the condition in (18), we know that h(t) = � n(·),n′(·)∈N E{Yi,n(·)(t) − Yi,n′(·)(t)}Pr{Ni1 = n(·), Ni0 = n′(·)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' It follows from Proposition 2 that h(t) = − � n(·),n′(·)∈N Pr{Ni1 = n(·), Ni0 = n′(·)} � t 0 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) � n(τ) − n′(τ) � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, we have h(t) = − � t 0 ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)E{Ni1(τ) − Ni0(τ)}dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 32 From Assumption 3, we know that E{Ni1(τ) − Ni0(τ)} = E{Ni(τ) | Zi = 1} − E{Ni(τ) | Zi = 0} = f(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Finally, using that ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ACER(t − τ) from Assumption 6, we have h(t) = − � t 0 ACER(t − τ)f(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The rest follows from the same argument as in §S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 Proof of Proposition 3 We first prove Proposition 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Consider a single-point treatment process Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From Assumption 8, we know (22) holds that, for any t ∈ [0, T], E �dYiτ(t) dt ���� Hit− � = E �dYi(t) dt ���� Ti = τ, H∗\\Ni it− � = λY (t), where the last equality follows from the definition of λY (t) in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Plugging in Model (23), we know that E �dYiτ(t) dt ���� Hit− � = µY + g(t − τ) + � t 0 ω(t − s)dYiτ(s) + ψUi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S10) Taking the expectation on both sides of (S10), we obtain E �dYiτ(t) dt � = µY + g(t − τ) + � t 0 ω(t − s)E{dYiτ(s)} + E{ψUi(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S11) Integrating both sides of (S11) from 0 to t yields E {Yiτ(t)} = µY t + � t 0 g(s − τ)ds + � t 0 � s 0 ω(s − ∆)E {dYiτ(∆)} ds + � t 0 E{ψUi(s)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S12) For the third term in (S12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(s − ∆)E {dYiτ(∆)} ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(s − ∆)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆′)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(s − ∆′)ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(∆′ = s − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆′)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='−dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆′ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(d(s − ∆) = −d∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆′)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆′ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆′)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆′ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='∆′ ω(∆′)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dsd∆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(Fubini’s Theorem) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='∆′ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYiτ(s − ∆′) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆)E{Yiτ(t − ∆)}d∆ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='ω(∆)d∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='E{Yiτ(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, E {Yiτ(t)} = µY t + � t 0 g(s − τ)ds + � t 0 ω(∆)E{Yiτ(t − ∆)}d∆ − �� t 0 ω(∆)d∆ � E{Yiτ(0)} + � t 0 E{ψUi(s)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking the derivative with respect to τ on both sides of the above equation yields ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ∂E{Yiτ(t)} ∂τ = − � t 0 ∂g(s − τ) ∂s ds − � t 0 ω(∆) ∂ ∂τ E{Yiτ(t − ∆)}d∆ = −g(t − τ) − � t 0 ω(∆)ACER(t − ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)d∆, where ∂E{Yiτ(0)}/∂τ = 0 follows from (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using the property of the Dirac δ function, we obtain, � t 0 {ω(∆) + δ(∆)}ACER(t − ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)d∆ = −g(t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S13) Taking the Fourier transform on both sides of (S13) yields, for all ν, {(Ψω)(ν) + 1}(ΨACERτ)(ν) = − exp(−iντ)(Ψg)(ν), where ΨACERτ is a short-hand notation for the Fourier transform of ACER(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) and the right- hand side follows from the Fourier shift theorem that Ψg(t − τ) = exp(−iντ)Ψg(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we have (ΨACERτ)(ν) = − exp(−iντ) (Ψg)(ν) (Ψω)(ν) + 1 = − exp(−iντ) �G(ν), where �G(ν) = � 1 + (Ψω)(ν) �−1� Ψg � (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By setting τ = 0, we obtain ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) = − � Ψ−1 �G � (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we have (ΨACERτ)(ν) = exp(−iντ)(ΨACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0))(ν), where ΨACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) is the Fourier transform of ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using the Fourier shift theorem, we obtain ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = ACER(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 34 We now prove Proposition 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The result is an application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, we know that Assumption 6 holds from Part (a), we need to verify the condition in (18), Assumption 7, and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Verifying the condition in (18): Define x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) ≡ E � Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By the law of total expectation, we have, for any fixed point processes n1(·) and n0(·), E � Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) � = E � E � Yi,n(·)(t) − Yi,n′(·)(t) | Hit−, Ni1 = n1(·), Ni0 = n0(·) � ��Ni1 = n1(·), Ni0 = n0(·) � = E � E � Yi,n(·)(t) | Hit− � − E � Yi,n′(·)(t) | Hit− � ��Ni1 = n1(·), Ni0 = n0(·) � , where the last equality holds since the Ni1 and Ni0 are fixed up to time t conditioning on the history Hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to the derivation of (S12), we can obtain E � Yi,n(·)(t) | Hit− � − E � Yi,n′(·)(t) | Hit− � = µY t + � t 0 � s 0 g(s − l)dn(l)ds + � t 0 ψUi(s)ds + � t 0 � s 0 ω(s − l)dYi,n(·)(l)ds −µY t − � t 0 � s 0 g(s − l)dn′(l)ds − � t 0 ψUi(s)ds − � t 0 � s 0 ω(s − l)dYi,n′(·)(l)ds = � t 0 � s 0 g(s − l)d{n(l) − n′(l)}ds + � t 0 � s 0 ω(s − l)d{Yi,n(·)(l) − Yi,n′(·)(l)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we have E � Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·) � = E � E � Yi,n(·)(t) | Hit− � − E � Yi,n′(·)(t) | Hit− � | Ni1 = n1(·), Ni0 = n0(·) � = � t 0 � s 0 g(s − l)d{n(l) − n′(l)}ds + � t 0 � s 0 ω(s − l)dx(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))ds, where � t 0 � s 0 ω(s − l)dx(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))ds = � t 0 � s 0 ω(s − l)dx(l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) dl dlds = � t 0 � s 0 ω(l′)dx(s − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) d(s − l′) d(s − l′)ds (l′ = s − l) = � t 0 � 0 s ω(l′)−dx(s − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) d(s − l′) dl′ds = � t 0 � s 0 ω(l′)dx(s − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) ds dl′ds 35 = � t 0 � t l′ ω(l′)dx(s − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) ds dsdl′ (Fubini’s Theorem) = � t 0 ω(l′) �� t l′ dx(s − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) ds ds � dl′ = � t 0 ω(l′)x(t − l′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))dl′ = � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, for t ∈ [0, T], x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = � t 0 � s 0 g(s − l)d{n(l) − n′(l)}ds + � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S14) Taking the derivative with respect to t on both sides of (S14) leads to ˙x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = � t 0 g(t − l)d{n(l) − n′(l)} + ω(t) ˙x(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) + � t 0 ω(s) ˙x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·))ds, where ˙x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) denotes the derivative of x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We know that ˙x(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, using the Fourier trans- form, we know that ˙x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = − � Ψ−1 ˜G � (t), where ˜G(ν) = � 1+(Ψω)(ν) �−1� Ψ˜g � (ν) and ˜g(t) ≡ � t 0 g(t − l)d{n(l) − n′(l)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, ˙x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) does not depend on n1(·) and n0(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Combining this with the fact that x(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = 0, we know that x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) does not depend on n1(·) and n0(·) and can write x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·), n1(·), n0(·)) = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking n1(·) = n(·) and n0(·) = n′(·), we have, E � Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n(·), Ni0 = n′(·) � = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From the law of total probability, we obtain E � Yi,n(·)(t) − Yi,n′(·)(t) � = � n1(·),n0(·)∈N E{Yi,n(·)(t) − Yi,n′(·)(t) | Ni1 = n1(·), Ni0 = n0(·)}Pr{Ni1 = n1(·), Ni0 = n0(·)} = � n1(·),n0(·)∈N x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·))Pr{Ni1 = n1(·), Ni0 = n0(·)} = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 36 As a result, the condition in (18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Verifying Assumption 7 (Additivity): We have shown that, for any n(·) and n′(·), E � Yi,n(·)(t) − Yi,n′(·)(t) � = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)), where x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)) is the solution of (S14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To verify additivity, we only need to show that, for all t, x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n1(·), n′ 1(·)) = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n2(·), n′ 2(·)), where, without loss of generality, n1(·) has events {τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τl}, n′ 1(·) has events {τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , τl−1}, n2(·) has a single event {τl}, and n′ 2(·) has no events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From (S14), we have x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·)) = � t 0 � s 0 g(s − l)d{n(l) − n′(l)}ds + � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·))ds = n(t) � j=1 � t τj g(s − τj)ds − n′(t) � j=1 � t τj g(s − τ ′ j)ds + � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n(·), n′(·))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n1(·), n′ 1(·)) is the solution of x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n1(·), n′ 1(·)) = � � � � � � t τl g(s − τj)ds + � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n1(·), n′ 1(·))ds t > τl 0 t ≤ τl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similarly, x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n2(·), n′ 2(·)) is the solution of x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n2(·), n′ 2(·)) = � � � � � � t τl g(s − τj)ds + � t 0 ω(s)x(t − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n2(·), n′ 2(·))ds t > τl 0 t ≤ τl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we have x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n1(·), n′ 1(·)) = x(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' n2(·), n′ 2(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Verifying Assumption 4: We only need to show that, for any t ≤ τ,∂E{Yiτ(t)}/∂τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From (S13), we have � t 0 {ω(∆) + δ(∆)}ACER(t − ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ)d∆ = 0 (S15) for t ≤ τ, where we use g(t − τ) = 0 for t ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For fixed τ, denote m(t) = ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) for t ≤ τ and x(t) = 0 for t > τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Then, for any t � t 0 {ω(∆) + δ(∆)}x(t − ∆)d∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S16) 37 Taking the Fourier transform on both sides yields, for all ν, {(Ψω)(ν) + 1}(Ψm)(ν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Recalling that {(Ψω)(ν) + 1} ̸= 0, we have Ψm = 0, and thus m(t) = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, ∂E{Yiτ(t)}/∂τ = ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = 0 for t ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='6 Proof of Proposition 4 Using a similar argument as in (S10), we have E �dYiτ(t) dt ���� Hit− � = φ {µY + g(t − τ)} + ψUi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking expectation and then integrating from 0 to t on both sides, we obtain E {Yiτ(t)} = � t 0 φ {µY + g(s − τ)} ds + � t 0 E {ψUi(s)} ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S17) Taking the derivative with respect to τ yields Part (a) of the proposition that ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = − � t 0 φ′ {µY + g(s − τ)} g′(s − τ)ds = φ(µY ) − φ{µY + g(t − τ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S18) Similar to the proof of Proposition 3, it suffices to verify Assumption 6, the condition in (12), and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From (S22), we have E{Yiτ1(t) − Yiτ2(t)} = � t τ1 φ {µY + g(s − τ1)} ds − � t τ2 φ {µY + g(s − τ2)} ds = � t−τ1 0 φ {µY + g(s)} ds − � t−τ1 τ2−τ1 φ {µY + g(s − τ2 + τ1)} ds = E{Y0(t − τ1) − Yτ2−τ1(t − τ1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' for τ1 ≤ τ2 ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Thus, Assumption 6 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We then verify the condition in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Similar to the proof of Proposition 3, we can obtain E �∂Yiτ(t) ∂τ 1(Ti1 ≤ τ) � = E �∂E{Yiτ(t) | Hit−} ∂τ 1(Ti1 ≤ τ) � = E � ∂E{Yi(t) | Ti = τ, H∗\\Ni it } ∂τ 1(Ti1 ≤ τ) � , where, from Model (24), ∂E{Yi(t) | Ti = τ, H∗\\Ni it } ∂τ = ∂ ∂τ � φ(µY )τ + � t τ φ{µY + g(s − τ)}ds + � t 0 ψUi(s)ds � 38 = φ(µY ) − φ(µY ) − � t τ g′(s − τ)φ′{µY + g(s − τ)}ds = φ(µY ) − φ(µY ) − [φ{µY + g(t − τ)} − φ(µY )] = φ(µY ) − φ{µY + g(t − τ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, we obtain E �∂Yiτ(t) ∂τ 1(Ti1 ≤ τ) � = [φ(µY ) − φ{µY + g(t − τ)}] Pr(Ti1 ≤ τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S19) Similarly, we obtain E �∂Yiτ(t) ∂τ 1(Ti0 ≤ τ) � = [φ(µY ) − φ{µY + g(t − τ)}] Pr(Ti0 ≤ τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S20) Combining (S19) and (S20) yields the condition in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' From (S18), we have ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = φ(µY ) − φ{µY + g(t − τ)} = φ(µY ) − φ(µY ) = 0 for t ≤ τ, where we use g(∆) = 0 for ∆ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, Assumption 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' □ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='7 Comment on the difficulties of the nonlinear non-additive model We now comment on the difficulties of the following nonlinear non-additive model λY (t) = φ � µY + � t 0 g(t − s)dNi(s) + � t 0 ω(t − s)dYi(s) + ψUi(t) � , (S21) where φ(·) is a non-negative link function and the rest are the same as in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Model (S21) is widely used in the analysis of neural data with different forms of φ (see among others Lawrence, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Kulkarni and Paninski, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Macke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Sussillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Zhao and Park, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Pandarinath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Compared with Models (23) and (24), it guarantees a non-negative intensity without additional restrictions on the unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' However, the impact of the unmeasured confounding is no longer additive under Model (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Even when the treatment is a single-point process, the following proposition shows that the causal effect depends on the choices of the link function and the distribution of Ui in Model (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proposition 5 Suppose that Assumptions 1, 2, and 8 hold and the underlying outcome satis- fies (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' When Ni is a single-point process, we have, for t, τ ∈ [0, T], ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = � t 0 E � ∂ ∂τ φ � µY + g(s − τ) + � s 0 ω(t − ∆)dYiτ(∆) + ψUi(s) �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 39 In Proposition 5, the ACER under Model (S21) does not satisfy the identification assumptions in Theorems 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Thus, we cannot use the identification result in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Under Model (S21), the identification of ACER is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Although it is strenuous to formally study its identifiability due to the complexity of Model (S21), econometricians have obtained negative results for the identification of non-separable models in the cross-sectional setting when both the treatment and the outcome are scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, Chesher (2003) gives sufficient conditions for nonparametric identification, which generally requires the instrumental variable to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Moreover, when the outcome is discrete, Chesher (2010) shows that the identification is typically not achieved even under parametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Therefore, the identification of Model (S21) is gloomy in a more complex context with point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Due to the dependence of the ACER on the model parameters and the distribution of Ui, its identification is also unpromising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We end this subsection with the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Using a similar argument as in (S10), we have E �dYiτ(t) dt ���� Hit � = φ � µY + g(t − τ) + � t 0 ω(t − s)dYi(s) + ψUi(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking expectation and then integrating from 0 to t on both sides, we obtain E {Yiτ(t)} = � t 0 E � φ � µY + g(s − τ) + � s 0 ω(t − l)dYiτ(l) + ψUi(s) �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S22) Taking the derivative with respect to τ yields ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) = � t 0 E � ∂ ∂τ φ � µY + g(s − τ) + � s 0 ω(t − l)dYiτ(l) + ψUi(s) �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S23) S3 Supplement of the numerical analysis S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 Derivation of ACER in simulation We know from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 that the ACER in the presence of non-additive confounding takes the form in (S23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For Scenario 3 in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1, we can derive that ∂ ∂τ φβ2 {µY + g(t − τ) + Ui(t − dU)} = � ∂ ∂τ g(t − τ) � β2{µY + g(t − τ) + Ui(t − dU)}β2−1 =β2bY a2 Y � a2 Y (t − τ) − 1 � exp{−aY (t − τ)}{µY + α(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' aY , bY ) + Ui(t − dU)}β2−1, where the last equality follows from ∂ ∂τ g(t − τ) = ∂ ∂τ α(t − τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' aY , bY ) 40 = ∂ ∂τ bY a2 Y (t − τ) exp{−aY (t − τ)} = bY a2 Y [− exp{−aY (t − τ)} + aY (t − τ) exp{−aY (t − τ)}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can calculate the true value of ACER(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' τ) using the Monte Carlo method by simulating the unmeasured confounding process Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 Additional information on the real data analysis In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2, we apply our methodology on the neural data to estimate the causal effect of neural activities in the olfactory bulbs on those in the piriform cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this section, we discuss more scientific backgrounds and conduct additional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In addition to the olfactory bulb (OB) mitral cells and the principal neurons (PN) in the piriform cortex (PCx), layer 1 feedforward interneurons (FFI) and layer 2/3 feedback interneurons (FBI) play important roles in the neural circuits for odor perception (Bolding and Franks, 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' They are inhibitory neurons that suppress the activities of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, OB mitral cells excite both the PN and FFI in PCx, which may result in an immediate excitation and a slightly delayed inhibition in the PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In addition, PN excites the FBI, which will in turn suppress future activities in PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The causal pathways among the aforementioned neurons are illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' OB Odor FFI PN FBI Figure 4: Causal diagram depicting the relationships among the olfactory bulb mitral cells (OB), principal neurons (PN), feedforward interneurons (FFI), and feedback interneurons (FBI) in the piriform cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' A red arrow represents an excitatory pathway while a blue arrow represents an inhibitory pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The dotted arrow from PN to FBI is disconnected when TeLC is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In the experiment analyzed here and in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2, the odor is replaced by light pulses, and only the neural activities of OB and PN are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The fitted ACER in Figure 3(c) shows that, overall, a spike in OB causes an immediate excitation in PN in PCx, and causes a relatively long-term suppression till the effect vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' This finding corroborates the causal pathways in Figure 4 that the immediate excitation may be due to the direct effects of OB cells on the PN, while the long-term suppression results from the induced activities from FFI and FBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' As a result, the stationarity of odor detection is maintained, as shown 41 in Figures 3(a) and (b), that the PN return to normal activity level quickly after a sharp increase in activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' To further investigate the roles of inhibitory pathways, Bolding and Franks (2018a) selectively expressed tetanus toxin light chain (TeLC) in PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The expression of TeLC prevents a principal neuron to excite any other neurons including the FBIs, while retaining PN’s excitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In other words, in a TeLC-expressed mouse, the causal pathway between PN to FBI no longer exists in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Figure 5 shows the resulting intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can see the consequence of the absence of the FBI, where the activities of PN remain at a higher-than-normal level till the end of the stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Applying the same estimation procedure to the data from the TeLC-expressed mice reveals a very different ACER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In this case, there are 15 TeLC-expressed mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' On each mouse, there are 10 trials in the treatment group and 10 in the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The confidence band is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Let {ˆgb : b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , B} be the estimates from the bootstrap samples, where trials are sampled with replacement in each treatment group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' For each t ∈ [0, T], we estimate the bootstrap mean ¯g(t) and the bootstrap standard deviation �σg(t) from the bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can then construct a (1 − α)% confidence band for ˆg as {(ˆg(t) − qαm−1/2ˆσg(t), ˆg(t) + qαm−1/2ˆσg(t)) : t ∈ [0, T]}, where qα is the (1 − α)% bootstrap quantile of Qb = maxt |ˆgb(t) − ¯g(t)|/ˆσg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The number of trials m is 300 in the TeLC experiment and 160 in the analysis in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can see that the causal effect lasts for a much shorter period, in the absence of self-excitation among PN and the inhibitions from FBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Note that an inhibition period still exists, which is likely the effect through the FFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Furthermore, the 5-fold cross-validation selected 18 knots in the TeLC- expressed mice, in sharp contrast to the 2 knots selected in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' By comparing the fitted ACER in Figure 5 and Figure 3 in the main text, we can see that the causal pathway between PN and FFI is a key mechanism in maintaining stationarity in odor perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We also notice that the nonparametric estimation procedure suffers from boundary effects, where a spurious inhibitory effect is estimated towards the right boundary of the support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The estimated effect from the observational analysis does not reflect the sharp inhibition following an event in OB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 Analysis assuming no unmeasured confounding In this section, we conduct an observational analysis of the real data assuming there is no unmeasured confounding between the treatment Ni and the outcome Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, we assume the outcome Yi follows the linear model λY (t) = E �dYi(t) dt ���� Hit− � = µ + � t 0 g(t − s)dNi(s), (S24) 42 OB Light U PN ACER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 −30 0 30 60 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0 500 1500 2500 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0 500 1500 2500 (a) Trials (b) Time (in seconds) Trials ∆ (in seconds) −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) (c) Time (in seconds) (d) Figure 5: Empirical intensities and fitted ACER on data from Bolding and Franks (2018b) on TeLC-expressed mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panels (a) and (b) show the empirical intensities of the neural activities of OB (Panel a) and PN (Panel b) in the stimulated (blue) and unstimulated (blue) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The solid curves represent the average intensity over 150 trials, and the shaped dashed curves demonstrate the empirical intensity from 20 randomly selected trials in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area in Panel (a) represents the duration of the light pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (c) shows the estimated −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) from the full data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area represents a 90% confidence band for visualizing the uncertainty of the estimates from 5000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Panel (d) shows the causal diagram for the relationship among the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 43 where g(·) is commonly interpreted as the effect of Ni on Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Here we do not include the self- dependence of Yi because we have shown in Proposition 3 that the function g is not comparable to ACER when the self-dependence is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Taking the expectation on both sides of (S24), we have E �dYi(t) dt � = µ + � t 0 g(t − s)E{dNi(s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(S25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='Integrating both sides of (S25) from 0 to t yields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='E{Yi(t)} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dYi(l) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(l − s)E{dNi(s)}dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dsdl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(∆)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dNi(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(t − ∆)dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(s = l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(∆)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='−dNi(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(∆)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dNi(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(∆)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dNi(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dld∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='(Fubini’s Theorem) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='�dNi(l − ∆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='dl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='d∆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(s)E{Ni(t − s)}ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='µt + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='g(t − s)E{Ni(s)}ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Define h′(t) = E{Yi(t)} and f′(t) = E{Ni(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We have h′(t) = µt + � t 0 g(t − s)f′(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S26) We take a similar estimation procedure as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' First, we estimate f′(t) and h′(t) using empirical cumulative intensities from all trials, denoted as ˆf′ and ˆh′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Second, we approximate g with truncated bases {ψj : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Finally, we obtain the estimator from ˆβ = arg min β∈RJ+1 ������ ˆh′ − βJ+1t − J � j=1 (ψj ∗ ˆf′)βj ������ 2 2 + η∥β∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' (S27) As with the proposed procedure, we set η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='01 and use cubic B-splines with evenly-spaced knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The number of knots is chosen using 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We construct 90% confidence bands 44 using the bootstrap to approximate the uncertainty as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' Results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' We can see that not using the instrumental variable method yields estimates inconsistent with the findings in Bolding and Franks (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In particular, in Figure 6(a), the observational analysis fails to capture the long-term inhibition between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 seconds that contributes to the stable response in PN, while displaying a false excitatory effect between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' In Figure 6(b), the observational analysis estimates a close-to-zero effect from OB to PN, contradicting the findings in Bolding and Franks (2018a) on TeLC-expressed mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 −30 0 30 60 90 ∆ (in seconds) −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content='5 −30 0 30 60 90 ∆ (in seconds) −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) (a) Figure 6: Estimated −ACER(∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 0) using the proposed instrumental variable method (black) and the estimated effect from the observational analysis (blue) using data from Bolding and Franks (2018b) on normal mice (Panel a) and TeLC-expressed mice (Panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' The shaded area represents a 90% confidence band for visualizing the uncertainty of the estimates from 5000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} +page_content=' 45' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfiAQh/content/2301.03246v1.pdf'} diff --git a/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/2301.11356v1.pdf.txt b/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/2301.11356v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c1bdad2c05abc8436578cf10cac2f0765538e5d --- /dev/null +++ b/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/2301.11356v1.pdf.txt @@ -0,0 +1,1658 @@ +THE AUTOMATED DISCOVERY OF KINETIC RATE MODELS - +METHODOLOGICAL FRAMEWORKS +A PREPRINT +Miguel Ángel de Carvalho Servia +Department of Chemical Engineering +Imperial College London +South Kensington, London, SW7 2AZ, UK +m.de-carvalho-servia21@imperial.ac.uk +Ilya Orson Sandoval +Department of Chemical Engineering +Imperial College London +South Kensington, London, SW7 2AZ, UK +o.sandoval-cardenas20@imperial.ac.uk +Klaus Hellgardt +Department of Chemical Engineering +Imperial College London +South Kensington, London, SW7 2AZ, UK +k.hellgardt@imperial.ac.uk +King Kuok (Mimi) Hii +Department of Chemistry +Imperial College London +White City, London, W12 0BZ, UK +mimi.hii@imperial.ac.uk +Dongda Zhang ∗ +Department of Chemical Engineering +The University of Manchester +Manchester, M13 9PL, UK +dongda.zhang@manchester.ac.uk +Ehecatl Antonio del Rio Chanona ∗ +Department of Chemical Engineering +Imperial College London +South Kensington, London, SW7 2AZ, UK +a.del-rio-chanona@imperial.ac.uk +January 30, 2023 +ABSTRACT +The industrialization of catalytic processes is of far more importance today than it has ever been before +and kinetic models are essential tools for their industrialization. Kinetic models affect the design, +the optimization and the control of catalytic processes, but they are not easy to obtain. Classical +paradigms, such as mechanistic modeling require substantial domain knowledge, while data-driven +and hybrid modeling lack interpretability. Consequently, a different approach called automated +knowledge discovery has recently gained popularity. Many methods under this paradigm have been +developed, where ALAMO, SINDy and genetic programming are notable examples. However, these +methods suffer from important drawbacks: they require assumptions about model structures, scale +poorly, lack robust and well-founded model selection routines, and they are sensitive to noise. To +overcome these challenges, the present work constructs two methodological frameworks, Automated +Discovery of Kinetics using a Strong/Weak formulation of symbolic regression, ADoK-S and ADoK- +W, for the automated generation of catalytic kinetic models. We leverage genetic programming for +model generation, a sequential optimization routine for model refinement, and a robust criterion for +model selection. Both frameworks are tested against three computational case studies of increasing +complexity. We showcase their ability to retrieve the underlying kinetic rate model with a limited +amount of noisy data from the catalytic system, indicating a strong potential for chemical reaction +engineering applications. +Keywords: chemical reaction engineering, kinetic model generation, automated knowledge discovery, information +criteria, machine learning +arXiv:2301.11356v1 [cs.SC] 26 Jan 2023 + +arXiv Template +A PREPRINT +1 +Introduction +Mathematical models are simplified descriptions of complex phenomena using a logic-based language. These models +are widely used in science and engineering, be it in physics to represent human mobility patterns Song et al. [2010], +Brockmann et al. [2006], in medicine to represent the metastatic spread of cancer Franssen et al. [2019], Margarit and +Romanelli [2016], or in chemical reaction engineering to represent the reaction kinetics of a specific process Schbib +et al. [1996], Battiston et al. [1982]. The wide usage of mathematical models across disciplines and fields is a direct +consequence of their high utility, both fundamentally and practically. Fundamentally, these models allow researchers to +simplify and distill complicated phenomena to quantitative mathematical expressions. Practically, they allow engineers +to develop industrial processes and researchers to investigate the kinetics of a chemical system. +Given the importance of models, it is imperative to answer: how can models be constructed? Classically, this +construction can be divided into three distinct paradigms: mechanistic, data-driven and hybrid modeling. Mechanistic +models (also referred to as white-box models) are constructed and derived purely using existing fundamental laws, such +as conservation equations and constitutive relations Baker et al. [2018]. Hypotheses are frequently made during the +construction of a mechanistic model with the trade-off between accuracy and simplicity in mind. These hypotheses are +usually manifested through equations of state, or empirical expressions Gernaey [2015]. Mechanistic models have a +great number of advantages, such as: being derived from prior expert knowledge, the parameters being physically +meaningful, having great extrapolatory properties, and being interpretable. For these reasons, these models are still +widely established in industry. Nevertheless, some of these advantages are double-edged swords. For example, +to construct these models via existing fundamental laws, a modeler must have strong background knowledge, the +construction is usually time-consuming, the level of nonlinearity may be high, and if the process is not perfectly +understood a mechanistic model may be infeasible. Also, having complicated nonlinear models results in increased +experimental effort to collect sufficient data for the estimation of parameters. +Whereas mechanistic models technically may not require data for their structure identification (they may still require +data for parameter estimation and validation), data-driven models solely use data for their construction. Unlike +mechanistic models, data-driven models can be designed relatively quickly, and their construction requires no +knowledge about the system being investigated. Their structure is highly flexible, and can be promptly repurposed to +account for more variables or to describe different processes. Data-driven approaches are also generally quicker to +evaluate than mechanistic ones, and as a result of this, the former have a lot of potential in real-time simulation Zhang +et al. [2019], del Rio-Chanona et al. [2018a], Park et al. [2021], Sun et al. [2022], optimization Petsagkourakis et al. +[2020], del Rio-Chanona et al. [2018b], Wu et al. [2023], Natarajan et al. [2021], and soft sensor development Mowbray +et al. [2022a], Kay et al. [2022], Kadlec et al. [2009]. However, since no physical knowledge is used to make the +black-box models, their prediction tends to be poor outside the range of data used to train them (i.e.: poor extrapolatory +abilities), which might classify their usage in certain scenarios as unsafe. Their performance is highly influenced by the +quantity and quality of data available, and data pre-treatment before training a model is frequently required. +The last class of models are the hybrid models. This type of models attempts to exploit the best of mechanistic and +data-driven modeling. In its essence, a hybrid model has a mechanistic backbone and a data-driven block which tries to +improve the fit of the backbone. There are two main approaches to hybrid modeling: parallel and sequential. In the +parallel approach, the data-driven block describes the model-data mismatch. In the sequential approach, the data-driven +block describes one or more parameters of the mechanistic backbone that may be dependent on multiple variables. +Or, the mechanistic backbone is integrated within the data-driven architecture. In either approach, generally, the +hybrid model will retain the extrapolation capabilities of a mechanistic model, whilst retain the flexibility and ease of +construction of a data-driven model Vega-Ramon et al. [2021], Mowbray et al. [2022b], Zhang et al. [2020]. +At face value, hybrid modeling seems like an elegant solution to the problems posed by mechanistic and data-driven +modeling, albeit not the only one. Another effective solution is to utilize existing data to automatically generate and +select mechanistic models by exploiting state-of-the-art statistical and machine learning methods. In this way, the +benefits of mechanistic models are maintained (e.g.: extrapolatory abilities, interpretability and physical meaning), +whilst some of their drawbacks are eliminated (e.g.: needing background knowledge, expensive and time-consuming). +This strategy/paradigm has been (loosely) named automated knowledge discovery. Formally, it is known as symbolic +regression Haider et al. [2023]. The methodology presented in this work follows this paradigm. +2 + +arXiv Template +A PREPRINT +Before going further we will set the mathematical notation that is shared between both optimization approaches. We +set a predefined time interval ∆t = [t0, tf] and denote the state variables as x(t) ∈ Rnx. The input dataset consists +of pairs (ti, yi) for a given set T of nt sampling times ti, where ti ∈ ∆t and yi ∈ Rnt. M is the set of symbolic +expressions reachable by the search procedure, where each model m ∈ M has a finite set of tunable parameters θm +whose dimension depends on each proposed model: m(·, θm). The predicted state variables are denoted by ˆx(t), and +L(ˆx(t), y(t)) is the discrepancy measure between our predictions and the measured values, which characterizes the +performance of the model (e.g.: least squares function, likelihood function). +Symbolic regression, in its strong formulation, aims to find the model that best maps the time to the state variables +directly: +ˆxm(t, θm) = m(t, θm). +(1) +In its weak formulation, it aims to find the model that best maps input variables to output variables as a differential +equation system: +˙x(t, θm) = m(x(t), θm) +(2a) +ˆxm(t, θm) = +� t +t0 +˙x(τ, θm)dτ +(2b) +The optimization problem for both formulations can be expressed as follows: +m⋆(t, θ⋆ +m) = +min +m∈M,θm +� +ti∈T +L (ˆxm(ti, θm), yi) . +(3) +A wide variety of methods have been proposed in literature to solve the symbolic regression problem. These include but +are not limited to: the ALAMO approach Wilson and Sahinidis [2017], the SINDy algorithm Brunton et al. [2016], +the work by Taylor and colleagues Taylor et al. [2021], the work by Neumann and colleagues Neumann et al. [2020], +and genetic programming Koza [1994]. Popular automated knowledge discovery frameworks frequently face at least +one of four challenges that may limit their ability to retrieve underlying ground-truth models, and consequently, their +real-world applicability. Firstly, they necessitate structural assumptions of the underlying data-generating model. +This is particularly true for some non-evolutionary strategies, as a design matrix (i.e.: a model library) needs to be +constructed for their execution Wilson and Sahinidis [2017], Brunton et al. [2016]. Secondly, they may display poor +scalability with respect to the number of state variables available (this is also particularly true of non-evolutionary +strategies) Wilson and Sahinidis [2017], Brunton et al. [2016], Taylor et al. [2021], Neumann et al. [2020]. Thirdly, +they lack a motivated and rigorous model selection routine (i.e.: their choice of model selection routine may not be +transparent and/or different routines are not tested) Wilson and Sahinidis [2017], Brunton et al. [2016], Taylor et al. +[2021], Neumann et al. [2020], Koza [1994]. Lastly, for the discovery of non-linear dynamics, they may be sensitive to +noisy data when rate measurements are not directly accessible Wilson and Sahinidis [2017], Brunton et al. [2016], +Taylor et al. [2021], Neumann et al. [2020], Koza [1994]. +In this section, we introduced the importance of mathematical modeling within chemical engineering, the challenges +of classical modeling paradigms, and the shortcomings of modern automated knowledge discovery methodologies. +This work aims to construct two generalizable and robust methodological frameworks that integrate a rigorous model +selection routine for the automated kinetic rate model discovery. The rest of the paper is organized as follows: in +Section 2 two proposed methods are motivated and described in detail; in Section 3 we introduce three case studies that +are used to analyze the performance of the proposed methodological frameworks; in Section 4 the results of the study +are presented and amply discussed along with the shortcomings of the proposed methodologies; and in Section 5 the +key findings are presented with a brief outlook on future research. +2 +Methodological Frameworks +The two proposed methodological frameworks are comprised of three stages: model generation, model refinement and +model selection. In this section we expand on each of these steps and present the distinctive characteristic of each of the +proposed frameworks. +Genetic programming (GP) is often considered one of the most generalizable and reliable model generation methods +3 + +arXiv Template +A PREPRINT +found in literature for an important reason: the flexibility to include prior knowledge. Its execution requires minimal +assumptions about the ground-truth model. As a direct contrast to ALAMO (note: ALAMO is formulated as a +cardinality constrained mixed-integer quadratic program, or more simply, as a mixed integer non-linear program) +and SINDy (note: SINDy was initially formulated as non-linear program, however, SINDy can also be solved via +mixed-integer optimization Bertsimas and Gurnee [2022]), GP does not need a design matrix. However, if knowledge +about the ground-truth model is available (e.g.: mass and energy balances), this can be provided via mathematical +constraints alike ALAMO and SINDy. +A especially attractive quality of GP when compared to mixed-integer based alternatives is the explicit control over +the levels of complexity in the resulting expressions. The generalization of a model has a close relationship with the +complexity of it in relation to the available data. Approaches based on a design matrix, which mixes expressions of +multiple complexities, does not control the resulting expression complexity, instead it controls the cardinality. An +explicit report of several expressions of increasing complexity provides more value to the modeler in order to commit +with one of the proposals. Here, complexity refers to the sum of all terms of a particular function. For instance, +f(x) = x2+2x−4 +5x +would have a complexity equal to 13, since each operator, constant and variable count as 1. As such, +we propose that GP should be used for the kinetic rate model generation stage of both frameworks. +The basic concept of GP is to specify a set of state variables (e.g.: temperature, pressure, concentration) and operators +(e.g.: ‘+’, ‘/’) that may be present in the underlying mechanistic model. This selection can be as relaxed or constrained +as the modeler decides. User-defined analytical functions (e.g.: +1 +k1x1+k2x2 ) may also be specified to be included within +generated models. With this, an initial population of models can be constructed. Quoting Darwin’s theory of evolution, +the best models — based on a specified performance metric — are evolved via genetic operations (e.g.: crossover and +mutation), and the worst models are discarded. This process is iterated until convergence is achieved or a termination +criterion is met. +By virtue of GP, model parameters are stochastically evolved through genetic mutations, using differential evolution, +and not by explicitly solving a parameter estimation problem. As such, a model refinement stage is needed as a +parameter estimation problem, where the error between the model’s response and the data are minimized by finding the +best set of kinetic parameters. +In GP, the most accurate generated model for each complexity level is output (the upper bound of complexity is +user-defined). Therefore, a model selection stage is needed to discern which kinetic rate model proposed is the most +appropriate for a particular dataset. +2.1 +Introduction to ADoK-S +The first proposed methodological framework, ADoK-S (Automated Discovery of Kinetics using a Strong formulation +of symbolic regression), uses GP to solve the strong formulation of the symbolic regression to find kinetic rate models. +A characteristic of the strong formulation of symbolic regression is that only models that can directly map the specified +state variables to the output variable are proposed. Therefore, in the context of kinetic rate model discovery, rate +measurements need to be provided to find the desired model; rate is defined as: +r = 1 +νi +dCi +dt +(4) +where νi and dCi +dt represent the stoichiometric coefficient and the rate of change of concentration with respect to time of +species i, respectively. +Realistically, seldom does a modeler have direct access to rate measurements, and therefore it would be practically +meaningless to assume that these measurements are indeed available. Classically, to construct a mechanistic kinetic rate +model of a chemical system, a modeler only has access to discrete measurements of the concentration of observed +species with respect to time (for batch reactors) or with respect to residence time (for continuous plug flow reactors). +With this dataset, assumptions about the kinetics of the system are made (e.g.: first order or second order kinetics), +models are generated, kinetic parameters are estimated by using the dynamic trajectories of the concentrations, and +a final model is proposed. Thus, it is fair to assume that concentration data with respect to time are available to a modeler. +4 + +arXiv Template +A PREPRINT +Under the assumption that dynamic trajectories of concentrations are available, and knowing that the strong formulation +of symbolic regression requires rate measurements of the chemical reaction, it is evident that these measurements must +be estimated from the available data. Assuming an isochoric and isothermal system, it can be claimed that Ci (the +concentration of species i ∈ Z+) is only dependent on time. Mathematically, under these conditions, the concentration +profile of an arbitrary species is a function only dependent on time, Ci = fi(t). Therefore, if fi(t) can be accurately +estimated, discrete rate measurements can also be estimated by numerically differentiating the concentration profile. To +avoid limiting assumptions, GP is used to estimate fi(t), as output measurements Ci and input measurements t are +available. +Finding an appropriate concentration profile follows the three stages referred earlier: model generation, model +refinement, and model selection. For the model generation stage, an implementation of GP done by Cranmer and +colleagues was used Cranmer [2020]. For the model refinement stage, posed as a parameter estimation problem, +the error between a model’s response and the data are minimized by finding the best set of kinetic parameters. This +optimization problem is solved by carrying out an initial screening of the kinetic parameter search space using the +artificial bee colony (ABC) algorithm. The best output from the ABC is used to warm-start the limited-memory +Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm. The ABC algorithm was chosen because of its excellent +explorative characteristics Cho et al. [2021]. Whereas, the LBFGS algorithm was chosen because of its excellent +performance in the parameter estimation task Malouf [2002], and optimization in general Liu and Nocedal [1989]. The +objective function used for the parameter estimation in this work was the negative log-likelihood (NLL), presented +below: +NLL(θ) = +� +i,j,k +� +(Ci,j,k − y(θ)i,j,k)2 +2ˆσ2 +i +− log +� +1 +� +2πˆσ2 +i +�� +(5) +where Ci,j,k is the measured concentration (i.e.: in-silico data) of species i ∈ S for dataset j ∈ D at time k ∈ T, where +S, D, and T represent the species set, data set and time set, respectively; y(θ)i,j,k is the concentration of species i ∈ S +for dataset j ∈ D at time k ∈ T proposed by an arbitrary model which is dependent on its parameters θ; ˆσ2 +i is the +variance of the noise that we assume the concentrations of species i ∈ S have. +For the model selection stage, the Akaike information criterion (AIC) is used to determine which model is the best ones. +The AIC was selected after a thorough analysis of the performance of different criteria (finite-sampled corrected AIC, +Bayesian information criteria and Hannan Quinn criterion) under several conditions (e.g.: different amounts of additive +noise, different amounts of data). This analysis concluded that AIC has a higher probability of selecting the correct +data-generating rate model than the other criteria tested. The AIC selects the model which has the lowest AIC value +calculated from the below expression: +AICm = 2NLL(θ)m + 2dm +(6) +here the subscript m serves to represent model m ∈ M, where M is the set of proposed models; dm represents the +number of parameters present in a model m ∈ M. +Once a concentration model has been successfully generated, refined and selected, this same model Ci = fi(t) is +numerically differentiated using the central difference method. In this way, a new training dataset is generated, +where the inputs are the discrete measurements of concentration through time and the outputs are the discrete +estimates of the rate measurements through time. Given that the rate is a function of the concentration of the species +(r = f(C1, C2, ...)), the same protocol outlined above can be executed again with the new training dataset. By doing +so, rate models that take as inputs the concentrations of the species observed in the chemical system and outputs the +rate of that same system are generated using GP. Then, using the ABC and the LBFGS algorithms, the models are +refined; using AIC, the best model is selected. +Once an optimized kinetic rate model is selected by AIC, this model is numerically integrated using the LSODA +algorithm Hindmarsh and Petzold [2005] ( +� +rdt = +� +1 +νi +dCi +dt dt = +1 +νi Ci(t)) implemented in SciPy python package +Virtanen et al. [2020]. The results from this integration are directly compared to the original datasets of the dynamical +trajectories of the observed concentrations. If the results are satisfactory to the modeler, the methodological framework +5 + +arXiv Template +A PREPRINT +is terminated. Otherwise, further data should be collected by applying (model-based) design of experiments. For the +sake of clarity and simplicity, the flowchart of the proposed methodological framework is presented in Figure 1. +Figure 1: The flowchart of ADoK-S (Automated Discovery of Kinetics using a Strong formulation of symbolic +regression); the red and blue dashed boxes represent the steps where rate measurements and rate models are estimated, +respectively. +2.2 +Introduction to ADoK-W +One of the primary reasons to solve the symbolic regression problem is to retrieve interpretable mathematical +expressions that experts can analyze and validate. But perhaps more importantly, expressions that experts can extract +knowledge from them. Clearly, the strong formulation of symbolic regression hinders this process within the catalytic +reaction engineering context. As explained in the previous section, to successfully implement the strong formulation +of symbolic regression, ADoK-S must first propose and refine concentration profiles, where the best one (selected +by AIC) is used to estimate rate measurements. Catalytic reaction engineers seldom know the model structure of a +concentration profile, especially because many kinetic rate models do not have a close-form solution. As such, experts +would find the task to validate concentration models strenuous, perhaps even impossible. Furthermore, having a model +that describes the dependence of the concentration of a species with respect to time does not provide significant and +actionable information to the experts. In other words, experts cannot extract much knowledge from these models +and we can conclude that the strong formulation of symbolic regression inevitably affects the interpretability of ADoK-S. +The strong formulation of symbolic regression may also affect the performance of ADoK-S with regards to noisy data. +The necessity to have the first step of ADoK-S is that rate measurements are required to retrieve a kinetic rate model (as +the strong formulation only allows the direct mapping of inputs to outputs). As previously mentioned, having direct +access to the rate measurements of a dynamical catalytic chemical system is a rarity and thus need to be estimated. As it +is explained in Bertsimas and Gurnee [2022], numerical differentiation exacerbates any noise inherent to the measured +data, and even robust differentiation techniques (e.g.: polynomial interpolation) may not be enough to provide usable +derivative data. In ADoK-S, these effects are minimized by selecting a concentration profile model with the AIC, but +the effects are not eradicated. Therefore, we can conclude that the strong formulation of symbolic regression may also +6 + +Start +Generate data: (t, C;(t)) +Estimating derivatives +Estimating rate models +With GP: C; = C;(t, 0) +With GP: r = r(C,Φ) +Parameter estimation:→* +Parameter estimation: Φ → Φ +Model selection with AIC +Model selection with AIC +- +Generate derivative data: (t, C (t, *) +Integrate rate model: J r(C, Φ*)dt +No +UseDoE/MBDoE +Satisfied? +Yes +StoparXiv Template +A PREPRINT +affect the performance of ADoK-S. +The innate drawbacks of the strong formulation for the automated kinetic rate discovery task motivated the construction +of a different methodological framework. ADoK-W (Automated Discovery of Kinetics using a Weak formulation of +symbolic regression) aims to combine the two steps from the previous framework into a single one by reformulating the +symbolic regression problem. The weak reformulation of symbolic regression bypasses altogether the construction +of concentration profile models, and the subsequent numerical differentiation to obtain estimations of the rate +measurements. Whereas the strong formulation provides a direct mapping between inputs and outputs, the weak +formulation of symbolic regression represents a mapping between input and output variables in the derivative space. +In other words, this reformulation generates models dependent on the state variables that can map the output via an +integration step. In this way, the GP algorithm receives the dynamic trajectories of concentration as inputs, proposes rate +models (r = f(C1, C2, ...) where Ci is the concentration of species i ∈ Z+), integrates the rate models with respect to +time at each given time-step where concentration data is available ( +� +rdt = +� +1 +νi +dCi +dt dt = 1 +νi (Ci(t = t) − Ci(t = 0))), +and compares the results from the integration with the original dataset. +Similarly to ADoK-S, due to the usage of a GP algorithm to solve the symbolic regression problem, the proposed rate +models may have under-evolved kinetic parameters and therefore necessitate a model refinement stage. As explained in +the previous section, this model refinement stage is posed as a parameter estimation problem, where Equation 5 is used +as the objective function. To solve this problem, the kinetic parameter search space is initially scanned by the ABC +algorithm, where its output is used as a warm-start for the LBFGS algorithm which will output the final set of kinetic +parameters. If desirable, this optimization routine can be repeated multiple times to improve the probability of reach- +ing a global optimum. Then, alike ADoK-S, the best model is selected based on the AIC value produced from Equation 6. +In identical fashion to the previously presented methodological framework, once an optimized kinetic rate model is +selected by AIC, this model is numerically integrated using the LSODA algorithm. The results from this integration are +directly compared to the original datasets of the dynamical trajectories of the observed concentrations. If the results are +satisfactory to the modeler, the methodological framework is terminated. Otherwise, further data should be collected by +applying (model-based) design of experiments. For the sake of clarity and simplicity, the flowchart of the proposed +methodological framework is presented in Figure 2. +3 +Catalytic Kinetic Case Studies +To showcase the performance of the proposed methodological frameworks, three illustrative catalytic kinetic case studies +were chosen: an isomerization reaction, the decomposition of nitrous oxide, and the toluene hydrodealkylation. Below, +their respective kinetic rate models are introduced, along with how the required datasets (i.e.: dynamic trajectories of +concentrations) are generated. +3.1 +Isomerization Reaction +The simplest case study presented in this work is a catalytic isomerization reaction, where A is transformed to B +reversibly over a catalytic active site. The reaction is shown below. +A ⇌ B +(7) +The kinetic rate model that describes the evolution of the concentrations of A and B through time is shown below. This +expression has been directly borrowed from the book by Marin and colleagues Marin et al. [2019]. +r = −dCA +dt += dCB +dt += +kACA − kBCB +kCCA + kDCB + kE +(8) +In Equation 8, CA and CB represent the concentration of reactant A and product B, respectively. The kinetic +parameters of the kinetic rate model are represented by ki where i ∈ [A, B, ..., E]. To generate the necessary +dataset to test both frameworks, three computational experiments are carried out, each with different initial +conditions. The computational experiments are run with the following initial conditions (in molar units, mol L−1): +(CA,0, CB,0) ∈ {(2, 0), (6, 1), (10, 2)}. For each computational experiment, the concentration of the reactant and +7 + +arXiv Template +A PREPRINT +Figure 2: The flowchart of ADoK-W (Automated Discovery of Kinetics using a Weak formulation of symbolic +regression). +product are recorded 10 times, at evenly spaced intervals between time t=0 s and t=10 s. For this particular case study, +the kinetic parameters were defined as: kA=7 M s−2, kB=3 M s−2, kC=4 s−1, kD=2 s−1 and kE=6 M s−1. +To approximate the behavior of a realistic chemical system, Gaussian noise was added to the computational experimental +measurements, yielding the needed dynamic trajectories of concentrations. The defined Gaussian noise has zero mean +and a standard deviation of 0.17 for the concentrations of A, and 0.18 for the concentrations of B (the standard deviations +represent 5% of the mean of the concentrations of A and B). For the parameter estimation task, it would be futile to +assume that, as modelers, the exact variance of the noise would be known. Thus, a conservative assumption is made by +setting ˆσA=0.35 and ˆσB=0.35 (i.e.: assuming a standard deviation of 10% of the mean of the concentrations of A and +B). The generated data of one of the computational experiments are presented in Figure 3. All the graphs presented in +this paper have been generated using the Matplotlib package implemented in Python Hunter [2007]. +3.2 +Decomposition of Nitrous Oxide +The second case study presented in this work is the catalytic decomposition of nitrous oxide, where nitrous oxide (N2O) +is transformed to nitrogen gas (N2) and oxygen gas (O2). The reaction is shown below. +2N2O ⇌ 2N2 + O2 +(9) +The kinetic rate model that describes the evolution of the concentrations of N2O, N2 and O2 through time is shown +below. This expression has been directly borrowed from the book by Levenspiel Levenspiel [1998]. +r = −2dCN2O +dt += 2dCN2 +dt += dCO2 +dt += +kAC2 +N2O +1 + kBCN2O +(10) +8 + +Start +Generate data: (t, C;(t)) +With GP: r = r(C,Φ) +Parameter estimation: Φ → Φ +Use DoE/MBDoE +Model selection with AIC +Integrate rate model: J r(C, Φ*)dt +No +Satisfied? +Yes +StoparXiv Template +A PREPRINT +In Equation 10, CN2O, CN2 and CO2 represent the concentration of reactant nitrous oxide, and of products nitrogen +gas and oxygen gas, respectively. The kinetic parameters of the kinetic rate model are represented by ki where +i ∈ [A, B]. To generate the necessary dataset to test both frameworks, three computational experiments are carried out, +each with different initial conditions. The experiments are run with the following initial conditions (in molar units): +(CN2O,0, CN2,0, CO2,0) ∈ {(5, 0, 1), (7.5, 1, 2), (10, 2, 3)}. For each experiment, the concentration of the reactant and +products are recorded 10 times, at evenly spaced intervals between time t=0 s and t=10 s. For this particular case study, +the kinetic parameters were defined as: kA=2 M−1 s−1 and kB=5 M−1. +To approximate the behavior of a realistic chemical system, Gaussian noise was added to the in-silico datasets, yielding +the needed dynamic trajectories of concentrations. The defined Gaussian noise has zero mean and a standard deviation +of 0.11 for the concentrations of N2O, 0.32 for the concentrations of N2, and 0.23 for the concentration of O2 (the +standard deviations represent 5% of the mean of the concentrations of N2O, N2 and O2). For the parameter estimation +task, it would be futile to assume that, as modelers, the exact variance of the noise would be known. Thus, a conservative +assumption is made by setting ˆσN2O=0.22, ˆσN2=0.63 and ˆσO2=0.46 (i.e.: assuming a standard deviation of 10% of +the mean of the concentrations of N2O, N2 and O2). The generated data of one of the computational experiments are +presented in Figure 3. +3.3 +Toluene Hydrodealkylation +The third and most complex case study (i.e.: case study with most species involved in the chemical reaction) presented +in this work is the catalytic toluene hydrodealkylation to benzene, where toluene (C6H5CH3) and hydrogen gas (H2) is +transformed to benzene (C6H6) and methane (CH4). The reaction is shown below. +C6H5CH3 + H2 ⇌ C6H6 + CH4 +(11) +The kinetic rate model that describes the evolution of the concentrations of C6H5CH3, H2, C6H6 and CH4 through time +is shown below. This expression has been directly borrowed from the book by Fogler Fogler [2016]. +r = −dCT +dt += −dCH +dt += dCB +dt += dCM +dt += +kACT CH +1 + kBCB + kCCT +(12) +In Equation 12, CT , CH, CB and CM represent the concentration of reactants toluene and hydrogen, and of products +benzene and methane, respectively. The kinetic parameters of the kinetic rate model are represented by ki where +i ∈ [A, B, C]. To generate the necessary dataset to test both frameworks, three computational experiments are carried +out, each with different initial conditions. The computational experiments are run with the following initial conditions +(in molar units): (CT,0, CH,0, CB,0, CM,0) ∈ {(1, 3, 0, 0.5), (3, 5.5, 1, 1.75), (5, 8, 2, 3)}. For this particular case +study, for each experiment, the concentration of the reactant and products are recorded 50 times, at evenly spaced +intervals between time t=0 s and t=10 s. The frequency of the sampling was increased due to the complexity of the +case study. It should be noted that this increment is still within the realistic range of experimental sampling frequency, +as these samples could have been gathered from a continuous-flow experiment Schrecker et al. [2023]. The kinetic +parameters were defined as: kA=2 M−1 s−1, kB=9 M−1 and kC=5 M−1. It should be noted that, for each of the +case studies, only the model structure is borrowed from literature. The value of the kinetic parameters are randomly +assigned. +To approximate the behavior of a realistic chemical system, Gaussian noise was added to the experimental results, +yielding the needed dynamic trajectories of concentrations. The defined Gaussian noise has zero mean and a standard +deviation of 0.09 for the concentrations of toluene, 0.16 for the concentrations of hydrogen gas, 0.07 for the concentration +of benzene, and 0.15 for the concentration of methane (the standard deviations represent 5% of the mean of the measured +concentrations of toluene, hydrogen gas, benzene and methane). For the parameter estimation task, same as before, +a conservative assumption is made by setting ˆσT =0.18, ˆσH=0.33, ˆσB=0.14, and ˆσM=0.30 (i.e.: assuming a standard +deviation of 10% of the mean of the concentrations of toluene, hydrogen gas, benzene and methane). The generated +results of one of the computational experiments are presented in Figure 3. +9 + +arXiv Template +A PREPRINT +Figure 3: Top left: The in-silico data of one of the computational experiments for the catalytic isomerisation reaction. +Top right: The generated data of one of the computational experiments for the catalytic decomposition of nitrous oxide +reaction. Bottom: The generated data of one of the computational experiments for the catalytic hydrodealkylation of +toluene reaction. +4 +Results and Discussions +4.1 +Isomerization Reaction — ADoK-S +As explained in Section 2.1, ADoK-S aims to solve a strong formulation of symbolic regression by implementing GP. +In this methodological framework, departing from the kinetic data generated (routine for data generation is detailed +in Section 3), the GP algorithm is used to generate concentration profile models (i.e.: models which represent the +evolution of concentration through time as a function of time, f(t)). The expression construction rules exclusively +included the arithmetic operators ‘+’, ‘−’, ‘×’, ‘/’ and ‘exp’. According to physical knowledge, this is a fair selection +of operators as the likelihood of these operators appearing is extremely high (to the best of the author’s knowledge, +trigonometric operators, for example, do not appear in concentration expressions). The generation of concentration +profiles is carried out for reactant A and product B at each computational experiment (i.e.: 2 concentration models +are generated for each experiment — (fA,i, fB,i) for i ∈ [1, 2, 3]). In a real case study, assuming that all of A gets +transformed to B without forming any side-product would not be a valid assumption. For that reason, it would be bad +practice to generate a model for the concentration of A (or B) and obtaining a model for B (or A) by simply solving the +mass balance. As such, we must propose models for A and B separately to ensure that the rates are well predicted. +Models must also be proposed for each experiment because, although the rate model is the same for each species, its +integration is dependent on initial conditions. In other words, for different initial conditions, the concentration profiles +have different functional forms. The same rational is applied to the subsequent case studies. +For the sake of brevity, only the results from the first experiment are presented, but the same routine is executed on all +the other computational experiments. It is also specified that the proposed models should only include ‘t’ as a variable. +As explained in Section 2.1, the concentration profiles, assuming isochoric and isothermal conditions, should only be +10 + +2.0 +B +Concentrations (M) +1.0 +0.5 +0.0 +0 +2 +4 +6 +8 +10 +Time (s)Nitrous Oxide +Nitrogen +Oxygen +4 +Concentrations (M) +3 +2 +L +0 +0 +2 +4 +6 +8 +10 +Time (s)3.0 +Toluene +Hydrogen +Benzene +2.5 +Methane +Concentrations (M) +2.0 +1.5 +1.0 +0.5 +0.0 +0 +2 +4 +6 +8 +10 +Time (s)arXiv Template +A PREPRINT +dependent on time. The GP algorithm proposed the model structures for the concentration of A and B in experiment 1 +shown below, where pℓ for ℓ ∈ [1, 2, 3, 4] are the parameters that can be estimated according to the concentration versus +time dataset and fi,j,k refers to the kth concentration model of species i in experiment j proposed by SR. +fA,1,1 = p1 +(13a) +fA,1,2 = exp (p1t) +(13b) +fA,1,3 = +p1 +t + p2 +(13c) +fA,1,4 = exp (p1 − t) + p2 +(13d) +fA,1,5 = p1 + p2 exp (−p3t) +(13e) +fA,1,6 = exp (−t)(p1t + p2) + p3 +(13f) +fA,1,7 = p1 + exp (−t +− exp (−p2t) + p3) +(13g) +fA,1,8 = +p1t + p2 +p3 + exp (t) + p4 +(13h) +fB,1,1 = p1 +(14a) +fB,1,2 = exp (p1t) +(14b) +fB,1,3 = p1 + p2t +(14c) +fB,1,4 = p1 − p2 exp(−t) +(14d) +fB,1,5 = p1 − +p2 +t + p3 +(14e) +fB,1,6 = p1 − +p2 +p3 + exp (t) +(14f) +fB,1,7 = exp (−t)(p1t − p2) + p3 +(14g) +fB,1,8 = −p1t − p2 +p3 + exp (t) + p4 +(14h) +For each of the proposed model structures shown above, parameter estimation is carried out by finding the parameters +which minimizes Equation 5 (as mentioned, the assumed variances for NLL are double of the real variances used to +generate the Gaussian noise). This problem is solved by employing the strategy outlined in Section 2.1 (i.e.: initial +parameter screening with ABC algorithm and solution refinement with LBFGS algorithm). +Table 1 displays the NLL values along with the AIC values for each of the models, showing that fA,1,4 is the best model +for the concentration profile of A (in experiment 1), and showing that fB,1,4 is the best model for the concentration +profile of B (in experiment 1). We can conclude from the proposed concentration models that the catalytic system under +investigation is not a complex one. This demonstrates that, although the first step of ADoK-S is mostly uninterpretable, +it can still provide the modeler with some level of insightful information pertaining to the complexity of the system. +Table 1: The negative log likelihood values and the AIC values of all concentration profile models proposed by the GP +algorithm for reactant A and product B for experiment 1. +Model +NLL Value +AIC Value +fA,1,1 +9.285 +20.570 +fA,1,2 +4.175 +10.351 +fA,1,3 +-2.343 +-0.686 +fA,1,4 +-2.850 +-1.701 +fA,1,5 +-3.221 +-0.442 +fA,1,6 +-3.235 +-0.470 +fA,1,7 +-3.237 +-0.474 +fA,1,8 +-3.235 +1.530 +Model +NLL Value +AIC Value +fB,1,1 +7.764 +17.528 +fB,1,2 +5.126 +12.252 +fB,1,3 +3.726 +11.451 +fB,1,4 +2.079 +8.159 +fB,1,5 +1.897 +9.793 +fB,1,6 +1.745 +9.490 +fB,1,7 +1.749 +9.497 +fB,1,8 +1.746 +11.492 +The concentration models proposed by the GP algorithm, optimized by the optimization routine and selected by the +AIC are shown in Figure 4 below. Qualitatively, the proposed concentration models fit the data quite well. Although +no physical constraints were imposed, it is important to mention that the chosen concentration profiles respect +the law of conservation of mass (i.e.: CA + CB = CA0 + CB0 = 2 + 0 = 2 M, substituting the chosen models, +CA+CB = exp k1 − t+k2+k3−k4 exp −t = exp k1 exp −t+k2+k3−k4 exp −t = k1 exp −t+k2+k3−k4 exp −t, +so if k1 = k4 and k2 + k3 = 2, then the law is respected) and their end-behavior are correct (i.e.: at time t = 0 s, +CA ≈ 2 M and CB ≈ 0 M; at time t → ∞, CA ≈ 0.6 M and CB ≈ 1.4 M). However, it should be noted that +this observation is only valid for this case study; it is not certain that this behavior is displayed in other case studies. +Notwithstanding, it is encouraging to see that ADoK-S, without enforcing constraints, recommends concentration +profiles that respect the conservation of mass and the reached equilibrium of the system. +Following the ADoK-S flowchart presented in Figure 1, the next step is to compute the derivatives of the concentration +profiles. In Figure 4, a plot of the numerical derivatives of concentration of A and B with respect to time is presented, +along with the (realistically inaccessible) rate measurements. +11 + +arXiv Template +A PREPRINT +Once the numerical derivative (i.e.: rate) data are estimated for each of the experiments, the GP algorithm is executed +again, but this time to find a unifying rate equation that minimizes the error from its evaluation and the rate data. For +this execution of the GP algorithm, it is specified that the proposed models should include the same operators as the +concentration profile models, with the exception of the ‘exp’ operator. That is, the proposed rate models may include: +‘+’, ‘−’, ‘×’ and ‘/’. According to physical knowledge this is, once again, a fair selection of operators as the likelihood +of these operators appearing in a rate model is very high. The flexibility of choosing different operators shows one way +in which prior domain knowledge can be injected into kinetic model discovery (another way would be to constrain the +algorithm, but this is outwith the scope of the investigation). Additionally, provided by prior knowledge, the proposed +rate models may include concentrations of the observed species as variables: ‘CA’ and ‘CB’ in this case. The GP +algorithm proposed the rate model structures shown below: +r1 = k1 +(15a) +r2 = k1CA +(15b) +r3 = +k1CA +k2CB + k3 +(15c) +r4 = k1CA − k2CB +k3CA +(15d) +r5 = k1CA − k2CB + k3 +k4CA + k5 +(15e) +r6 = k1CA − k2CB − k3 +k4CA + k5CB + k6 +(15f) +r7 = k1C2 +A + k2CACB − k3CA − k4C2 +B +k5C2 +A + k6CACB + k7CA + k8C2 +B +(15g) +where ki for i ∈ [1, 2, ..., 8] are the parameters that are estimated according to the concentration data. To do so, the +proposed rate models are integrated and evaluated at each time-step, given the initial conditions CA,0 and CB,0. Once +again, the parameter estimation problem is solved by minimizing the NLL using the optimization routine comprised of +the ABC and LBFGS algorithms. Table 2 clearly shows the NLL value along with the AIC values for each of the rate +models, showing that r6 is the best model for to represent the dynamical catalytic reactive system under investigation. +Table 2: The negative log likelihood values and the AIC values of all proposed rate models by ADoK-S. +Model +NLL Value +AIC Value +r1 +530.768 +1063.535 +r2 +61.304 +124.608 +r3 +5.187 +16.373 +r4 +1.301 +8.601 +r5 +-1.124 +7.751 +r6 +-2.419 +7.161 +r7 +-2.349 +11.301 +The response of the rate model proposed by the GP algorithm and selected by the AIC is shown in Figure 4. Qualitatively, +once again, the proposed rate model fits the data extremely well, especially considering that only three computational +experiments with 10 time-steps each were ran (i.e.: 60 datapoints). +Nevertheless, recalling the correct model structure presented in Equation 8, it is evident that the rate model selected has +one extra parameter in the numerator. In fact, none of the rate equations proposed by SR retrieves the exact structure of +the data-generating model. After performing parameter estimation on the selected model, the parameters obtained are +presented in Table 3. +The first thing that can be perceived from Table 3 is that the extra parameter in the numerator of model r6 is estimated +to be zero. In other words, the GP algorithm was not able to retrieve the exact model structure, but the optimization +routine determined that the extra parameter should be non-existent, arguably retrieving the correct model structure. +12 + +arXiv Template +A PREPRINT +Figure 4: The conditions for the first computational experiment are CA,0 = 2 M and CB,0 = 0 M. Top left: Data +from the first computational experiment and the selected concentration profiles selected by AIC. Top right: Numerical +derivatives of the concentration profiles for and the true rate measurements (which realistically are inaccessible) for the +first computational experiment. Bottom: Response of the selected GP-proposed rate model using ADoK-S for the first +computational experiment. +Table 3: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S +selected model. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +5.897 M s−2 +7.000 M s−2 +k2 / kB +2.302 M s−2 +3.000 M s−2 +k3 +0.000 M2 s−2 +N/a +k4 / kC +2.613 s−1 +4.000 s−1 +k5 / kD +3.182 s−1 +2.000 s−1 +k6 / kE +5.033 M s−1 +6.000 M s−1 +The second thing that can be perceived is that the estimated kinetic parameters are slightly different from the kinetic +parameters that were used to generate the data. Of course, this is unsurprising due to the additive Gaussian noise that +was introduced in the dataset. Had the parameter uncertainty been calculated, chances are that the true parameters +would lie within the 95% confidence interval, given the small difference between the true and the estimated values +(notwithstanding, this is pure conjecture as parameter uncertainty was outside the scope of this investigation). +To briefly summarize, the results demonstrated that ADoK-S is robust to noise, regardless of the fact that derivatives +needed to be calculated numerically. The main reason for the robustness of the method stems from utilizing GP and AIC +to generate and select, respectively, good concentration models that do not overfit the data. In this way, the numerical +derivatives continue to have a high degree of accuracy. The final rate model output by ADoK-S had a near identical +13 + +fA, 1, 4 +A +B +Concentrations (M) +0 +0 +2 +4 +6 +8 +10 +Time (s)True Rate Measurements A +True Rate Measurements B +1.0 +Estimated Rate Measurements A +Estimated Rate Measurements B +0.5 +(t-SW) +0.0 +-0.5 +-1.0 +0 +2 +4 +6 +8 +10 +Time (t)Concentrations (M) +r6 for A +r6 for B +A +B +0 +0 +2 +4 +6 +8 +10 +Time (s)arXiv Template +A PREPRINT +structure to the data-generating one, differing only by a single parameter in the numerator. However, after performing +parameter estimation, the extra parameter was determined to be practically zero, and therefore should not appear in the +proposed model. The other estimated kinetic parameters, although not identical to the true values, they were close to +them. All in all, ADoK-S is capable of retrieving the underlying kinetic rate model of a catalytic isomerization reaction +with realistic data. +4.2 +Isomerization Reaction — ADoK-W +As explained in Section 2.2, ADoK-W aims to solve a weak formulation of symbolic regression by implementing GP. +In this methodological framework, departing from the kinetic data generated, the GP algorithm instead of generating +concentration profile models and then kinetic rate models, it automatically evolves rate models by integrating and +comparing them with the concentration data available directly. Note that the law of conservation of mass is satisfied +by construction under this integrating scheme. The expression construction rules exclusively included the arithmetic +operators ‘+’, ‘-’, ‘×’ and ‘/’, since rates including other operators are less common. Alike in ADoK-S, the kinetic rate +models generated are allowed to be a function of the species whose concentrations were measured, r(CA, CB) in this +case. The best expression proposed by the GP algorithm, sorted by degree of complexity (i.e.: the number of operators +and variables), are shown below. +r1 = k1 +(16a) +r2 = k1CA +(16b) +r3 = k1CA − k2CB +k3CA +(16c) +r4 = k1CA − k2CB − k3 +k4CA +(16d) +r5 = k1CA − k2CB − k3 +k4CA + k5 +(16e) +r6 = k1CA − k2CB − k3 +k4CA + k5CB + k6 +(16f) +r7 = k1C2 +ACB − k2CAC2 +B − k3CA + k4CB +k5C2 +ACB − k6CA +(16g) +The estimation of each kinetic parameter, ki for i ∈ [1, 2, ..., 6], is carried out as explained previously: NLL is used +as the objective function where the assumed variances are double of the real variances used to generate the additive +Gaussian noise, and this is solved by deploying the optimization routine consisting of the ABC and the LBFGS +algorithms. Table 4 clearly shows the NLL value along with the AIC values for each of the rate models, showing that r6 +is the best model for the given dataset. +Table 4: The negative log likelihood values and the AIC values of all proposed rate models by ADoK-W. +Model +NLL Value +AIC Value +r1 +540.410 +1082.821 +r2 +53.857 +109.715 +r3 +-0.831 +4.337 +r4 +-0.988 +6.025 +r5 +-2.724 +4.551 +r6 +-4.275 +3.450 +r7 +-2.289 +7.422 +The response of the selected model is shown in Figure 5. For the sake of brevity, only one of the experiments is +presented. The final rate model output by the proposed framework had a near identical structure to the data-generating +one, differing only by a single parameter in the numerator, displaying an identical result to ADoK-S. And once again, +the optimization routine used to solve the parameter estimation problem is able to determine that the extra parameter is +practically zero, and therefore may be non-existent in the actual model. In this case, since the data-generating model is +known, we can indeed validate this conclusion which would have naturally been made regardless of having access to +14 + +arXiv Template +A PREPRINT +the underlying ground-truth model or not. +Figure 5: Response of the selected GP-proposed rate model using ADoK-W for the first computational experiment +where CA,0 = 2 M and CB,0 = 0 M. +Similar to the estimated kinetic parameter values presented in Table 3, Table 5 also shows that the estimated kinetic +parameters are slightly different from the true values. As previously explained, this discrepancy is caused by the additive +Gaussian noise. It should be noted that the the value of NLL for r6 in Table 2 and in Table 4 (r6 is identical for both +approaches) are slightly different. The reason being that the estimated parameters are different for both approaches. +This finding can be attributed to the optimization algorithm not finding the global optimum and getting stuck in different +local optima in different runs. +Table 5: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W +selected model. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +5.557 M s−2 +7.000 M s−2 +k2 / kB +2.335 M s−2 +3.000 M s−2 +k3 +0.000 M2 s−2 +N/a +k4 / kC +3.114 s−1 +4.000 s−1 +k5 / kD +1.379 s−1 +2.000 s−1 +k6 / kE +6.033 M s−1 +6.000 M s−1 +One thing that should be mentioned about ADoK-W is the computational time. On one hand, the GP-steps of ADoK-S +(i.e.: proposing concentration profiles and rate models) takes in the order of minutes to be completed. For this particular +case study, it took around 10 minutes to propose 6 concentration models (2 concentration profiles, one for A and one for +B, for each of the three experiments) and 1 rate model using an Apple MacBook Air (M1, 2020). On the other hand, +the GP-step in ADoK-W (i.e.: proposing rate models) take in the order of hours to be completed. For this particular +case study, it took around 5 hours to propose 1 rate model using the high performance computing cluster with 8-core +CPU configured with 64GB of RAM. This is unsurprising, as in ADoK-W, to calculate the fitness value of each of the +proposed rate models, numerical integration needs to be performed. For ADoK-S, the fitness values are calculated by +merely evaluating a function. Therefore, ADoK-W builds resilience to noise by making it more computationally in- +tensive. This could have ramifications as to the amount of data that this approach could take before it becomes intractable. +The results demonstrated that ADoK-W, as expected from a weak formulation, is robust to noise. The final rate model +output by ADoK-W has a near identical structure to the data-generating one, differing only by a single parameter +in the numerator. However, the optimization algorithm used for parameter estimation is able to determine that the +15 + +2.0 +1.5 +Concentrations (M) +r6 for A +r6 for B +1.0 +A +B +0.5 +0.0 +0 +2 +4 +6 +8 +10 +Time (s)arXiv Template +A PREPRINT +extra parameter was practically zero, and therefore should be removed from the model. The other estimated kinetic +parameters are, although not identical to the true values, close to them. In this way, ADoK-W and ADoK-S were +identical. However, in computational time, they were noticeably different, where ADoK-S took in the order of minutes +to terminate and ADoK-W took in the order of hours. It has been hypothesized that ADoK-W might be successful in +highly noisy environments where ADoK-S might fail, but it might also be intractable in the high-data regime where +ADoK-S might still be tractable. All in all, ADoK-W, alike ADoK-S, is capable of retrieving the underlying kinetic rate +model of a catalytic isomerization reaction with realistic data. +4.3 +Decomposition of Nitrous Oxide — ADoK-S +As shown in Figure 1, the first step of ADoK-S is to generate concentration profiles, perform parameter estimation +on the the proposed models, and with AIC select the best model. For the catalytic decomposition of nitrous oxide, +ADoK-S selected the following concentration profiles for each species in each experiment. +fN2O,1 = exp (1.587 − 0.357t) +(17a) +fN2,1 = t − exp (0.192t) + 1.683 +(17b) +fO2,1 = 0.395 exp (2.320179 − t) + 3.630 +(17c) +fN2O,2 = exp (2.023 − 0.386t) +(17d) +fN2,2 = exp +� +exp +� +t +t +0.908 + 1.147 +�� +− 1.198 +(17e) +fO2,2 = +5.516 +exp (exp (−0.586t)) +(17f) +fN2O,3 = exp (2.023 − 0.386t) +(17g) +fN2,3 = 3.081t + 1.540 − (0.240 − 0.010t)(t − exp (1.643 − 2t)) +(17h) +fO2,3 = exp (2.058 − exp (−0.597t)) +(17i) +For the sake of brevity, the concentration models proposed by the GP algorithm, optimized by the optimization routine +and selected by the AIC are shown only for the second experiment in Figure 6 below. The selected concentration +profiles, qualitatively, fit the concentration data well. +Once established the selected concentration models, the rate measurements (i.e.: derivatives of the concentration +profiles) are estimated. Below, a plot of the numerical derivatives and (realistically inaccessible) rate measurements is +presented. From this plot, we can conclude that the rate measurements are indeed accurately estimated. +After estimating the rate measurements of the catalytic system, rate models are proposed using the GP algorithm and +optimized. The best model is then selected by AIC, which for this case study, it is shown below. +r = +C2 +N2O +k′ +1 + k′ +2CN2O += +k1C2 +N2O +1 + k2CN2O +(18) +The response of the selected model is shown in Figure 6. Qualitatively, once again, the proposed rate model fits the data +extremely well, especially considering that only three computational experiments with 10 time-steps each were ran (i.e.: +90 datapoints). Furthermore, recalling the correct model structure presented in Equation 10, with some simple algebraic +manipulation (multiplying numerator and denominator by +1 +k′ +1 ), the same form is retrieved, as shown in Equation 18. +Table 6 shows the comparison between the kinetic parameters used to generate the data of the homogeneous decomposi- +tion of nitrous oxide and the estimated values of these parameters. Alike the previous case study, the estimated kinetic +parameters are slightly different from the true ones, which is unsurprising given the additive Gaussian noise. All in all, +ADoK-S is capable of retrieving the underlying kinetic rate model of the catalytic decomposition of nitrous oxide with +realistic data. +16 + +arXiv Template +A PREPRINT +Figure 6: The conditions for the second computational experiment are CN2O,0 = 7.5M, CN2,0 = 1M and CO2,0 = 2M. +Top left: Data from the second computational experiment and the selected concentration profiles selected by AIC. +Top right: Numerical derivatives of the concentration profiles and the true rate measurements (which realistically are +inaccessible) for the second computational experiment. Bottom: Response of the selected GP-proposed rate model +using ADoK-S for the second computational experiment. +Table 6: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S +selected model for the catalytic decomposition of nitrous oxide. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +2.332 M −1 s−1 +2.000 M −1 s−1 +k2 / kB +5.811 M −1 +5.000 M −1 +4.4 +Decomposition of Nitrous Oxide — ADoK-W +As shown in Figure 2, ADoK-W combines the two steps of the other proposed methodology into a single one by solving +the symbolic regression problem in his weak formulation. The selected model by the approach is: +The response of the selected model is shown in Figure 7. Identically to the results presented in the previous section, +recalling the correct model structure presented in Equation 10, with some simple algebraic manipulation (multiplying +numerator and denominator by +1 +k′ +1 ), the same form is retrieved, as shown in Equation 18. +Table 7 shows the comparison between the kinetic parameters used to simulate the homogeneous decomposition +of nitrous oxide and the estimated values of these parameters. Alike the previous case study, the estimated kinetic +parameters are slightly different from the true ones, which is unsurprising given the additive Gaussian noise. All in all, +the ADoK-W is capable of retrieving the underlying kinetic rate model of the catalytic decomposition of nitrous oxide +17 + +8 +Concentrations (M) +6 +4 +fNO2, 2 +fN2,2 +fo2,2 +2 +Nitrous Oxide +Nitrogen +Oxygen +0 +2 +4 +6 +8 +10 +Time (s)6 +True Rate Measurements NO2 +True Rate Measurements N2 +5 +True Rate Measurements O2 +4 +Estimated Rate Measurements NO2 +3 +Estimated Rate Measurements N2 +(Ms-1) +Estimated Rate Measurements O2 +2 +1 +0 +-1 +-2 +-3 +-4 +0 +2 +4 +6 +8 +10 +Time (t)8 +Concentrations (M) +6 +4 +r for NO2 +r for N2 +r for O2 +2 +NO2 +N2 +02 +0 +0 +2 +4 +6 +8 +10 +Time (s)arXiv Template +A PREPRINT +Figure 7: Response of the selected GP-proposed rate model using ADoK-W for the second computational experiment +where CN2O,0 = 7.5 M, CN2,0 = 1 M and CO2,0 = 2 M. +with realistic data. +Table 7: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W +selected model for the catalytic decomposition of nitrous oxide. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +2.322 M −1 s−1 +2.000 M −1 s−1 +k2 / kB +5.811 M −1 +5.000 M −1 +4.5 +Toluene Hydrodealkylation — ADoK-S +For the catalytic hydrodealkylation of toluene to benzene, ADoK-S selected the following concentration profiles for +each species in each experiment. +fT,1 = +1.115 +0.316t2 + t + 1.074 +(19a) +fH,1 = exp (−0.510t) + 1.975 +(19b) +fB,1 = exp +� +−3.160 +t + exp (t) +� +(19c) +fM,1 = −0.719 +exp (t) + 1.407 +(19d) +fT,2 = exp (t + 0.195 exp (0.568 − t) + 0.781) +(19e) +fH,2 = exp +� +exp +� +1.132 − t +4.222 +� ++ 1.794 +� +(19f) +fB,2 = +t +t +3.658 + 0.900 + 1.012 +(20a) +fM,2 = t − 4.885 +t + 1.971 + 3.985 +(20b) +fT,3 = exp (exp (−0.099(2t − 4.791))) − 0.042t (20c) +fH,3 = exp (exp (−0.218(t + 0.971)) + 1.267) +(20d) +fB,3 = +t +exp +� +t +10.001 +� + 2.361 +(20e) +fM,3 = −16.919 +t + 2.985 + 8.488 +(20f) +In the expressions above, T, H, B and M refer to toluene, hydrogen, benzene and methane, respectively. For the sake +of brevity, the concentration models proposed by the GP algorithm, optimized by the optimization routine and selected +by the AIC are only shown for the third experiment in Figure 8 below. The selected concentration profiles, qualitatively, +fit the concentration data well. +As shown in Figure 1, once the concentration profiles are modeled, the parameters estimated, and the best model +selected, the rate measurements (i.e.: derivatives of the concentration profiles) are approximated. Below, a plot of the +18 + +8 +Concentrations (M) +6 +4 +r for NO2 +r for N2 +r for O2 +2 +NO2 +N2 +02 +0 +0 +2 +4 +6 +8 +10 +Time (s)arXiv Template +A PREPRINT +numerical derivatives and (realistically inaccessible) rate measurements is presented. From this plot, we can conclude +that the rate measurements are fairly well estimated. However, significant discrepancies can be appreciated at the start +of the graph for the rate of production of benzene, showing that the initial curvature of the concentration profile for +benzene is not well captured. +After estimating the rate measurements of the catalytic system, rate models are proposed using the GP algorithm and +optimized. The best model is then selected by AIC, which for this case study, it is shown below. +r = +k1CT CH +k2 + k3CB + k4CH +(21) +Recalling the correct model structure presented in Equation 12, the proposed kinetic rate model from ADoK-S is not +identical to it. Although the proposed model is very similar to the correct one, it differs only in one variable in the de- +nominator (CH should be CT ). This result is unsurprising, since the profile of hydrogen and toluene are nearly identical, +simply differing by a constant scaling factor (as both species have the same stoichiometric coefficients but different initial +concentrations). As such, knowing the data-generating model a priori, as modelers, we are not satisfied and should then +perform additional experiments. For the purpose of this investigation, we limit the experimental budget to 4 experiments. +To demonstrate the potential of the case study, model-based design of experiments is conducted. Specifically, the +Hunter-Reiner criterion is used, where an experimental point is found which maximizes the difference between two +models’ responses Hunter and Reiner [1965]. Alike the parameter estimation, the optimization routine outlined which +harnesses the output of the ABC algorithm to warm-start the LBFGS algorithm is used to solve the Hunter-Reiner +criterion, where the two models under question are Equation 12 (ground-truth model) and Equation 21 (ADoK-S +proposed model). The optimized experimental point is (CT,0, CH,0, CB,0, CM,0) = (5, 3, 0, 2.276). Realistically, +this exercise could not be undertaken, since the underlying dynamic model would not be available. Regardless, as +mentioned before, this exercise is meant to investigate the potential of the methodological framework proposed. One +thing should be mentioned however: the initial experiments conducted are randomly selected, which is not an optimal +methodology. In practice, statistic design of experiments should be initially used to carefully select experiments, so that +the generated dynamic trajectories are sufficiently different, the process information gain can be maximized and the +experimental expense can be minimized. This would improve the effectiveness of the proposed methodologies. +As per the flowchart presented in Figure 1, after generating new experiments, the algorithm must be repeated for the +appended computational experiment. The resulted new best rate model selected by AIC is shown below. +r = +k′ +1CT CH +k′ +2CB + CT + k′ +3 += +k1CT CH +1 + k2CB + k3CT +(22) +By undergoing simple algebraic manipulation (multiplying the numerator and denominator by a factor of +1 +k′ +3 ), the +proposed model (Equation 22) is identical to the data-generating kinetic rate model (Equation 12). The response of the +selected kinetic rate model is shown below in Figure 8, whilst the estimation of the corresponding kinetic parameters +are presented in Table 8. +Table 8: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S +selected model for the catalytic hydrodealkylation of toluene to benzene. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +1.922 M −1 s−1 +2.000 M −1 s−1 +k2 / kB +8.726 M −1 +9.000 M −1 +k3 / kC +4.656 M −1 +5.000 M −1 +From these results we can conclude that, although ADoK-S is not able to return the correct kinetic rate model from a +small dataset composed of three random experiments, it is still able to return a model very similar to the ground-truth +one. Notwithstanding, provided more data from a single extra experiment, ADoK-S is able to select an identical kinetic +rate model whilst estimating the corresponding kinetic parameters reasonably well. +19 + +arXiv Template +A PREPRINT +Figure 8: The conditions for the third computational experiment are CT,0 = 5 M, CH,0 = 8 M, CB,0 = 2 M and +CM,0 = 3 M. Top left: Data from the third computational experiment and the selected concentration profiles selected by +AIC. Top right: Numerical derivatives of the concentration profiles and the true rate measurements (which realistically +are inaccessible) for the third computational experiment. Bottom: Response of the selected GP-proposed rate model +using ADoK-S for the third computational experiment. +4.6 +Toluene Hydrodealkylation — ADoK-W +Similarly to the behavior of ADoK-S on the catalytic hydrodealkylation of toluene to benzene, the 1-step approach is +unable to produce the true underlying kinetic model from the three initial computational experiments. The initially +selected model is presented below. +r = k1(CH + k2)(CT − k3) +CB + CT +(23) +Recalling the correct model structure presented in Equation 12, the proposed kinetic rate model from ADoK-W is +quite different from it. As such, like in the previous example, we are not satisfied with the model and should then +perform additional experiments. For the purpose of this investigation, we limit the experimental budget to 4 experiments. +Model-based design of experiments is conducted yet again using the Hunter-Reiner criterion and the optimization +routine to solve it. The optimized experimental point is (CT,0, CH,0, CB,0, CM,0) = (5, 7.302, 0, 2.247). Realistically, +this exercise could not be undertaken, since the underlying dynamic model would not be available. Regardless, as +mentioned, this exercise is meant to investigate the potential of the methodological framework proposed. +As per the flowchart presented in Figure 2, after generating new experiments, the algorithm must be repeated for the +appended computational experiment, just like for ADoK-S. The new best rate model selected by AIC is shown below. +As demonstrated, with one additional computational experiment, ADoK-W is also able to retrieve the dynamics of +20 + +8 +Concentrations (M) +6 +fT,3 +fH,3 +4 +fb,3 +fm,3 +Toluene +2 +Hydrogen +Benzene +Methane +0 +0 +6 +8 +10 +2 +4 +Time (s)Estimated/True Rate Measurements Toluene +2 +Estimated/True Rate Measurements Hydrogen +Estimated/True Rate Measurements Benzene +Estimated/True Rate Measurements Methane +1 +(t-sW) +0 +-1 +0 +2 +6 +8 +10 +4 +Time (s)8 +Concentrations (M) +6 +rfor T +rfor H +rfor B +rfor M +2 +H +B +M +0 +6 +8 +10 +0 +2 +4 +Time (s)arXiv Template +A PREPRINT +the catalytic system. This allows us to conclude that as the complexity of a system under investigation increases, the +required amount of data provided to both approaches must increase accordingly. In this context, ‘complexity’ refers +not only to the number of species that can be directly observed, but also to the level of convolution of the underlying +dynamics. For completeness, Figure 9 presents the behavior of the model selected by ADoK-W, along with Table 9 +which presents the estimated values (and the true values) of the kinetic parameters. +r = +k′ +1CT CH +k′ +2CB + CT + k′ +3 += +k1CT CH +1 + k2CB + k3CT +(24) +Figure 9: Response of the selected GP-proposed rate model using ADoK-W for the second computational experiment +where CT,0 = 5 M, CH,0 = 8 M, CB,0 = 2 M and CM,0 = 3 M. +Table 9: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W +selected model for the catalytic hydrodealkylation of toluene to benzene. +Kinetic Parameter +Estimated Value +True Value +k1 / kA +2.190 M −1 s−1 +2.000 M −1 s−1 +k2 / kB +10.000 M −1 +9.000 M −1 +k3 / kC +5.340 M −1 +5.000 M −1 +5 +Conclusions +Kinetic rate models are indispensable for the successful development of catalytic processes. Classical modeling +paradigms offer strategies to construct these models, some of which are well established in industry, others which +are becoming more prevalent. Regardless, these classical paradigms demonstrate some noteworthy challenges: the +construction of mechanistic models are expensive and time-consuming, whilst data-driven and hybrid models are +mostly uninterpretable and usually lack the ability to extrapolate. Automated knowledge discovery, a newer paradigm, +aims to harness machine learning and computational advances to generate closed-form models and circumvent these +challenges. Nevertheless, popularized methodologies within automated knowledge discovery also demonstrate notable +drawbacks: usually requiring limiting assumptions about an underlying model structure, a general lack of a robust and +motivated model selection routine, and sensitivity to noisy data. +These limitations motivated the presented work to explore and propose two different methodological frameworks for +the automated discovery of kinetic rate models. The first one, ADoK-S, explores how to solve a strong formulation of +the symbolic regression problem. It does so by exploiting a genetic programming algorithm to automatically generate +closed-form concentration profile models, a sequential optimization routine to refine the most promising generated +kinetic models, and the Akaike information criterion to selected the best model (it is important to mention that this +criterion has been selected due to a thorough analysis conducted on different model selection criteria which concluded +21 + +8 +Concentrations (M) +rfor T +r for H +rfor B +rfor M +2 +H +B +M +0 +0 +6 +8 +10 +2 +4 +Time (s)arXiv Template +A PREPRINT +that AIC is the most robust one). Once the best concentration model is selected, rate measurements are estimated via +numerical differentiation, and the mentioned routine of model generation, model refinement and model selection is +repeated to discover kinetic rate models. +As referred, ADoK-S needs to estimate rate measurements through numerical differentiation, which can be problematic +under high-noise environments. To guard against this potential downside of ADoK-S, another methodological +framework was proposed, ADoK-W, which explores how to solve a weak formulation of the symbolic regression +problem. By reformulation the problem to its weak form, the step to propose concentration models and estimate rate +measurements is bypassed. Instead, kinetic rate models can be proposed from the start by working in the derivative +hyperspace. In ADoK-W, alike ADoK-S, models are generated with a genetic programming algorithm, the most +promising models are refined using a sequential optimization routine, and the best kinetic rate model is selected by AIC. +To benchmark both approaches, three catalytic case studies (of increasing complexity) were carefully selected: a simple +isomerization reaction, the decomposition of nitrous oxide, and the hydrodealkylation of toluene. Both approaches +successfully retrieved the underlying dynamics of the isomerization reaction and of the decomposition of nitrous +oxide with three pseudo-randomized computational experiments. However, neither approach was able to retrieve +the ground-truth kinetic model for the hydrodealkylation of toluene case study. Nonetheless, to prove the potential +of the methodological frameworks, an extra experiment was appended to the original dataset. This experiment was +determined by solving the Hunter-Reiner criterion between the data-generating model and the methodology-selected +model. By adding one extra experiment, both approaches successfully recovered the kinetics, concluding that as the +complexity of the system increases, so must the data (information) provided to ADoK-S and ADoK-W. As previously +referred, the methodologies could be more efficient by implementing statistic design of experiments in the first step of +the frameworks, so that the experiments generated are optimal rather than random. +As such, it is fair to conclude that the proposed methodologies build-on from previously presented approaches to +automated knowledge discovery by tackling identified shortcomings directly. From this investigations, new angles of +research arose which will be investigated in future work. For instance: the frameworks should be stress-tested with +respect to noise and quantity of data; the introduction of physical and mathematical constraints should be explored +and assessed with respect to the efficacy of ADoK-S and ADoK-W; investigate possible ways to approximate optimal +experiments without having direct access to the underlying dynamics model; lastly, research how the methodological +frameworks can be harnessed to uncover how temperature (and other variables, such as the morphology of a catalyst) +affect the rate of a reaction. But for now, with the provided study, we believe that these methodologies can be +successfully applied to aid chemical reaction engineers in solving current and future problems. +References +R. E. Baker, J. M. Peña, J. Jayamohan, and A. Jérusalem. Mechanistic models versus machine learning, a fight worth +fighting for the biological community? Biol. Lett., 14(5):20170660, May 2018. doi:10.1098/rsbl.2017.0660. +G.C. Battiston, L. 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Bioeng., 117(11):3356–3367, July 2020. +doi:10.1002/bit.27512. +24 + diff --git a/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/load_file.txt b/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18a0d7dacd2c18b88053de9c6eb27c12aff1be29 --- /dev/null +++ b/q9FIT4oBgHgl3EQfxSv4/content/tmp_files/load_file.txt @@ -0,0 +1,1462 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf,len=1461 +page_content='THE AUTOMATED DISCOVERY OF KINETIC RATE MODELS - METHODOLOGICAL FRAMEWORKS A PREPRINT Miguel Ángel de Carvalho Servia Department of Chemical Engineering Imperial College London South Kensington, London, SW7 2AZ, UK m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='de-carvalho-servia21@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk Ilya Orson Sandoval Department of Chemical Engineering Imperial College London South Kensington, London, SW7 2AZ, UK o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='sandoval-cardenas20@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk Klaus Hellgardt Department of Chemical Engineering Imperial College London South Kensington, London, SW7 2AZ, UK k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='hellgardt@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk King Kuok (Mimi) Hii Department of Chemistry Imperial College London White City, London, W12 0BZ, UK mimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='hii@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk Dongda Zhang ∗ Department of Chemical Engineering The University of Manchester Manchester, M13 9PL, UK dongda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='zhang@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk Ehecatl Antonio del Rio Chanona ∗ Department of Chemical Engineering Imperial College London South Kensington, London, SW7 2AZ, UK a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='del-rio-chanona@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='uk January 30, 2023 ABSTRACT The industrialization of catalytic processes is of far more importance today than it has ever been before and kinetic models are essential tools for their industrialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic models affect the design, the optimization and the control of catalytic processes, but they are not easy to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Classical paradigms, such as mechanistic modeling require substantial domain knowledge, while data-driven and hybrid modeling lack interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Consequently, a different approach called automated knowledge discovery has recently gained popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Many methods under this paradigm have been developed, where ALAMO, SINDy and genetic programming are notable examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, these methods suffer from important drawbacks: they require assumptions about model structures, scale poorly, lack robust and well-founded model selection routines, and they are sensitive to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To overcome these challenges, the present work constructs two methodological frameworks, Automated Discovery of Kinetics using a Strong/Weak formulation of symbolic regression, ADoK-S and ADoK- W, for the automated generation of catalytic kinetic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' We leverage genetic programming for model generation, a sequential optimization routine for model refinement, and a robust criterion for model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Both frameworks are tested against three computational case studies of increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' We showcase their ability to retrieve the underlying kinetic rate model with a limited amount of noisy data from the catalytic system, indicating a strong potential for chemical reaction engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Keywords: chemical reaction engineering, kinetic model generation, automated knowledge discovery, information criteria, machine learning arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='11356v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='SC] 26 Jan 2023 arXiv Template A PREPRINT 1 Introduction Mathematical models are simplified descriptions of complex phenomena using a logic-based language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' These models are widely used in science and engineering, be it in physics to represent human mobility patterns Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2010], Brockmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2006], in medicine to represent the metastatic spread of cancer Franssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2019], Margarit and Romanelli [2016], or in chemical reaction engineering to represent the reaction kinetics of a specific process Schbib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [1996], Battiston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [1982].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The wide usage of mathematical models across disciplines and fields is a direct consequence of their high utility, both fundamentally and practically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Fundamentally, these models allow researchers to simplify and distill complicated phenomena to quantitative mathematical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Practically, they allow engineers to develop industrial processes and researchers to investigate the kinetics of a chemical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Given the importance of models, it is imperative to answer: how can models be constructed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Classically, this construction can be divided into three distinct paradigms: mechanistic, data-driven and hybrid modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Mechanistic models (also referred to as white-box models) are constructed and derived purely using existing fundamental laws, such as conservation equations and constitutive relations Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Hypotheses are frequently made during the construction of a mechanistic model with the trade-off between accuracy and simplicity in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' These hypotheses are usually manifested through equations of state, or empirical expressions Gernaey [2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Mechanistic models have a great number of advantages, such as: being derived from prior expert knowledge, the parameters being physically meaningful, having great extrapolatory properties, and being interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For these reasons, these models are still widely established in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Nevertheless, some of these advantages are double-edged swords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For example, to construct these models via existing fundamental laws, a modeler must have strong background knowledge, the construction is usually time-consuming, the level of nonlinearity may be high, and if the process is not perfectly understood a mechanistic model may be infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Also, having complicated nonlinear models results in increased experimental effort to collect sufficient data for the estimation of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Whereas mechanistic models technically may not require data for their structure identification (they may still require data for parameter estimation and validation), data-driven models solely use data for their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Unlike mechanistic models, data-driven models can be designed relatively quickly, and their construction requires no knowledge about the system being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Their structure is highly flexible, and can be promptly repurposed to account for more variables or to describe different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Data-driven approaches are also generally quicker to evaluate than mechanistic ones, and as a result of this, the former have a lot of potential in real-time simulation Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2019], del Rio-Chanona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2018a], Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2022], optimization Petsagkourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020], del Rio-Chanona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2018b], Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2023], Natarajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], and soft sensor development Mowbray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2022a], Kay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2022], Kadlec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, since no physical knowledge is used to make the black-box models, their prediction tends to be poor outside the range of data used to train them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : poor extrapolatory abilities), which might classify their usage in certain scenarios as unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Their performance is highly influenced by the quantity and quality of data available, and data pre-treatment before training a model is frequently required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The last class of models are the hybrid models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This type of models attempts to exploit the best of mechanistic and data-driven modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In its essence, a hybrid model has a mechanistic backbone and a data-driven block which tries to improve the fit of the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' There are two main approaches to hybrid modeling: parallel and sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In the parallel approach, the data-driven block describes the model-data mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In the sequential approach, the data-driven block describes one or more parameters of the mechanistic backbone that may be dependent on multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Or, the mechanistic backbone is integrated within the data-driven architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In either approach, generally, the hybrid model will retain the extrapolation capabilities of a mechanistic model, whilst retain the flexibility and ease of construction of a data-driven model Vega-Ramon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], Mowbray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2022b], Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' At face value, hybrid modeling seems like an elegant solution to the problems posed by mechanistic and data-driven modeling, albeit not the only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Another effective solution is to utilize existing data to automatically generate and select mechanistic models by exploiting state-of-the-art statistical and machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this way, the benefits of mechanistic models are maintained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : extrapolatory abilities, interpretability and physical meaning), whilst some of their drawbacks are eliminated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : needing background knowledge, expensive and time-consuming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This strategy/paradigm has been (loosely) named automated knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Formally, it is known as symbolic regression Haider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The methodology presented in this work follows this paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2 arXiv Template A PREPRINT Before going further we will set the mathematical notation that is shared between both optimization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' We set a predefined time interval ∆t = [t0, tf] and denote the state variables as x(t) ∈ Rnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The input dataset consists of pairs (ti, yi) for a given set T of nt sampling times ti, where ti ∈ ∆t and yi ∈ Rnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' M is the set of symbolic expressions reachable by the search procedure, where each model m ∈ M has a finite set of tunable parameters θm whose dimension depends on each proposed model: m(·, θm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The predicted state variables are denoted by ˆx(t), and L(ˆx(t), y(t)) is the discrepancy measure between our predictions and the measured values, which characterizes the performance of the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : least squares function, likelihood function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Symbolic regression, in its strong formulation, aims to find the model that best maps the time to the state variables directly: ˆxm(t, θm) = m(t, θm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' (1) In its weak formulation, it aims to find the model that best maps input variables to output variables as a differential equation system: ˙x(t, θm) = m(x(t), θm) (2a) ˆxm(t, θm) = � t t0 ˙x(τ, θm)dτ (2b) The optimization problem for both formulations can be expressed as follows: m⋆(t, θ⋆ m) = min m∈M,θm � ti∈T L (ˆxm(ti, θm), yi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' (3) A wide variety of methods have been proposed in literature to solve the symbolic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' These include but are not limited to: the ALAMO approach Wilson and Sahinidis [2017], the SINDy algorithm Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2016], the work by Taylor and colleagues Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], the work by Neumann and colleagues Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020], and genetic programming Koza [1994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Popular automated knowledge discovery frameworks frequently face at least one of four challenges that may limit their ability to retrieve underlying ground-truth models, and consequently, their real-world applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Firstly, they necessitate structural assumptions of the underlying data-generating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This is particularly true for some non-evolutionary strategies, as a design matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : a model library) needs to be constructed for their execution Wilson and Sahinidis [2017], Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Secondly, they may display poor scalability with respect to the number of state variables available (this is also particularly true of non-evolutionary strategies) Wilson and Sahinidis [2017], Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2016], Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Thirdly, they lack a motivated and rigorous model selection routine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : their choice of model selection routine may not be transparent and/or different routines are not tested) Wilson and Sahinidis [2017], Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2016], Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020], Koza [1994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Lastly, for the discovery of non-linear dynamics, they may be sensitive to noisy data when rate measurements are not directly accessible Wilson and Sahinidis [2017], Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2016], Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021], Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020], Koza [1994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this section, we introduced the importance of mathematical modeling within chemical engineering, the challenges of classical modeling paradigms, and the shortcomings of modern automated knowledge discovery methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This work aims to construct two generalizable and robust methodological frameworks that integrate a rigorous model selection routine for the automated kinetic rate model discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Section 2 two proposed methods are motivated and described in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' in Section 3 we introduce three case studies that are used to analyze the performance of the proposed methodological frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' in Section 4 the results of the study are presented and amply discussed along with the shortcomings of the proposed methodologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' and in Section 5 the key findings are presented with a brief outlook on future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2 Methodological Frameworks The two proposed methodological frameworks are comprised of three stages: model generation, model refinement and model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this section we expand on each of these steps and present the distinctive characteristic of each of the proposed frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Genetic programming (GP) is often considered one of the most generalizable and reliable model generation methods 3 arXiv Template A PREPRINT found in literature for an important reason: the flexibility to include prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Its execution requires minimal assumptions about the ground-truth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As a direct contrast to ALAMO (note: ALAMO is formulated as a cardinality constrained mixed-integer quadratic program, or more simply, as a mixed integer non-linear program) and SINDy (note: SINDy was initially formulated as non-linear program, however, SINDy can also be solved via mixed-integer optimization Bertsimas and Gurnee [2022]), GP does not need a design matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, if knowledge about the ground-truth model is available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : mass and energy balances), this can be provided via mathematical constraints alike ALAMO and SINDy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' A especially attractive quality of GP when compared to mixed-integer based alternatives is the explicit control over the levels of complexity in the resulting expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The generalization of a model has a close relationship with the complexity of it in relation to the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Approaches based on a design matrix, which mixes expressions of multiple complexities, does not control the resulting expression complexity, instead it controls the cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' An explicit report of several expressions of increasing complexity provides more value to the modeler in order to commit with one of the proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Here, complexity refers to the sum of all terms of a particular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For instance, f(x) = x2+2x−4 5x would have a complexity equal to 13, since each operator, constant and variable count as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, we propose that GP should be used for the kinetic rate model generation stage of both frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The basic concept of GP is to specify a set of state variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : temperature, pressure, concentration) and operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : ‘+’, ‘/’) that may be present in the underlying mechanistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This selection can be as relaxed or constrained as the modeler decides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' User-defined analytical functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : 1 k1x1+k2x2 ) may also be specified to be included within generated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' With this, an initial population of models can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Quoting Darwin’s theory of evolution, the best models — based on a specified performance metric — are evolved via genetic operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : crossover and mutation), and the worst models are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This process is iterated until convergence is achieved or a termination criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' By virtue of GP, model parameters are stochastically evolved through genetic mutations, using differential evolution, and not by explicitly solving a parameter estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, a model refinement stage is needed as a parameter estimation problem, where the error between the model’s response and the data are minimized by finding the best set of kinetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In GP, the most accurate generated model for each complexity level is output (the upper bound of complexity is user-defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Therefore, a model selection stage is needed to discern which kinetic rate model proposed is the most appropriate for a particular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 Introduction to ADoK-S The first proposed methodological framework, ADoK-S (Automated Discovery of Kinetics using a Strong formulation of symbolic regression), uses GP to solve the strong formulation of the symbolic regression to find kinetic rate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' A characteristic of the strong formulation of symbolic regression is that only models that can directly map the specified state variables to the output variable are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Therefore, in the context of kinetic rate model discovery, rate measurements need to be provided to find the desired model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' rate is defined as: r = 1 νi dCi dt (4) where νi and dCi dt represent the stoichiometric coefficient and the rate of change of concentration with respect to time of species i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Realistically, seldom does a modeler have direct access to rate measurements, and therefore it would be practically meaningless to assume that these measurements are indeed available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Classically, to construct a mechanistic kinetic rate model of a chemical system, a modeler only has access to discrete measurements of the concentration of observed species with respect to time (for batch reactors) or with respect to residence time (for continuous plug flow reactors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' With this dataset, assumptions about the kinetics of the system are made (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : first order or second order kinetics), models are generated, kinetic parameters are estimated by using the dynamic trajectories of the concentrations, and a final model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Thus, it is fair to assume that concentration data with respect to time are available to a modeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 4 arXiv Template A PREPRINT Under the assumption that dynamic trajectories of concentrations are available, and knowing that the strong formulation of symbolic regression requires rate measurements of the chemical reaction, it is evident that these measurements must be estimated from the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Assuming an isochoric and isothermal system, it can be claimed that Ci (the concentration of species i ∈ Z+) is only dependent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Mathematically, under these conditions, the concentration profile of an arbitrary species is a function only dependent on time, Ci = fi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Therefore, if fi(t) can be accurately estimated, discrete rate measurements can also be estimated by numerically differentiating the concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To avoid limiting assumptions, GP is used to estimate fi(t), as output measurements Ci and input measurements t are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Finding an appropriate concentration profile follows the three stages referred earlier: model generation, model refinement, and model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the model generation stage, an implementation of GP done by Cranmer and colleagues was used Cranmer [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the model refinement stage, posed as a parameter estimation problem, the error between a model’s response and the data are minimized by finding the best set of kinetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This optimization problem is solved by carrying out an initial screening of the kinetic parameter search space using the artificial bee colony (ABC) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The best output from the ABC is used to warm-start the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The ABC algorithm was chosen because of its excellent explorative characteristics Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Whereas, the LBFGS algorithm was chosen because of its excellent performance in the parameter estimation task Malouf [2002], and optimization in general Liu and Nocedal [1989].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The objective function used for the parameter estimation in this work was the negative log-likelihood (NLL), presented below: NLL(θ) = � i,j,k � (Ci,j,k − y(θ)i,j,k)2 2ˆσ2 i − log � 1 � 2πˆσ2 i �� (5) where Ci,j,k is the measured concentration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : in-silico data) of species i ∈ S for dataset j ∈ D at time k ∈ T, where S, D, and T represent the species set, data set and time set, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' y(θ)i,j,k is the concentration of species i ∈ S for dataset j ∈ D at time k ∈ T proposed by an arbitrary model which is dependent on its parameters θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' ˆσ2 i is the variance of the noise that we assume the concentrations of species i ∈ S have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the model selection stage, the Akaike information criterion (AIC) is used to determine which model is the best ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The AIC was selected after a thorough analysis of the performance of different criteria (finite-sampled corrected AIC, Bayesian information criteria and Hannan Quinn criterion) under several conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : different amounts of additive noise, different amounts of data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This analysis concluded that AIC has a higher probability of selecting the correct data-generating rate model than the other criteria tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The AIC selects the model which has the lowest AIC value calculated from the below expression: AICm = 2NLL(θ)m + 2dm (6) here the subscript m serves to represent model m ∈ M, where M is the set of proposed models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' dm represents the number of parameters present in a model m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Once a concentration model has been successfully generated, refined and selected, this same model Ci = fi(t) is numerically differentiated using the central difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this way, a new training dataset is generated, where the inputs are the discrete measurements of concentration through time and the outputs are the discrete estimates of the rate measurements through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Given that the rate is a function of the concentration of the species (r = f(C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=')), the same protocol outlined above can be executed again with the new training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' By doing so, rate models that take as inputs the concentrations of the species observed in the chemical system and outputs the rate of that same system are generated using GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Then, using the ABC and the LBFGS algorithms, the models are refined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' using AIC, the best model is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Once an optimized kinetic rate model is selected by AIC, this model is numerically integrated using the LSODA algorithm Hindmarsh and Petzold [2005] ( � rdt = � 1 νi dCi dt dt = 1 νi Ci(t)) implemented in SciPy python package Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The results from this integration are directly compared to the original datasets of the dynamical trajectories of the observed concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' If the results are satisfactory to the modeler, the methodological framework 5 arXiv Template A PREPRINT is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Otherwise, further data should be collected by applying (model-based) design of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the sake of clarity and simplicity, the flowchart of the proposed methodological framework is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Figure 1: The flowchart of ADoK-S (Automated Discovery of Kinetics using a Strong formulation of symbolic regression);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' the red and blue dashed boxes represent the steps where rate measurements and rate models are estimated, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 Introduction to ADoK-W One of the primary reasons to solve the symbolic regression problem is to retrieve interpretable mathematical expressions that experts can analyze and validate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' But perhaps more importantly, expressions that experts can extract knowledge from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Clearly, the strong formulation of symbolic regression hinders this process within the catalytic reaction engineering context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As explained in the previous section, to successfully implement the strong formulation of symbolic regression, ADoK-S must first propose and refine concentration profiles, where the best one (selected by AIC) is used to estimate rate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Catalytic reaction engineers seldom know the model structure of a concentration profile, especially because many kinetic rate models do not have a close-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, experts would find the task to validate concentration models strenuous, perhaps even impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Furthermore, having a model that describes the dependence of the concentration of a species with respect to time does not provide significant and actionable information to the experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In other words, experts cannot extract much knowledge from these models and we can conclude that the strong formulation of symbolic regression inevitably affects the interpretability of ADoK-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The strong formulation of symbolic regression may also affect the performance of ADoK-S with regards to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The necessity to have the first step of ADoK-S is that rate measurements are required to retrieve a kinetic rate model (as the strong formulation only allows the direct mapping of inputs to outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As previously mentioned, having direct access to the rate measurements of a dynamical catalytic chemical system is a rarity and thus need to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As it is explained in Bertsimas and Gurnee [2022], numerical differentiation exacerbates any noise inherent to the measured data, and even robust differentiation techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : polynomial interpolation) may not be enough to provide usable derivative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In ADoK-S, these effects are minimized by selecting a concentration profile model with the AIC, but the effects are not eradicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Therefore, we can conclude that the strong formulation of symbolic regression may also 6 Start Generate data: (t, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='(t)) Estimating derivatives Estimating rate models With GP: C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' = C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='(t, 0) With GP: r = r(C,Φ) Parameter estimation:→* Parameter estimation: Φ → Φ Model selection with AIC Model selection with AIC Generate derivative data: (t, C (t, *) Integrate rate model: J r(C, Φ*)dt No UseDoE/MBDoE Satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Yes StoparXiv Template A PREPRINT affect the performance of ADoK-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The innate drawbacks of the strong formulation for the automated kinetic rate discovery task motivated the construction of a different methodological framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' ADoK-W (Automated Discovery of Kinetics using a Weak formulation of symbolic regression) aims to combine the two steps from the previous framework into a single one by reformulating the symbolic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The weak reformulation of symbolic regression bypasses altogether the construction of concentration profile models, and the subsequent numerical differentiation to obtain estimations of the rate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Whereas the strong formulation provides a direct mapping between inputs and outputs, the weak formulation of symbolic regression represents a mapping between input and output variables in the derivative space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In other words, this reformulation generates models dependent on the state variables that can map the output via an integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this way, the GP algorithm receives the dynamic trajectories of concentration as inputs, proposes rate models (r = f(C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=') where Ci is the concentration of species i ∈ Z+), integrates the rate models with respect to time at each given time-step where concentration data is available ( � rdt = � 1 νi dCi dt dt = 1 νi (Ci(t = t) − Ci(t = 0))), and compares the results from the integration with the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Similarly to ADoK-S, due to the usage of a GP algorithm to solve the symbolic regression problem, the proposed rate models may have under-evolved kinetic parameters and therefore necessitate a model refinement stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As explained in the previous section, this model refinement stage is posed as a parameter estimation problem, where Equation 5 is used as the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To solve this problem, the kinetic parameter search space is initially scanned by the ABC algorithm, where its output is used as a warm-start for the LBFGS algorithm which will output the final set of kinetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' If desirable, this optimization routine can be repeated multiple times to improve the probability of reach- ing a global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Then, alike ADoK-S, the best model is selected based on the AIC value produced from Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In identical fashion to the previously presented methodological framework, once an optimized kinetic rate model is selected by AIC, this model is numerically integrated using the LSODA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The results from this integration are directly compared to the original datasets of the dynamical trajectories of the observed concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' If the results are satisfactory to the modeler, the methodological framework is terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Otherwise, further data should be collected by applying (model-based) design of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the sake of clarity and simplicity, the flowchart of the proposed methodological framework is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 3 Catalytic Kinetic Case Studies To showcase the performance of the proposed methodological frameworks, three illustrative catalytic kinetic case studies were chosen: an isomerization reaction, the decomposition of nitrous oxide, and the toluene hydrodealkylation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Below, their respective kinetic rate models are introduced, along with how the required datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : dynamic trajectories of concentrations) are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 Isomerization Reaction The simplest case study presented in this work is a catalytic isomerization reaction, where A is transformed to B reversibly over a catalytic active site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The reaction is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' A ⇌ B (7) The kinetic rate model that describes the evolution of the concentrations of A and B through time is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This expression has been directly borrowed from the book by Marin and colleagues Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = −dCA dt = dCB dt = kACA − kBCB kCCA + kDCB + kE (8) In Equation 8, CA and CB represent the concentration of reactant A and product B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The kinetic parameters of the kinetic rate model are represented by ki where i ∈ [A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=', E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To generate the necessary dataset to test both frameworks, three computational experiments are carried out, each with different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The computational experiments are run with the following initial conditions (in molar units, mol L−1): (CA,0, CB,0) ∈ {(2, 0), (6, 1), (10, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For each computational experiment, the concentration of the reactant and 7 arXiv Template A PREPRINT Figure 2: The flowchart of ADoK-W (Automated Discovery of Kinetics using a Weak formulation of symbolic regression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' product are recorded 10 times, at evenly spaced intervals between time t=0 s and t=10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this particular case study, the kinetic parameters were defined as: kA=7 M s−2, kB=3 M s−2, kC=4 s−1, kD=2 s−1 and kE=6 M s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To approximate the behavior of a realistic chemical system, Gaussian noise was added to the computational experimental measurements, yielding the needed dynamic trajectories of concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The defined Gaussian noise has zero mean and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='17 for the concentrations of A, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='18 for the concentrations of B (the standard deviations represent 5% of the mean of the concentrations of A and B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the parameter estimation task, it would be futile to assume that, as modelers, the exact variance of the noise would be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Thus, a conservative assumption is made by setting ˆσA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='35 and ˆσB=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='35 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : assuming a standard deviation of 10% of the mean of the concentrations of A and B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The generated data of one of the computational experiments are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' All the graphs presented in this paper have been generated using the Matplotlib package implemented in Python Hunter [2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 Decomposition of Nitrous Oxide The second case study presented in this work is the catalytic decomposition of nitrous oxide, where nitrous oxide (N2O) is transformed to nitrogen gas (N2) and oxygen gas (O2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The reaction is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2N2O ⇌ 2N2 + O2 (9) The kinetic rate model that describes the evolution of the concentrations of N2O, N2 and O2 through time is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This expression has been directly borrowed from the book by Levenspiel Levenspiel [1998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = −2dCN2O dt = 2dCN2 dt = dCO2 dt = kAC2 N2O 1 + kBCN2O (10) 8 Start Generate data: (t, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='(t)) With GP: r = r(C,Φ) Parameter estimation: Φ → Φ Use DoE/MBDoE Model selection with AIC Integrate rate model: J r(C, Φ*)dt No Satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Yes StoparXiv Template A PREPRINT In Equation 10, CN2O, CN2 and CO2 represent the concentration of reactant nitrous oxide, and of products nitrogen gas and oxygen gas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The kinetic parameters of the kinetic rate model are represented by ki where i ∈ [A, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To generate the necessary dataset to test both frameworks, three computational experiments are carried out, each with different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The experiments are run with the following initial conditions (in molar units): (CN2O,0, CN2,0, CO2,0) ∈ {(5, 0, 1), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5, 1, 2), (10, 2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For each experiment, the concentration of the reactant and products are recorded 10 times, at evenly spaced intervals between time t=0 s and t=10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this particular case study, the kinetic parameters were defined as: kA=2 M−1 s−1 and kB=5 M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To approximate the behavior of a realistic chemical system, Gaussian noise was added to the in-silico datasets, yielding the needed dynamic trajectories of concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The defined Gaussian noise has zero mean and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='11 for the concentrations of N2O, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='32 for the concentrations of N2, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='23 for the concentration of O2 (the standard deviations represent 5% of the mean of the concentrations of N2O, N2 and O2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the parameter estimation task, it would be futile to assume that, as modelers, the exact variance of the noise would be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Thus, a conservative assumption is made by setting ˆσN2O=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='22, ˆσN2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='63 and ˆσO2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='46 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : assuming a standard deviation of 10% of the mean of the concentrations of N2O, N2 and O2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The generated data of one of the computational experiments are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 Toluene Hydrodealkylation The third and most complex case study (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : case study with most species involved in the chemical reaction) presented in this work is the catalytic toluene hydrodealkylation to benzene, where toluene (C6H5CH3) and hydrogen gas (H2) is transformed to benzene (C6H6) and methane (CH4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The reaction is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' C6H5CH3 + H2 ⇌ C6H6 + CH4 (11) The kinetic rate model that describes the evolution of the concentrations of C6H5CH3, H2, C6H6 and CH4 through time is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This expression has been directly borrowed from the book by Fogler Fogler [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = −dCT dt = −dCH dt = dCB dt = dCM dt = kACT CH 1 + kBCB + kCCT (12) In Equation 12, CT , CH, CB and CM represent the concentration of reactants toluene and hydrogen, and of products benzene and methane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The kinetic parameters of the kinetic rate model are represented by ki where i ∈ [A, B, C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To generate the necessary dataset to test both frameworks, three computational experiments are carried out, each with different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The computational experiments are run with the following initial conditions (in molar units): (CT,0, CH,0, CB,0, CM,0) ∈ {(1, 3, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5), (3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='75), (5, 8, 2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this particular case study, for each experiment, the concentration of the reactant and products are recorded 50 times, at evenly spaced intervals between time t=0 s and t=10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The frequency of the sampling was increased due to the complexity of the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It should be noted that this increment is still within the realistic range of experimental sampling frequency, as these samples could have been gathered from a continuous-flow experiment Schrecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The kinetic parameters were defined as: kA=2 M−1 s−1, kB=9 M−1 and kC=5 M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It should be noted that, for each of the case studies, only the model structure is borrowed from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The value of the kinetic parameters are randomly assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To approximate the behavior of a realistic chemical system, Gaussian noise was added to the experimental results, yielding the needed dynamic trajectories of concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The defined Gaussian noise has zero mean and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='09 for the concentrations of toluene, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='16 for the concentrations of hydrogen gas, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='07 for the concentration of benzene, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='15 for the concentration of methane (the standard deviations represent 5% of the mean of the measured concentrations of toluene, hydrogen gas, benzene and methane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the parameter estimation task, same as before, a conservative assumption is made by setting ˆσT =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='18, ˆσH=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='33, ˆσB=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='14, and ˆσM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='30 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : assuming a standard deviation of 10% of the mean of the concentrations of toluene, hydrogen gas, benzene and methane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The generated results of one of the computational experiments are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 9 arXiv Template A PREPRINT Figure 3: Top left: The in-silico data of one of the computational experiments for the catalytic isomerisation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top right: The generated data of one of the computational experiments for the catalytic decomposition of nitrous oxide reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Bottom: The generated data of one of the computational experiments for the catalytic hydrodealkylation of toluene reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 4 Results and Discussions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 Isomerization Reaction — ADoK-S As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1, ADoK-S aims to solve a strong formulation of symbolic regression by implementing GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this methodological framework, departing from the kinetic data generated (routine for data generation is detailed in Section 3), the GP algorithm is used to generate concentration profile models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : models which represent the evolution of concentration through time as a function of time, f(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The expression construction rules exclusively included the arithmetic operators ‘+’, ‘−’, ‘×’, ‘/’ and ‘exp’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' According to physical knowledge, this is a fair selection of operators as the likelihood of these operators appearing is extremely high (to the best of the author’s knowledge, trigonometric operators, for example, do not appear in concentration expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The generation of concentration profiles is carried out for reactant A and product B at each computational experiment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : 2 concentration models are generated for each experiment — (fA,i, fB,i) for i ∈ [1, 2, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In a real case study, assuming that all of A gets transformed to B without forming any side-product would not be a valid assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For that reason, it would be bad practice to generate a model for the concentration of A (or B) and obtaining a model for B (or A) by simply solving the mass balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, we must propose models for A and B separately to ensure that the rates are well predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Models must also be proposed for each experiment because, although the rate model is the same for each species, its integration is dependent on initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In other words, for different initial conditions, the concentration profiles have different functional forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The same rational is applied to the subsequent case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the sake of brevity, only the results from the first experiment are presented, but the same routine is executed on all the other computational experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It is also specified that the proposed models should only include ‘t’ as a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1, the concentration profiles, assuming isochoric and isothermal conditions, should only be 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 B Concentrations (M) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0 2 4 6 8 10 Time (s)Nitrous Oxide Nitrogen Oxygen 4 Concentrations (M) 3 2 L 0 0 2 4 6 8 10 Time (s)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 Toluene Hydrogen Benzene 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 Methane Concentrations (M) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0 2 4 6 8 10 Time (s)arXiv Template A PREPRINT dependent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The GP algorithm proposed the model structures for the concentration of A and B in experiment 1 shown below, where pℓ for ℓ ∈ [1, 2, 3, 4] are the parameters that can be estimated according to the concentration versus time dataset and fi,j,k refers to the kth concentration model of species i in experiment j proposed by SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 = p1 (13a) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 = exp (p1t) (13b) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 = p1 t + p2 (13c) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 = exp (p1 − t) + p2 (13d) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 = p1 + p2 exp (−p3t) (13e) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 = exp (−t)(p1t + p2) + p3 (13f) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='7 = p1 + exp (−t − exp (−p2t) + p3) (13g) fA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='8 = p1t + p2 p3 + exp (t) + p4 (13h) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 = p1 (14a) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 = exp (p1t) (14b) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 = p1 + p2t (14c) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 = p1 − p2 exp(−t) (14d) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 = p1 − p2 t + p3 (14e) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 = p1 − p2 p3 + exp (t) (14f) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='7 = exp (−t)(p1t − p2) + p3 (14g) fB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='8 = −p1t − p2 p3 + exp (t) + p4 (14h) For each of the proposed model structures shown above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' parameter estimation is carried out by finding the parameters which minimizes Equation 5 (as mentioned,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' the assumed variances for NLL are double of the real variances used to generate the Gaussian noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This problem is solved by employing the strategy outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : initial parameter screening with ABC algorithm and solution refinement with LBFGS algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 1 displays the NLL values along with the AIC values for each of the models, showing that fA,1,4 is the best model for the concentration profile of A (in experiment 1), and showing that fB,1,4 is the best model for the concentration profile of B (in experiment 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' We can conclude from the proposed concentration models that the catalytic system under investigation is not a complex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This demonstrates that, although the first step of ADoK-S is mostly uninterpretable, it can still provide the modeler with some level of insightful information pertaining to the complexity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 1: The negative log likelihood values and the AIC values of all concentration profile models proposed by the GP algorithm for reactant A and product B for experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Model NLL Value AIC Value fA,1,1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='285 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='570 fA,1,2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='175 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='351 fA,1,3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='686 fA,1,4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='850 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='701 fA,1,5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='442 fA,1,6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='470 fA,1,7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='474 fA,1,8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='235 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='530 Model NLL Value AIC Value fB,1,1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='764 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='528 fB,1,2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='126 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='252 fB,1,3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='726 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='451 fB,1,4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='079 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='159 fB,1,5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='897 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='793 fB,1,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='745 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='490 fB,1,7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='749 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='497 fB,1,8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='746 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='492 The concentration models proposed by the GP algorithm, optimized by the optimization routine and selected by the AIC are shown in Figure 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Qualitatively, the proposed concentration models fit the data quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Although no physical constraints were imposed, it is important to mention that the chosen concentration profiles respect the law of conservation of mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : CA + CB = CA0 + CB0 = 2 + 0 = 2 M, substituting the chosen models, CA+CB = exp k1 − t+k2+k3−k4 exp −t = exp k1 exp −t+k2+k3−k4 exp −t = k1 exp −t+k2+k3−k4 exp −t, so if k1 = k4 and k2 + k3 = 2, then the law is respected) and their end-behavior are correct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : at time t = 0 s, CA ≈ 2 M and CB ≈ 0 M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' at time t → ∞, CA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 M and CB ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, it should be noted that this observation is only valid for this case study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' it is not certain that this behavior is displayed in other case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Notwithstanding, it is encouraging to see that ADoK-S, without enforcing constraints, recommends concentration profiles that respect the conservation of mass and the reached equilibrium of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Following the ADoK-S flowchart presented in Figure 1, the next step is to compute the derivatives of the concentration profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In Figure 4, a plot of the numerical derivatives of concentration of A and B with respect to time is presented, along with the (realistically inaccessible) rate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 11 arXiv Template A PREPRINT Once the numerical derivative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : rate) data are estimated for each of the experiments, the GP algorithm is executed again, but this time to find a unifying rate equation that minimizes the error from its evaluation and the rate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this execution of the GP algorithm, it is specified that the proposed models should include the same operators as the concentration profile models, with the exception of the ‘exp’ operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' That is, the proposed rate models may include: ‘+’, ‘−’, ‘×’ and ‘/’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' According to physical knowledge this is, once again, a fair selection of operators as the likelihood of these operators appearing in a rate model is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The flexibility of choosing different operators shows one way in which prior domain knowledge can be injected into kinetic model discovery (another way would be to constrain the algorithm, but this is outwith the scope of the investigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Additionally, provided by prior knowledge, the proposed rate models may include concentrations of the observed species as variables: ‘CA’ and ‘CB’ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The GP algorithm proposed the rate model structures shown below: r1 = k1 (15a) r2 = k1CA (15b) r3 = k1CA k2CB + k3 (15c) r4 = k1CA − k2CB k3CA (15d) r5 = k1CA − k2CB + k3 k4CA + k5 (15e) r6 = k1CA − k2CB − k3 k4CA + k5CB + k6 (15f) r7 = k1C2 A + k2CACB − k3CA − k4C2 B k5C2 A + k6CACB + k7CA + k8C2 B (15g) where ki for i ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=', 8] are the parameters that are estimated according to the concentration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To do so, the proposed rate models are integrated and evaluated at each time-step, given the initial conditions CA,0 and CB,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Once again, the parameter estimation problem is solved by minimizing the NLL using the optimization routine comprised of the ABC and LBFGS algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 2 clearly shows the NLL value along with the AIC values for each of the rate models, showing that r6 is the best model for to represent the dynamical catalytic reactive system under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 2: The negative log likelihood values and the AIC values of all proposed rate models by ADoK-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Model NLL Value AIC Value r1 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='768 1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='535 r2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='304 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='608 r3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='187 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='373 r4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='301 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='601 r5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='124 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='751 r6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='419 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='161 r7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='349 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='301 The response of the rate model proposed by the GP algorithm and selected by the AIC is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Qualitatively, once again, the proposed rate model fits the data extremely well, especially considering that only three computational experiments with 10 time-steps each were ran (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : 60 datapoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Nevertheless, recalling the correct model structure presented in Equation 8, it is evident that the rate model selected has one extra parameter in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In fact, none of the rate equations proposed by SR retrieves the exact structure of the data-generating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' After performing parameter estimation on the selected model, the parameters obtained are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The first thing that can be perceived from Table 3 is that the extra parameter in the numerator of model r6 is estimated to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In other words, the GP algorithm was not able to retrieve the exact model structure, but the optimization routine determined that the extra parameter should be non-existent, arguably retrieving the correct model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 12 arXiv Template A PREPRINT Figure 4: The conditions for the first computational experiment are CA,0 = 2 M and CB,0 = 0 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top left: Data from the first computational experiment and the selected concentration profiles selected by AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top right: Numerical derivatives of the concentration profiles for and the true rate measurements (which realistically are inaccessible) for the first computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Bottom: Response of the selected GP-proposed rate model using ADoK-S for the first computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 3: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S selected model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='897 M s−2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−2 k2 / kB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='302 M s−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−2 k3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M2 s−2 N/a k4 / kC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='613 s−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 s−1 k5 / kD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='182 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 s−1 k6 / kE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='033 M s−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−1 The second thing that can be perceived is that the estimated kinetic parameters are slightly different from the kinetic parameters that were used to generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Of course, this is unsurprising due to the additive Gaussian noise that was introduced in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Had the parameter uncertainty been calculated, chances are that the true parameters would lie within the 95% confidence interval, given the small difference between the true and the estimated values (notwithstanding, this is pure conjecture as parameter uncertainty was outside the scope of this investigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To briefly summarize, the results demonstrated that ADoK-S is robust to noise, regardless of the fact that derivatives needed to be calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The main reason for the robustness of the method stems from utilizing GP and AIC to generate and select, respectively, good concentration models that do not overfit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this way, the numerical derivatives continue to have a high degree of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The final rate model output by ADoK-S had a near identical 13 fA, 1, 4 A B Concentrations (M) 0 0 2 4 6 8 10 Time (s)True Rate Measurements A True Rate Measurements B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 Estimated Rate Measurements A Estimated Rate Measurements B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 (t-SW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0 2 4 6 8 10 Time (t)Concentrations (M) r6 for A r6 for B A B 0 0 2 4 6 8 10 Time (s)arXiv Template A PREPRINT structure to the data-generating one, differing only by a single parameter in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, after performing parameter estimation, the extra parameter was determined to be practically zero, and therefore should not appear in the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The other estimated kinetic parameters, although not identical to the true values, they were close to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' All in all, ADoK-S is capable of retrieving the underlying kinetic rate model of a catalytic isomerization reaction with realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 Isomerization Reaction — ADoK-W As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2, ADoK-W aims to solve a weak formulation of symbolic regression by implementing GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this methodological framework, departing from the kinetic data generated, the GP algorithm instead of generating concentration profile models and then kinetic rate models, it automatically evolves rate models by integrating and comparing them with the concentration data available directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Note that the law of conservation of mass is satisfied by construction under this integrating scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The expression construction rules exclusively included the arithmetic operators ‘+’, ‘-’, ‘×’ and ‘/’, since rates including other operators are less common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Alike in ADoK-S, the kinetic rate models generated are allowed to be a function of the species whose concentrations were measured, r(CA, CB) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The best expression proposed by the GP algorithm, sorted by degree of complexity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : the number of operators and variables), are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r1 = k1 (16a) r2 = k1CA (16b) r3 = k1CA − k2CB k3CA (16c) r4 = k1CA − k2CB − k3 k4CA (16d) r5 = k1CA − k2CB − k3 k4CA + k5 (16e) r6 = k1CA − k2CB − k3 k4CA + k5CB + k6 (16f) r7 = k1C2 ACB − k2CAC2 B − k3CA + k4CB k5C2 ACB − k6CA (16g) The estimation of each kinetic parameter, ki for i ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=', 6], is carried out as explained previously: NLL is used as the objective function where the assumed variances are double of the real variances used to generate the additive Gaussian noise, and this is solved by deploying the optimization routine consisting of the ABC and the LBFGS algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 4 clearly shows the NLL value along with the AIC values for each of the rate models, showing that r6 is the best model for the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 4: The negative log likelihood values and the AIC values of all proposed rate models by ADoK-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Model NLL Value AIC Value r1 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='410 1082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='821 r2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='857 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='715 r3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='831 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='337 r4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='988 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='025 r5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='724 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='551 r6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='275 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='450 r7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='289 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='422 The response of the selected model is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the sake of brevity, only one of the experiments is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The final rate model output by the proposed framework had a near identical structure to the data-generating one, differing only by a single parameter in the numerator, displaying an identical result to ADoK-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' And once again, the optimization routine used to solve the parameter estimation problem is able to determine that the extra parameter is practically zero, and therefore may be non-existent in the actual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this case, since the data-generating model is known, we can indeed validate this conclusion which would have naturally been made regardless of having access to 14 arXiv Template A PREPRINT the underlying ground-truth model or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Figure 5: Response of the selected GP-proposed rate model using ADoK-W for the first computational experiment where CA,0 = 2 M and CB,0 = 0 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Similar to the estimated kinetic parameter values presented in Table 3, Table 5 also shows that the estimated kinetic parameters are slightly different from the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As previously explained, this discrepancy is caused by the additive Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It should be noted that the the value of NLL for r6 in Table 2 and in Table 4 (r6 is identical for both approaches) are slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The reason being that the estimated parameters are different for both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This finding can be attributed to the optimization algorithm not finding the global optimum and getting stuck in different local optima in different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 5: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W selected model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='557 M s−2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−2 k2 / kB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='335 M s−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−2 k3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M2 s−2 N/a k4 / kC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='114 s−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 s−1 k5 / kD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='379 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 s−1 k6 / kE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='033 M s−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M s−1 One thing that should be mentioned about ADoK-W is the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' On one hand, the GP-steps of ADoK-S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : proposing concentration profiles and rate models) takes in the order of minutes to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this particular case study, it took around 10 minutes to propose 6 concentration models (2 concentration profiles, one for A and one for B, for each of the three experiments) and 1 rate model using an Apple MacBook Air (M1, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' On the other hand, the GP-step in ADoK-W (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : proposing rate models) take in the order of hours to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For this particular case study, it took around 5 hours to propose 1 rate model using the high performance computing cluster with 8-core CPU configured with 64GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This is unsurprising, as in ADoK-W, to calculate the fitness value of each of the proposed rate models, numerical integration needs to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For ADoK-S, the fitness values are calculated by merely evaluating a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Therefore, ADoK-W builds resilience to noise by making it more computationally in- tensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This could have ramifications as to the amount of data that this approach could take before it becomes intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The results demonstrated that ADoK-W, as expected from a weak formulation, is robust to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The final rate model output by ADoK-W has a near identical structure to the data-generating one, differing only by a single parameter in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, the optimization algorithm used for parameter estimation is able to determine that the 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 Concentrations (M) r6 for A r6 for B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 A B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 0 2 4 6 8 10 Time (s)arXiv Template A PREPRINT extra parameter was practically zero, and therefore should be removed from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The other estimated kinetic parameters are, although not identical to the true values, close to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this way, ADoK-W and ADoK-S were identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, in computational time, they were noticeably different, where ADoK-S took in the order of minutes to terminate and ADoK-W took in the order of hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It has been hypothesized that ADoK-W might be successful in highly noisy environments where ADoK-S might fail, but it might also be intractable in the high-data regime where ADoK-S might still be tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' All in all, ADoK-W, alike ADoK-S, is capable of retrieving the underlying kinetic rate model of a catalytic isomerization reaction with realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 Decomposition of Nitrous Oxide — ADoK-S As shown in Figure 1, the first step of ADoK-S is to generate concentration profiles, perform parameter estimation on the the proposed models, and with AIC select the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the catalytic decomposition of nitrous oxide, ADoK-S selected the following concentration profiles for each species in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' fN2O,1 = exp (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='587 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='357t) (17a) fN2,1 = t − exp (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='192t) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='683 (17b) fO2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='395 exp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='320179 − t) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='630 (17c) fN2O,2 = exp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='023 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='386t) (17d) fN2,2 = exp � exp � t t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='908 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='147 �� − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='198 (17e) fO2,2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='516 exp (exp (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='586t)) (17f) fN2O,3 = exp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='023 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='386t) (17g) fN2,3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='081t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='540 − (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='240 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='010t)(t − exp (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='643 − 2t)) (17h) fO2,3 = exp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='058 − exp (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='597t)) (17i) For the sake of brevity, the concentration models proposed by the GP algorithm, optimized by the optimization routine and selected by the AIC are shown only for the second experiment in Figure 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The selected concentration profiles, qualitatively, fit the concentration data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Once established the selected concentration models, the rate measurements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : derivatives of the concentration profiles) are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Below, a plot of the numerical derivatives and (realistically inaccessible) rate measurements is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' From this plot, we can conclude that the rate measurements are indeed accurately estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' After estimating the rate measurements of the catalytic system, rate models are proposed using the GP algorithm and optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The best model is then selected by AIC, which for this case study, it is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = C2 N2O k′ 1 + k′ 2CN2O = k1C2 N2O 1 + k2CN2O (18) The response of the selected model is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Qualitatively, once again, the proposed rate model fits the data extremely well, especially considering that only three computational experiments with 10 time-steps each were ran (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : 90 datapoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Furthermore, recalling the correct model structure presented in Equation 10, with some simple algebraic manipulation (multiplying numerator and denominator by 1 k′ 1 ), the same form is retrieved, as shown in Equation 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 6 shows the comparison between the kinetic parameters used to generate the data of the homogeneous decomposi- tion of nitrous oxide and the estimated values of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Alike the previous case study, the estimated kinetic parameters are slightly different from the true ones, which is unsurprising given the additive Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' All in all, ADoK-S is capable of retrieving the underlying kinetic rate model of the catalytic decomposition of nitrous oxide with realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 16 arXiv Template A PREPRINT Figure 6: The conditions for the second computational experiment are CN2O,0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5M, CN2,0 = 1M and CO2,0 = 2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top left: Data from the second computational experiment and the selected concentration profiles selected by AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top right: Numerical derivatives of the concentration profiles and the true rate measurements (which realistically are inaccessible) for the second computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Bottom: Response of the selected GP-proposed rate model using ADoK-S for the second computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 6: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S selected model for the catalytic decomposition of nitrous oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='332 M −1 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 s−1 k2 / kB 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='811 M −1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 Decomposition of Nitrous Oxide — ADoK-W As shown in Figure 2, ADoK-W combines the two steps of the other proposed methodology into a single one by solving the symbolic regression problem in his weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The selected model by the approach is: The response of the selected model is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Identically to the results presented in the previous section, recalling the correct model structure presented in Equation 10, with some simple algebraic manipulation (multiplying numerator and denominator by 1 k′ 1 ), the same form is retrieved, as shown in Equation 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 7 shows the comparison between the kinetic parameters used to simulate the homogeneous decomposition of nitrous oxide and the estimated values of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Alike the previous case study, the estimated kinetic parameters are slightly different from the true ones, which is unsurprising given the additive Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' All in all,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' the ADoK-W is capable of retrieving the underlying kinetic rate model of the catalytic decomposition of nitrous oxide 17 8 Concentrations (M) 6 4 fNO2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 2 fN2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 fo2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Nitrous Oxide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Nitrogen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Oxygen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Time (s)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='True Rate Measurements NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='True Rate Measurements N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='True Rate Measurements O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Estimated Rate Measurements NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Estimated Rate Measurements N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='(Ms-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Estimated Rate Measurements O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Time (t)8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Concentrations (M) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='r for NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='r for N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='r for O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='NO2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Time (s)arXiv Template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='A PREPRINT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='Figure 7: Response of the selected GP-proposed rate model using ADoK-W for the second computational experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='where CN2O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 M, CN2,0 = 1 M and CO2,0 = 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' with realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 7: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W selected model for the catalytic decomposition of nitrous oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='322 M −1 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 s−1 k2 / kB 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='811 M −1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='5 Toluene Hydrodealkylation — ADoK-S For the catalytic hydrodealkylation of toluene to benzene, ADoK-S selected the following concentration profiles for each species in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' fT,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='316t2 + t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='074 (19a) fH,1 = exp (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='510t) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='975 (19b) fB,1 = exp � −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='160 t + exp (t) � (19c) fM,1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='719 exp (t) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='407 (19d) fT,2 = exp (t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='195 exp (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='568 − t) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='781) (19e) fH,2 = exp � exp � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='132 − t 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='222 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='794 � (19f) fB,2 = t t 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='658 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='900 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='012 (20a) fM,2 = t − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='885 t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='971 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='985 (20b) fT,3 = exp (exp (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='099(2t − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='791))) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='042t (20c) fH,3 = exp (exp (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='218(t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='971)) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='267) (20d) fB,3 = t exp � t 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='001 � + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='361 (20e) fM,3 = −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='919 t + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='985 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='488 (20f) In the expressions above, T, H, B and M refer to toluene, hydrogen, benzene and methane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the sake of brevity, the concentration models proposed by the GP algorithm, optimized by the optimization routine and selected by the AIC are only shown for the third experiment in Figure 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The selected concentration profiles, qualitatively, fit the concentration data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As shown in Figure 1, once the concentration profiles are modeled, the parameters estimated, and the best model selected, the rate measurements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' : derivatives of the concentration profiles) are approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Below, a plot of the 18 8 Concentrations (M) 6 4 r for NO2 r for N2 r for O2 2 NO2 N2 02 0 0 2 4 6 8 10 Time (s)arXiv Template A PREPRINT numerical derivatives and (realistically inaccessible) rate measurements is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' From this plot, we can conclude that the rate measurements are fairly well estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, significant discrepancies can be appreciated at the start of the graph for the rate of production of benzene, showing that the initial curvature of the concentration profile for benzene is not well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' After estimating the rate measurements of the catalytic system, rate models are proposed using the GP algorithm and optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The best model is then selected by AIC, which for this case study, it is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = k1CT CH k2 + k3CB + k4CH (21) Recalling the correct model structure presented in Equation 12, the proposed kinetic rate model from ADoK-S is not identical to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Although the proposed model is very similar to the correct one, it differs only in one variable in the de- nominator (CH should be CT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This result is unsurprising, since the profile of hydrogen and toluene are nearly identical, simply differing by a constant scaling factor (as both species have the same stoichiometric coefficients but different initial concentrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, knowing the data-generating model a priori, as modelers, we are not satisfied and should then perform additional experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the purpose of this investigation, we limit the experimental budget to 4 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To demonstrate the potential of the case study, model-based design of experiments is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Specifically, the Hunter-Reiner criterion is used, where an experimental point is found which maximizes the difference between two models’ responses Hunter and Reiner [1965].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Alike the parameter estimation, the optimization routine outlined which harnesses the output of the ABC algorithm to warm-start the LBFGS algorithm is used to solve the Hunter-Reiner criterion, where the two models under question are Equation 12 (ground-truth model) and Equation 21 (ADoK-S proposed model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The optimized experimental point is (CT,0, CH,0, CB,0, CM,0) = (5, 3, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='276).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Realistically, this exercise could not be undertaken, since the underlying dynamic model would not be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Regardless, as mentioned before, this exercise is meant to investigate the potential of the methodological framework proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' One thing should be mentioned however: the initial experiments conducted are randomly selected, which is not an optimal methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In practice, statistic design of experiments should be initially used to carefully select experiments, so that the generated dynamic trajectories are sufficiently different, the process information gain can be maximized and the experimental expense can be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This would improve the effectiveness of the proposed methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As per the flowchart presented in Figure 1, after generating new experiments, the algorithm must be repeated for the appended computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The resulted new best rate model selected by AIC is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = k′ 1CT CH k′ 2CB + CT + k′ 3 = k1CT CH 1 + k2CB + k3CT (22) By undergoing simple algebraic manipulation (multiplying the numerator and denominator by a factor of 1 k′ 3 ), the proposed model (Equation 22) is identical to the data-generating kinetic rate model (Equation 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The response of the selected kinetic rate model is shown below in Figure 8, whilst the estimation of the corresponding kinetic parameters are presented in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 8: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-S selected model for the catalytic hydrodealkylation of toluene to benzene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='922 M −1 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 s−1 k2 / kB 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='726 M −1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 k3 / kC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='656 M −1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 From these results we can conclude that, although ADoK-S is not able to return the correct kinetic rate model from a small dataset composed of three random experiments, it is still able to return a model very similar to the ground-truth one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Notwithstanding, provided more data from a single extra experiment, ADoK-S is able to select an identical kinetic rate model whilst estimating the corresponding kinetic parameters reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 19 arXiv Template A PREPRINT Figure 8: The conditions for the third computational experiment are CT,0 = 5 M, CH,0 = 8 M, CB,0 = 2 M and CM,0 = 3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top left: Data from the third computational experiment and the selected concentration profiles selected by AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Top right: Numerical derivatives of the concentration profiles and the true rate measurements (which realistically are inaccessible) for the third computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Bottom: Response of the selected GP-proposed rate model using ADoK-S for the third computational experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='6 Toluene Hydrodealkylation — ADoK-W Similarly to the behavior of ADoK-S on the catalytic hydrodealkylation of toluene to benzene, the 1-step approach is unable to produce the true underlying kinetic model from the three initial computational experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The initially selected model is presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = k1(CH + k2)(CT − k3) CB + CT (23) Recalling the correct model structure presented in Equation 12, the proposed kinetic rate model from ADoK-W is quite different from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, like in the previous example, we are not satisfied with the model and should then perform additional experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For the purpose of this investigation, we limit the experimental budget to 4 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Model-based design of experiments is conducted yet again using the Hunter-Reiner criterion and the optimization routine to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The optimized experimental point is (CT,0, CH,0, CB,0, CM,0) = (5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='302, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='247).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Realistically, this exercise could not be undertaken, since the underlying dynamic model would not be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Regardless, as mentioned, this exercise is meant to investigate the potential of the methodological framework proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As per the flowchart presented in Figure 2, after generating new experiments, the algorithm must be repeated for the appended computational experiment, just like for ADoK-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The new best rate model selected by AIC is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As demonstrated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' with one additional computational experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' ADoK-W is also able to retrieve the dynamics of 20 8 Concentrations (M) 6 fT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 fH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 4 fb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 fm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='3 Toluene 2 Hydrogen Benzene Methane 0 0 6 8 10 2 4 Time (s)Estimated/True Rate Measurements Toluene 2 Estimated/True Rate Measurements Hydrogen Estimated/True Rate Measurements Benzene Estimated/True Rate Measurements Methane 1 (t-sW) 0 1 0 2 6 8 10 4 Time (s)8 Concentrations (M) 6 rfor T rfor H rfor B rfor M 2 H B M 0 6 8 10 0 2 4 Time (s)arXiv Template A PREPRINT the catalytic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This allows us to conclude that as the complexity of a system under investigation increases, the required amount of data provided to both approaches must increase accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In this context, ‘complexity’ refers not only to the number of species that can be directly observed, but also to the level of convolution of the underlying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For completeness, Figure 9 presents the behavior of the model selected by ADoK-W, along with Table 9 which presents the estimated values (and the true values) of the kinetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' r = k′ 1CT CH k′ 2CB + CT + k′ 3 = k1CT CH 1 + k2CB + k3CT (24) Figure 9: Response of the selected GP-proposed rate model using ADoK-W for the second computational experiment where CT,0 = 5 M, CH,0 = 8 M, CB,0 = 2 M and CM,0 = 3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Table 9: The data-generating kinetic parameters and the estimated values of the kinetic parameters of the ADoK-W selected model for the catalytic hydrodealkylation of toluene to benzene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetic Parameter Estimated Value True Value k1 / kA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='190 M −1 s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 s−1 k2 / kB 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 k3 / kC 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='340 M −1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='000 M −1 5 Conclusions Kinetic rate models are indispensable for the successful development of catalytic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Classical modeling paradigms offer strategies to construct these models, some of which are well established in industry, others which are becoming more prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Regardless, these classical paradigms demonstrate some noteworthy challenges: the construction of mechanistic models are expensive and time-consuming, whilst data-driven and hybrid models are mostly uninterpretable and usually lack the ability to extrapolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Automated knowledge discovery, a newer paradigm, aims to harness machine learning and computational advances to generate closed-form models and circumvent these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Nevertheless, popularized methodologies within automated knowledge discovery also demonstrate notable drawbacks: usually requiring limiting assumptions about an underlying model structure, a general lack of a robust and motivated model selection routine, and sensitivity to noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' These limitations motivated the presented work to explore and propose two different methodological frameworks for the automated discovery of kinetic rate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' The first one, ADoK-S, explores how to solve a strong formulation of the symbolic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' It does so by exploiting a genetic programming algorithm to automatically generate closed-form concentration profile models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' a sequential optimization routine to refine the most promising generated kinetic models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' and the Akaike information criterion to selected the best model (it is important to mention that this criterion has been selected due to a thorough analysis conducted on different model selection criteria which concluded 21 8 Concentrations (M) rfor T r for H rfor B rfor M 2 H B M 0 0 6 8 10 2 4 Time (s)arXiv Template A PREPRINT that AIC is the most robust one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Once the best concentration model is selected, rate measurements are estimated via numerical differentiation, and the mentioned routine of model generation, model refinement and model selection is repeated to discover kinetic rate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As referred, ADoK-S needs to estimate rate measurements through numerical differentiation, which can be problematic under high-noise environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To guard against this potential downside of ADoK-S, another methodological framework was proposed, ADoK-W, which explores how to solve a weak formulation of the symbolic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' By reformulation the problem to its weak form, the step to propose concentration models and estimate rate measurements is bypassed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Instead, kinetic rate models can be proposed from the start by working in the derivative hyperspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' In ADoK-W, alike ADoK-S, models are generated with a genetic programming algorithm, the most promising models are refined using a sequential optimization routine, and the best kinetic rate model is selected by AIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' To benchmark both approaches, three catalytic case studies (of increasing complexity) were carefully selected: a simple isomerization reaction, the decomposition of nitrous oxide, and the hydrodealkylation of toluene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Both approaches successfully retrieved the underlying dynamics of the isomerization reaction and of the decomposition of nitrous oxide with three pseudo-randomized computational experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' However, neither approach was able to retrieve the ground-truth kinetic model for the hydrodealkylation of toluene case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Nonetheless, to prove the potential of the methodological frameworks, an extra experiment was appended to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' This experiment was determined by solving the Hunter-Reiner criterion between the data-generating model and the methodology-selected model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' By adding one extra experiment, both approaches successfully recovered the kinetics, concluding that as the complexity of the system increases, so must the data (information) provided to ADoK-S and ADoK-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As previously referred, the methodologies could be more efficient by implementing statistic design of experiments in the first step of the frameworks, so that the experiments generated are optimal rather than random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' As such, it is fair to conclude that the proposed methodologies build-on from previously presented approaches to automated knowledge discovery by tackling identified shortcomings directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' From this investigations, new angles of research arose which will be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' For instance: the frameworks should be stress-tested with respect to noise and quantity of data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' the introduction of physical and mathematical constraints should be explored and assessed with respect to the efficacy of ADoK-S and ADoK-W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' investigate possible ways to approximate optimal experiments without having direct access to the underlying dynamics model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' lastly, research how the methodological frameworks can be harnessed to uncover how temperature (and other variables, such as the morphology of a catalyst) affect the rate of a reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' But for now, with the provided study, we believe that these methodologies can be successfully applied to aid chemical reaction engineers in solving current and future problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' References R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='coche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='100702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Petsagkourakis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Sandoval, E.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Schbib, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' García, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Gígola, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Errazu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Kinetics of Front-End Acetylene Hydrogenation in Ethylene Production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Res.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Biotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' Bioeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=', 117(11):3356–3367, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='1002/bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content='27512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FIT4oBgHgl3EQfxSv4/content/2301.11356v1.pdf'} diff --git a/qtE3T4oBgHgl3EQfMAny/content/tmp_files/2301.04370v1.pdf.txt b/qtE3T4oBgHgl3EQfMAny/content/tmp_files/2301.04370v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..26191cd4465b8d570b5d615aaa60c464bf882dde --- /dev/null +++ b/qtE3T4oBgHgl3EQfMAny/content/tmp_files/2301.04370v1.pdf.txt @@ -0,0 +1,1655 @@ +Order-Preserving Database Encryption with Secret Sharing +Dongfang Zhao +University of Nevada, Reno +United States +ABSTRACT +The order-preserving encryption (OPE) problem was initially for- +mulated by the database community in 2004 soon after the para- +digm database-as-a-service (DaaS) was coined in 2002. Over the +past two decades, OPE has drawn tremendous research interest +from communities of databases, cryptography, and security; we +have witnessed significant advances in OPE schemes both theoret- +ically and systematically. All existing OPE schemes assume that +the outsourced database is modeled as a single semi-honest adver- +sary who should learn nothing more than the order information +of plaintext messages up to a negligible probability. This paper +addresses the OPE problem from a new perspective: instead of mod- +eling the outsourced database as a single semi-honest adversary, +we assume the outsourced database service compromises a cluster +of non-colluding servers, which is a practical assumption as all +major cloud vendors support multiple database instances deployed +to exclusive sub-networks or even to distinct data centers. This as- +sumption allows us to design a new stateless OPE protocol, namely +order-preserving database encryption with secret sharing (ODES), +by employing secret-sharing schemes among those presumably +non-colluding servers. We will demonstrate that ODES guarantees +the latest security level, namely IND-FAOCPA, and outperforms +the state-of-the-art scheme by orders of magnitude. +ACM Reference Format: +Dongfang Zhao. 2023. Order-Preserving Database Encryption with Secret +Sharing. In Proceedings of ACM Conference (Conference’17). ACM, New York, +NY, USA, 13 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +1.1 +Inception of Order-Preserving Encryption +For two decades, we have witnessed the inception and prosperity of +database as a service (DaaS) since the publication of the seminal pa- +per [14] in ICDE’02. As of the writing of this paper, all major cloud +computing vendors (e.g., Amazon Web Services, Google Cloud Plat- +form, Microsoft Azure) support DaaS with pay-as-you-go business +models, which enables users to avoid the upfront cost of managing +their in-house databases. From the user’s perspective, the DaaS +can be thought of as an outsourced database maintained by cloud +computing vendors. As with any outsourced service, the security +of outsourced databases has been one of the top concerns for users: +the threat in an outsourced database comes not only from outside +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +attackers but also inside adversaries, e.g., developers and adminis- +trators of the cloud computing vendor. Among others, one avenue of +research to address the above security issue is to encrypt the user’s +sensitive data before uploading them to the outsourced database. +One key challenge of DaaS lies in the management of those en- +crypted data, such as building indexes, because the index must be +associated with the plaintext to speed up the query and modifica- +tion requests from the user and yet all the server can learn about +is the ciphertext. To that end, in SIGMOD’04, Agrawal et al. [1] +proposed to encode the plaintext in the outsourced database while +retaining the numerical order of the plaintext. The paper demon- +strated that it was possible to achieve the best of both worlds: the +so-called order-preservation encryption (OPE) can ensure both the +confidentiality and the ordinal of the outsourced data. The work +quickly drew a lot of research interest from the database and the +security/cryptography communities. +1.2 +Brief Timeline of OPE Security +In EuroCrypt’09, Boldyreva et al. [8] presented the first security +definition of OPE. Following the convention of cryptography, the +definition is based on the canonical structure of an encryption +scheme: (i) the security goal is computational indistinguishability, +(ii) the threat model is to allow the adversary to obtain a polyno- +mial number of ciphertexts of arbitrary plaintexts, i.e., the so-called +chosen-plaintext attack (CPA), and (iii) a simulation-based reduc- +tion to prove that distinguishability is negligible. +Unfortunately, it was shown that it is impossible to achieve +indistinguishability under the standard CPA attack because the CPA +definition is overly strong and can be violated if the adversary can +learn about the ordinal of the plaintexts. The good news was that +a new security notion, namely indistinguishability under ordered +chosen-plaintext attack (IND-OCPA), was proposed by [8]. IND- +OCPA is as strong as IND-CPA except for allowing the adversary +to only learn about the ordinal of the plaintext. +Not long after IND-OCPA was proposed, Popa et al. [26] in +Oakland’13 pointed out that IND-OCPA is insufficient for the well- +known frequency attack. The frequency attack is due to the deter- +ministic ciphertexts, which are known to be insecure under the +conventional CPA attack as well. +A stronger security notion was then proposed in CCS’15, namely +indistinguishability under frequency-analyzing and ordered chosen- +plaintext attack (IND-FAOCPA) [19]. Multiple subsequent schemes +claimed to meet IND-FAOCPA, such as [27]. As of the writing of +this paper, IND-FAOCPA remains the strongest security notion for +order-preserving encryption schemes. +1.3 +System Research of OPE Schemes +While the cryptography and security communities spent tremen- +dous effort in properly defining and proving security from a theoret- +ical perspective, the database and system communities are equally +1 +arXiv:2301.04370v1 [cs.CR] 11 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +interested in other metrics such as performance and costs. Many +leading cloud vendors are now supporting encrypted database ser- +vices, such as Microsoft Azure [2] as reported in SIGMOD’20. +An evaluation paper [7] in VLDB’19 summarized the pros and +cons of major OPE schemes as of 2019. The metrics include encryp- +tion complexity, comparison complexity, ciphertext size, I/O cost, +and communication cost. There was no clear winner based on the +reported numbers in the paper. +In VLDB’21, Li et al. [22] presented a new frequency-hiding OPE +scheme with a 128-bit AES (OPEA). OPEA outperforms existing +OPE schemes in almost all aspects: +• OPEA is IND-FAOCPA secure; +• OPEA incurs a constant number of interactions between +clients and servers; +• OPEA completes both insertion and query requests signifi- +cantly faster than the counterparts; +• OPEA incurs O(𝑁) client storage space, where 𝑁 denotes +the number of distinct plaintexts. +The only limitation of OPEA lies in the client storage: as a stateful +scheme, OPEA requires the client to maintain a local table to keep +track of the plaintext orders. Although duplicate plaintexts only +need to be stored one time, counterparts (e.g., POPE [27]) could +take constant client storage. (However, POPE suffers the problem +of possible incomparable elements) +1.4 +Motivation and Challenges of This Work +The goal of this work is to eliminate the shortcoming of OPEA [22] +while retaining its advantages compared with existing solutions. +This implies that we want to achieve both strong security levels and +high performance. By high performance, we mean lower processing +time, which is contributed by computational time, communication +time, and I/O time. Clearly, the O(𝑁) client storage of OPEA is a +performance bottleneck. +On the other hand, achieving both strong security guarantees +and high performance is very challenging with currently available +cryptographic primitives and our conventional wisdom of system +optimization. As discussed above, client storage is necessitated by +the stateful coordination between the client and the server, because +a stateful mechanism is believed to achieve higher efficiency for +maintaining the tuple orders. As of the writing of this paper, we +are only aware of one work [30] taking a stateless approach; but no +experimental results were reported. In addition, 128-AES is widely +believed to be one of the most efficient and secure symmetric en- +cryption schemes nowadays; therefore, it is unlikely to improve +the performance of OPEA by upgrading or optimizing the crypto- +graphic subsystems. Something more drastic is in need should we +aim to further improve the performance without trading off the +security level. +1.5 +Proposed Solution +This work proposes a new OPE scheme by employing secret-sharing +primitives. While secret-sharing can be thought of as an encryption +scheme (in a broad sense) its internal machinery works quite differ- +ently than single-node encryption. Instead of placing the encrypted +test on a single machine, we now assume a cluster of machines that +would not collude. The ciphertext is now distributed into multiple +User +Client +Server +Conventional Order-preserving Encryption +Client +Cluster of Servers +Proposed Order Preservation w/ Multiparty Secrets +Log on +Figure 1: Proposed multiparty secrets vs. conventional +single-node encryption. +database servers as secret shares and cannot be decrypted without +the authorization of the data owner. Therefore, confidentiality is +achieved. +To achieve the ordinal of plaintexts, the secret-sharing primitives +should allow us to compare the corresponding plaintext values by +asking the server not to share its local secret shares. If we can +achieve this comparison merely through some local computation +of the secret shares, the resulting scheme would be stateless and +save us some I/O costs. +As we will demonstrate in the latter sections, a specific type of +secret-sharing scheme does allow us to achieve both confidentiality +and ordinal of encoded data. To make matters more concrete, Fig- +ure 1 illustrate the high-level difference between the conventional +OPE schemes and the proposed scheme, which we coin as ordered +database encryption with secret-sharing (ODES). +1.6 +Contributions +In summary, this work makes the following technical contributions: +• We propose the very first stateless order-preserving encryp- +tion scheme for outsourced databases with secret sharing, +namely ordered database encryption with secret-sharing +(ODES); +• We demonstrate that ODES guarantees a strong security +level, i.e., IND-FAOCPA; +• We design various database protocols for leveraging the +proposed ODES scheme; +• We implement the ODES scheme and database protocols on +top of SQLite databases; and +• We conduct a thorough evaluation of ODES by comparing +it with state-of-the-art schemes on top of the TPC-H bench- +mark and three real-world applications on a 10-node cluster. +2 +BACKGROUND AND RELATED WORK +2.1 +Order-Preserving Encryption +The concept of order-preserving encryption (OPE) was originally +proposed in the database community [1]. The motivation is evident: +how could we achieve both the confidentiality and the ordinals of +sensitive data in an outsourced database? The confidentiality part +is obvious and the ordinal requirement is also well justified: it is +very common for database systems to build indexes to speed up +the query and insertion requests and being able to sort or order the +outsourced data sets is essential to achieve this goal. +2 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +The early-state solution to achieve the dual goals is somewhat +straightforward: the plaintexts are encoded with the help of some +statistical distribution such that the encoded values remain in the +same order as the plaintext. There are a few issues with this ap- +proach. First, the encoded values are deterministic. This means the +encoding cannot be secure if the adversary can somehow obtain +the encoding of some chosen plaintexts, i.e., the so-called chosen- +plaintext attack (CPA). It can be argued that in outsourced databases +we do not need security as strong as CPA, but from a security point +of view, a practical database system should always provide CPA +security as the minimum [2]. +We then run into a dilemma between CPA security and ordi- +nal encoding: CPA security implies the randomness of ciphertexts +which cannot retain the order of plaintexts. The solution is to intro- +duce some function for the database to order the encrypted tuples +without relying on the raw values of ciphertexts, which is called +order-revealing encryption (ORE) [9]. Accordingly, a new security +notion was proposed to allow the adversary to learn about the +orders of plaintexts, resulting in the so-called indistinguishabil- +ity under ordered chosen-plaintext attack (IND-OCPA). Obviously, +there are many options to calculate the order values but they can +be categorized into two categories: (i) a stateful scheme where the +client and the server coordinate to maintain the order information +of encrypted records in the database [26], and (ii) a stateless scheme +where the order information can be retrieved on the fly [30]. Most +OPE works in the literature focus on the stateful approach; as we +will see in the latter sections, the proposed ODES is a stateless +scheme. +It turned out that there were new issues for ORE and IND- +OCPA: Many IND-OCPA schemes [8, 26] are vulnerable to attacks +that leverage the access patterns of the queries. To that end, a +newer notion is defined, the indistinguishability under frequency- +analyzing ordered chosen-plaintext attack (IND-FAOCPA). Multiple +IND-FAOCPA schemes have been proposed in the literature, such +as [19, 22, 27]. A relatively recent evaluation paper [7] reports the +performance of some of the most popular OPE schemes, including +what have not been mentioned in this paper: [10, 11, 21]. As of the +writing of this paper, the OPE scheme proposed in [22] achieves +the best performance in almost all the metrics (e.g., client storage, +query rounds), and we will primarily compare the proposed ODES +with the protocol proposed in [22]. +2.2 +Secret Sharing +The idea of a secret sharing scheme (SSS) is straightforward: a +given plaintext 𝑝𝑡 is converted into a set of encoded bytes 𝑐𝑡’s such +that only a specific subset of 𝑐𝑡’s can reconstruct the original 𝑝𝑡. +The goal of SSS is to reduce the risk of disclosing the plaintext; +instead of compromising the holder of the plaintext, the malicious +adversary needs to compromise multiple entities before any of the +shareholders detect the attack. Even for weaker attacks where only +semi-honest adversaries are assumed, dispersing the secret shares +to more parties raises the bar of a successful eavesdropping attack. +The SSS can be tuned by the subset size. Formally, a 𝑡-out-of-𝑛 +threshold SSS (TSSS) is defined as follows. +Definition 1 (TSSS). A TSSS is comprised of two algorithms: +• Share: a randomized algorithm that takes as input a plaintext +𝑝𝑡 and returns a sequence 𝑆 = (𝑠1, . . . ,𝑠𝑛) of shares. +• Reconstruct: a deterministic algorithm that takes a set of at +least 𝑡 shares and returns the plaintext. +The number 𝑡 is called the threshold of the TSSS. Let 𝑈 of size 𝑡 be a +subset of 𝑛 shares, |𝑈 | ≥ 𝑡 and 𝑈 ⊆ {𝑠1, . . . ,𝑠𝑛}, we require that a +TSSS holds the following property: +𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡(𝑈 ) = 𝑝𝑡. +As we will see in the next section §2.3, the definition of TSSS +leads to a slightly different security definition compared with the +conventional encryption schemes. +The canonical example of TSSS is due to Shamir [29], in which +the secrets are revealed through a (𝑡 − 1)-degree polynomial. In +essence, each share can reconstruct the coefficient of a specific de- +gree of unknowns through the LaGrange polynomials. In addition +to Shamir’s construction, other schemes exist. Ito et al. [17] pro- +posed the replicated secret-sharing scheme, which is based on finite +fields where each share is a vector. One nice property of replicated +secret-sharing is its linearity: the addition and subtraction of local +shares equal the addition and subtraction of the plaintext. A simpler +variant of replicated secret-sharing is additive secret sharing, where +each share is a scalar value and the threshold 𝑡 is set to 𝑛. Indeed, +this is a building block of our proposed ODES protocol; we will +see how the linearity of the additive secret-sharing allows us to +preserve the orders of plaintexts in §3.3. +SSS has a tight connection with secure multiparty computation +(MPC) [3], which has a long history [32]. The goal of MPC is more +ambitious than SSS: in addition to keeping the plaintext confidential, +we want to calculate an arbitrary function of the original plaintexts +by touching on only the encoded data on multiple parties. The +original problem was solved by the so-called garbled circuits [32], +whose idea was pretty simple: we can ask each party to encode the +input with its private key, shuffle the encrypted ciphertexts, and +then enumerate all the keys to decrypt the result. Since we assume +the encryption scheme is secure, the only way that the result can be +revealed is that the correct combination of private keys is applied +to one of the garbled outputs. This is indeed a feasible solution, at +least theoretically; in practice, the circuits may grow exponentially +and result in efficiency issues. There are many more efficient MPC +solutions, such as [4, 20, 24, 33]. +2.3 +Provable Security +When employing an encryption scheme in an application, it is +highly desirable to demonstrate its security provably. Formally, we +need to identify the following three important pieces for the prov- +able security of a given encryption scheme: security goal, threat +model, and assumptions. The security goal spells out the desired +effect when the application is under attack; the threat model ar- +ticulates what an adversary can do with the attack, such as what +information of the plaintext/ciphertext can be collected and the +resource/time limitation of the attack; the assumption lists the +presumed specifics of the subsystems or components of the crypto- +graphic scheme, which is usually an important building block for +security proof, e.g., reduction. The security goal and threat model +are usually called security definition collectively. +3 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +One well-accepted security definition with a good balance be- +tween efficiency and security is that the adversary can launch a +chosen-plaintext attack (CPA), defined as follows. +Definition 2 (Chosen-Plaintext Attack). Given a security +parameter 𝑛, i.e., the bitstring length of the key, an adversary can +obtain up to 𝑝𝑜𝑙𝑦(𝑛) of plaintext-ciphertext pairs (𝑚,𝑐), where 𝑚 +is arbitrarily chosen by the adversary and 𝑝𝑜𝑙𝑦(·) is a polynomial +function on 𝑛. With such information, the adversary tries to decrypt a +𝑐′ that is not included in the polynomial number of known ciphertexts. +The polynomial requirement is only for practical reasons, as we +usually assume that the adversary should only be able to run a +polynomial algorithm without unlimited resources. Accordingly, +we want to design encryption schemes that are CPA secure: even if +the adversary A can obtain those extra pieces of information, A +should not be able to decode the ciphertext better than a random +guess up to a very small probability. To quantify the degree of this +small probability, negligible function is defined as below. +Definition 3. A function 𝜇(·) is called negligible if for all poly- +nomials 𝑝𝑜𝑙𝑦(𝑛) the inequality 𝜇(𝑛) < +1 +𝑝𝑜𝑙𝑦(𝑛) holds for sufficiently +large 𝑛’s. +For completeness, we list the following lemmas for negligible +functions that will be used in later sections. We state them without +the proofs, which can be found in introductory cryptography or +complexity theory texts. +Lemma 1 (Summation of two negligible functions is a neg- +ligible function). Let 𝜇1(𝑛) and 𝜇2(𝑛) be both negligible func- +tions. Then 𝜇(𝑛) is a negligible function that is defined as 𝜇(𝑛) +def= +𝜇1(𝑛) + 𝜇2(𝑛). +Lemma 2 (Quotient of a polynomial function over an ex- +ponential function is a negligible function). 𝑝𝑙𝑜𝑦(𝑛) +2𝑛 +is a +negligible function. That is, ∃𝑁 ∈ N, ∀𝑛 ≥ 𝑁 : +𝑝𝑙𝑜𝑦(𝑛) +2𝑛 +< +1 +𝑝𝑜𝑙𝑦(𝑛) , +where N denotes natural numbers. +The canonical method to prove the security of a proposed encryp- +tion scheme, such as IND-CPA, is through reduction [23]. Usually, +breaking the scheme is reduced to a hard mathematical problem, +which means that if an attack is possible for the scheme then the +mathematical problem would be efficiently solved. That is, the en- +cryption scheme is not easier than the mathematical problem. The +scheme is modeled as a subroutine, whose inputs are simulated such +that the adversary cannot tell whether it is being involved in an +attack or in a subroutine to help solve the math problem. Although +forward proof is possible, the more commonly used technique is +the contradiction: by assuming that the adversary could distinguish +some designed experiments with a non-negligible advantage, the re- +duction would lead to a non-negligible probably to efficiently solve +the hard mathematical problem that is believed to be intractable, +thus leading to a contradiction. +Unfortunately, in the context of order-preserving encryption, it +has been proven that the conventional IND-CPA is impossible [8]. +Therefore, the cryptography community proposed a relaxed nota- +tion called indistinguishability under ordered chosen-plaintext attack +(IND-OCPA) [8]. However, it was shown that [5] effective attacks +can be launched on IND-OCPA security caused by the access pat- +terns. The root cause of this issue lies in the deterministic cipher- +text in early-stage order-preserving encryption schemes. Indeed, +it is well known that a deterministic encryption scheme can be +impossibly secure against CPA attacks. As a result, modern order- +preserving encryption schemes are all randomized, which implies +that the ciphertexts are not directly comparable and necessities +indirect comparison between ciphertexts. Such indirection com- +parison is usually coined as order-revealing encryption (ORE) [9] +that generalizes the original notion of OPE. In a more general +sense, some so-called frequency-hiding order-preserving encryp- +tion schemes [19, 27] were proposed. Accordingly, a new security +notion was proposed, namely indistinguishability under frequency- +analyzing ordered chosen-plaintext attack (IND-FAOCPA) [19]. +IND-FAOCPA is the latest security definition in this area and our +proposed ODES scheme is IND-FAOCPA secure. +The above review of provable security assumes that the cipher- +text is a single entity and does not consider the scenario where the +ciphertext is a set, which is the case for secret sharing. The provable +security of secret sharing takes a slightly different approach because +of the additional assumption that not all shares will be accessible +to the adversary per the definition. While it is true that if we can +prove the entire set of shares is secure then any subset of shares +is also secure, a more common approach to proving the semantic +security of secret shares is through interchangeable libraries [28]. +The key idea is to model the scheme as a library with the input of +either an 𝐿 or 𝑅 plaintext input, and then the proof will show that +the library with 𝐿 input eventually looks identical to the library +with the 𝑅 input through a series of interchangeable operations. +We will see how this technique is used in §3.5 +3 +ORDER-PRESERVING DATABASE +ENCRYPTION WITH SECRET SHARING +3.1 +Overview +The intuitive idea behind the proposed secret-sharing-based order +preservation is straightforward: we leverage the multiplicity of a +cluster of database servers such that no plaintext is leaked while +maintaining the comparative order among the plaintexts. That is, +we somehow break the original plaintext into multiple shares, each +of which is allocated to a distinct server in a database cluster. The +nodes in the cluster are assumed to be non-colluding, which can be +implemented by deploying the servers into different sub-networks +or different data centers. The order of the plaintext can be retrieved +and updated by an aggregation of local functions on each server. +To make matters more concrete, Figure 2 illustrate the idea in +an oversimplified scenario where two servers are available to store +encoded and ordered data. Assume that the data owner wants to +save the balance table into a remote database service. The table is +as simple as a key-value store with the year-month as the key and +the U.S. dollar amount as the value for his business. +Each of the two database servers stores some seemingly random +numbers that will help keep the real amounts confidential. We must +be able to reconstruct the original plaintext from the shares stored +by the servers because otherwise, the user would lose the data. In +this example, the original balance amount can be reconstructed +by simply adding up the shares from distinct database servers. It +4 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +Balance Plaintext +YY-MON +Amount +22-JAN +$15,000 +22-FEB +$12,000 +22-MAR +$13,000 +22-APR +$14,000 +Balance Share0 +YY-MON +Amount +22-JAN +$6,000 +22-FEB +$14,000 +22-MAR +$11,000 +22-APR +$-6,000 +Balance Share1 +YY-MON +Amount +22-JAN +$9,000 +22-FEB +$-2,000 +22-MAR +$2,000 +22-APR +$20,000 +No Collusion +Balance Index +YY-MON +Order +22-JAN +4 +22-FEB +1 +22-MAR +2 +22-APR +3 +Cloud Services +User +Figure 2: Example of two shares with order preservation. +should be noted that the reconstruction should only happen on the +client side as the outsourced databases are not fully trusted. That +is, the database servers are not supposed to share their local data. +We will formally define what we mean by “fully trusted” later; but +for now, we keep our discussion at a non-technical level. +Another important piece of information is the index metadata +for keeping track of the order information of the plaintext. The +index table will be useful when a new record is inserted into the +databases: it allows us to do a binary search to locate the correct +order of the newly inserted record in the database. In addition, it +would facilitate the effectiveness of order-related queries. +Here is another example for illustrating the plaintext comparison +through two sets of secret shares. Let’s say the user recently ob- +tained a new record for (22-MAY, $10,000) and would like to know +whether the balance is higher than the previous month, 22-APR. +Note that ODES is a stateless protocol, so the client cannot simply +compare $10,000 to (22-APR, $14,000) since the latter does not exist +after being secretly shared with the two database servers. What +the client would do is split $11,000 into two random numbers, say +$3,000 and $8,000, which are combined with the key and sent to +the two database servers. In our example of Fig. 2, the plus server +receives (22-MAY, $3,000) record and the minus server receives (22- +MAY, $8,000) record. Both servers then carry out local computations +of the 22-APR and 22-MAY records; for example, on the plus server, +it calculates the following delta (𝛿0, 3000-(-6000) = 9000). Similarly, +the minus server calculates (𝛿1, 8000-20000 = -12000). Both servers +then broadcast their local 𝛿𝑗 to all other servers. Note that this +sharing would not reveal any information other than the orders +and is therefore allowed (as opposed to the secret share itself). Now, +each database server has obtained all the 𝛿𝑗’s and then applies +an aggregation over them. For example, the plus server calculates +9000+(-12000) = -3000 < 0, which means the balance of 22-MAY is +lower than 22-APR. The server can also update the index metadata +accordingly by conducting a binary search on the index file; in this +Tuples +SQLite 0 +SQLite i +SQLite m +... +... +Ordered +Tuples +Provider +Customer +Share Index +Tuple 1 +... +Tuple n +Outsourced Database Services +Figure 3: The proposed architecture of preserving tuple or- +ders with secret shares in outsourced SQLite databases. +example, (22-MAY, 1) will be inserted into the index table and some +existing orders will be incremented by one. +3.2 +Architecture +As shown in Figure 3, we envision a cluster of 𝑚 outsourced data- +base servers that (i) do not share their local data and (ii) can access +an index for the orders of the secret shares. Since our system proto- +type is implemented with SQLite, we assume there are 𝑚 SQLite +instances. As long as at least one of the 𝑚 SQLite instances does not +collude with others, the outsourced data is secure, which overcomes +the so-called dishonest majority problem in the literature. This im- +plies that more databases imply a higher security level; however, +this is at the cost of more computational and I/O overhead in the +system. +The architecture also differentiates two different user roles: a +data provider (on the left) and a data customer (on the right). The +key difference is that the provider may modify the secret shares +in the database cluster and possibly update the index metadata as +well, while the customer only makes read-only queries. We will see +that both modification and read-only queries will be facilitated by +the share index in the proposed protocols. +3.3 +Primitives +We are now ready to formally present the primitives of ODES to +allow order preservation. These primitives will be used as building +blocks in various protocols later (§3.4). We assume that the plaintext +is a string of 𝑙 bits and there are overall 𝑚 database servers. +Share. We start with the Share function that splits a given plain- +text into 𝑚 secret shares. +5 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +Definition 4 (Share). A Share() function is defined as +𝑆ℎ𝑎𝑟𝑒 : {0, 1}𝑙 → +� +{0, 1}𝑙 �𝑚 +, +𝑝𝑡 ↦→ {𝑠𝑖}, 0 ≤ 𝑖 < 𝑚, +where 𝑠𝑖 is a random number when 𝑖 ≠ 0 and 𝑠0 is calculated as +𝑠0 ≔ +𝑚−1 +∑︁ +𝑖=1 +𝑠𝑖. +Reconstruct. The Reconstruct function is the reverse of Share. +Definition 5 (Reconstruct). A Reconstruct() function is defined +as +𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡 : +� +{0, 1}𝑙 �𝑚 +→ {0, 1}𝑙, +{𝑠0, . . . ,𝑠𝑚−1} ↦→ +𝑚−1 +∑︁ +𝑖=0 +𝑠𝑖. +Compare. A cluster of 𝑚 servers aims to compare two plaintexts +by local computations of two sets of secret shares. As the name +suggests, a server is not allowed to disclose its local share. Let +𝑆𝐿 def += +� +𝑠𝐿 +0, . . . ,𝑠𝐿 +𝑚−1 +� +denote the secret shares of the first plaintext +𝐿 and 𝑆𝑅 def += +� +𝑠𝑅 +0 , . . . ,𝑠𝑅 +𝑚−1 +� +denote the secret shares of the second +plaintext 𝑅. +Definition 6 (Compare). A Compare is a function: +𝐶𝑜𝑚𝑝𝑎𝑟𝑒 : +� +{0, 1}𝑙 �𝑚 +× +� +{0, 1}𝑙 �𝑚 +→ {−1, 0, 1}, +(𝑆𝐿,𝑆𝑅) ↦→ + + +−1, +𝐿 < 𝑅 +0, +𝐿 = 𝑅 +1, +𝐿 > 𝑅 +, +where Compare is assigned the same arithmetic sign as the following +summation +𝑚−1 +∑︁ +𝑖=0 +� +𝑠𝐿 +𝑖 − 𝑠𝑅 +𝑖 +� +. +Note that the terms in the above equation are calculated by each +database server independently. The servers broadcast their local +differences to the entire cluster such that each server can decide +the ordinal information of the plaintexts. +3.4 +Protocols +We present three ODES protocols in this section. Various primitives +will be used in the protocols. To make the protocols self-contained, +we include some high-level implementation details of the primitives; +when we do so, we make comments to the pseudocode to remind +the readers that certain primitives are being called (e.g., Line 3 in +Alg. 1). We follow the cryptographic convention to use ≔ to denote +a deterministic assignment and ← to denote a uniformly sampled +value from a randomized algorithm or distribution. +3.4.1 +Initialization. When the system is deployed for the first time, +we assume that the data provider has an initial list of data sets +that will be encoded and uploaded to the remote database service. +Several tasks must be completed before the proposed ODES system +goes into operation, including: +• An initial index file is built by the data provider based on +the local plaintexts; +• Each plaintext field is decomposed into a list of shares; +• The shares of the same plaintext should be randomized and +distributed to distinct database servers. +All of those tasks are completed in the initialization phase, as +illustrated in Alg. 1. For the 𝑖-th record 𝑟, the client (i.e., the provider +in Fig. 3) splits it into 𝑚 pieces (Lines 3–6): the last 𝑚 − 1 pieces +𝑠𝑖 [1 : 𝑗] are simply random values and the first piece is calculated as +the difference between 𝑟 and �𝑠𝑖 [1 : 𝑗]. To add more randomness +to the encoded ciphertext, we apply permutation to the 𝑚 shares +as well (Line 7). In Line 9, the client sends each of the 𝑚 shares to a +distinct database server; in Line 10, the server receives the share +and inserts it into its local share. After processing the record 𝑟, the +index metadata is updated in Line 12. After completing all the 𝑛 +records, the client broadcasts the index table to all the database +servers in Lines 14–16. +Algorithm 1: ODES Init +Input: A relation 𝑅 of cardinality 𝑛, i.e., there are |𝑅| = 𝑛 +plaintext tuples; a set of 𝑚 non-colluding nodes 𝑁; +function 𝑅𝑛𝑑() returning a random number; +Output: Each node 𝑁𝑗 holds a list of ciphertext shares 𝑅𝑗, +|𝑅𝑗 | = 𝑛; the index 𝑖𝑑𝑥 holding the orders of 𝑅; +1 for i = 0; i < n; i++ do +2 +r = R[i] +3 +for j = 1; j < m; j++ do +// Share() +4 +𝑠𝑖 [𝑗] ← 𝑅𝑛𝑑() +5 +𝑠𝑖 [0] � 𝑟 − 𝑠𝑖 [𝑗] +6 +end +7 +Permute elements in 𝑠𝑖 +8 +for j = 0; j < m; j++ do +9 +Send 𝑠𝑖 [𝑗] to 𝑁𝑗 +10 +𝑅𝑗 [𝑖] � 𝑠𝑖 [𝑗] +// On server 𝑁𝑗 +11 +end +12 +Update 𝑖𝑑𝑥 +13 end +14 for j = 0; j < m; j++ do +15 +Send 𝑖𝑑𝑥 to 𝑁𝑗 +16 end +The complexity of Alg. 1 is as follows. Lines 3–6 take O(𝑚) +steps, Line 7 takes O(𝑚!) steps, and Lines 8–11 take O(𝑚) steps. +Therefore, Lines 1–13 take O(𝑛𝑚!). Lines 14–16 trivially take O(𝑚) +steps and can be ignored. The total complexity of Alg. 1 is therefore +O(𝑛𝑚!). Although the O(𝑚) factor in the asymptotic complexity +seems costly, it is usually not an issue in practice because𝑚 is taken +at a relatively small value, e.g., 2, 4, and 8. +3.4.2 +Insertion. The insertion protocol assumes that the cluster of +database servers already holds secret shares and will take in a new +record from the client. In this context, the protocol comprises two +phases: (i) the client prepares the secret shares of the new record +6 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +and sends them to the cluster of servers, and (ii) the servers update +their local shares and the index metadata for ordering information. +We depict both phases in Alg. 2. +Client protocol. Similarly to the initialization protocol, the client +splits the given record 𝑟 into 𝑚 shares in Lines 1–4. The client +then permutes the shares and then sends each of them to a distinct +database server (Lines 5–8). +Server protocol. A server always inserts the received share into +its local table, as shown in Line 9. Lines 10–25 work on updating +the index through a binary search for the correct order of the new +record. Line 13 computes the difference between the received share +and the share whose corresponding plaintext is the median. Line +14 allows all database servers to learn about the differences so that +they can decide whether to move to the smaller or the larger half +of the sorted shares (Lines 16–19). If there exists a duplicate value +in the database, then the protocol will end up at Line 21 (𝛿 = 0); +otherwise, Line 24 sets the order of the new record. +Complexity analysis. The complexity of Alg. 2 is as follows. Lines +1–4 take O(𝑚) steps, Line 5 takes O(𝑚!) steps, and Lines 6–8 take +O(𝑚) steps. Therefore, the client takes overall O(𝑚!) steps. Lines +11–12 and 16–21 imply O(log𝑛) iterations. Line 14 takes O(𝑚) +steps. Lines 21 and 24 take constant steps. Therefore, the server +takes overall O(𝑚 log𝑛) steps. +3.4.3 +Query. We represent the order-related predicate as𝑂𝑃, which +is part of the query. There are three phases in the query protocol: +• In the first phase of the query protocol, the client (i.e., the +provider in Fig. 3) simply broadcasts the predicate to all +database servers. +• The second phase happens on each database server, which +works on the local shares that qualify for the 𝑂𝑃 predicate +and send them back to the client. +• In the third phase, the client (i.e., the customer in Fig. 3) col- +lects all the qualified shares and reconstructs the plaintexts. +These three phases are summarized in Alg. 3. Lines 5–8 check every +indexed order to see whether the local share qualifies for 𝑂𝑃 and +if so, the server 𝑁𝑗 replies its local share 𝑅𝑗 [𝑘] to the client. Lines +13–19 reconstruct the list of plaintexts by aggregating the shares +into each of the tuples. +The complexity of Alg. 3 is as follows. For the server, the overall +complexity is O(𝑛) (Lines 5–10). For the client, Lines 1–3 take O(𝑚) +steps. Line 11 itself takes O(𝑛𝑚) steps because 𝐾 ≤ 𝑛. Lines 13–19 +take O(𝑛𝑚) also. Therefore, the overall complexity of the client is +O(𝑛𝑚). +3.4.4 +Deletion and Modification. The deletion of ODES is more +straightforward than insertion. The first phase is asking every data- +base server to remove the corresponding local shares. The second +phase is to remove the order information of the to-be-removed +record in the index file and update the index file by decrement- +ing (by one) the order values of those records whose orders are +larger than the order of the deleted record. We omit the detailed +description of this protocol in this conference paper. +The modification of ODES can be trivially implemented by first +deleting the record and then inserting the new value. We skip the +formal protocol description of the modification in this paper. +Algorithm 2: ODES Insert +Input: A new record 𝑟 of record identifier 𝑟𝑖𝑑 to be +inserted into a relation 𝑅, |𝑅| = 𝑛; 𝑅 is not directly +accessible and can only be reconstructed from 𝑅𝑗, +each of which is stored at node 𝑁𝑗, 1 ≤ 𝑗 ≤ 𝑚; a +global index 𝑖𝑑𝑥 holding the order information of 𝑅; +Output: 𝑅𝑗 is updated with an additional record; 𝐼𝑛𝑥 is +updated to reflect the new order; +/* On client +*/ +1 for j = 1; j < m; j++ do +// Share() +2 +𝑠[𝑗] ← 𝑅𝑛𝑑() +3 +𝑠[0] � 𝑟 − 𝑠[𝑗] +4 end +5 Permute elements in 𝑠 +6 for j = 0; j < m; j++ do +7 +Send 𝑠[𝑗] to 𝑁𝑗 +8 end +/* On server 𝑁𝑗 +*/ +9 𝑅𝑗 � 𝑅𝑗 ∪ {𝑠[𝑗]} +// Update the data share +10 𝑙𝑜 � 0, ℎ𝑖 � 𝑛 − 1, 𝛿 ≔ 0 +11 while 𝑙𝑜 < ℎ𝑖 do +// Binary search for order position of 𝑟 +12 +𝑚𝑖𝑑 � ⌊ 𝑙𝑜+ℎ𝑖 +2 +⌋ +13 +𝛿𝑗 � 𝑠[𝑗] − 𝑅𝑗 [𝑖𝑑𝑥[𝑚𝑖𝑑]] +14 +Broadcast 𝛿𝑗 to 𝑁’s +// Sharing 𝛿 is fine, but not 𝑠 [𝑗 ] +15 +𝛿 � �𝑚 +𝑗=0 𝛿𝑗 +16 +if 𝛿 > 0 then +// Compare() +17 +𝑙𝑜 � 𝑚𝑖𝑑 + 1 +18 +else if 𝛿 < 0 then +// Compare() +19 +ℎ𝑖 � 𝑚𝑖𝑑 − 1 +20 +else +21 +insert(𝑖𝑑𝑥, 𝑚𝑖𝑑, 𝑟𝑖𝑑) +// Found duplicate values in 𝑅 +22 end +23 if 𝛿 ≠ 0 then +24 +insert(𝑖𝑑𝑥, 𝑙𝑜𝑤, 𝑟𝑖𝑑) +// No existing values found in 𝑅 +25 end +3.5 +Security Analysis +3.5.1 +Threat Model. The database servers are assumed to be semi- +honest, meaning that they are not interested in launching active +attacks such as compromising the secret shares. However, those +servers may eavesdrop on the messages and share passively; for +example, the servers may manage to collect secret shares of arbi- +trary plaintexts, which essentially implies a chosen-plaintext attack +(CPA). +In the context of order-preserving data confidentiality, however, +achieving the conventional CPA indistinguishability (IND-CPA) is +impossible because the ordering information does allow the adver- +sary to make a guess much better than a random one. Therefore, we +assume the adversary server is interested in learning information +other than the orders of the plaintext. This is also called IND-OCPA +in the literature. A stronger security notion was proposed called +7 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +Algorithm 3: ODES Query +Input: An order-predicate 𝑂𝑃(·) based on which a set of +records are returned from 𝑅, |𝑅| = 𝑛; each of share of +𝑅𝑗 is stored at node 𝑁𝑗, 1 ≤ 𝑗 ≤ 𝑚; +Output: The client receives a set of records 𝑟’s whose +order 𝑟𝑜𝑟𝑑 satisfying 𝑂𝑃(𝑟𝑜𝑟𝑑) = 𝑇𝑟𝑢𝑒; +/* On client +*/ +1 for j = 0; j < m; j++ do +2 +Send 𝑂𝑃(·) to 𝑅𝑗 +3 end +/* On server 𝑁𝑗 +*/ +4 Receive 𝑂𝑃(·) from client +5 for i = 0; i < n; i++ do +6 +if 𝑂𝑃(𝑖) then +7 +𝑘 ≔ 𝑖𝑑𝑥[𝑖] +8 +Send 𝑅𝑗 [𝑘] to the client +9 +end +10 end +/* On client +*/ +11 Collect 𝐾 values from each of 𝑚 nodes +12 res = [] +13 for k = 0; k < K; k++ do +14 +𝑣𝑎𝑙 ≔ 0 +15 +for j = 0; j < m; j++ do +// Reconstruct() +16 +𝑣𝑎𝑙 += 𝑅𝑗 [𝑘] +17 +end +18 +res.append(𝑣𝑎𝑙) +19 end +20 return res +indistinguishability under frequency-analyzing ordered chosen- +plaintext attack (IND-FAOCPA) [19], implying that the access pat- +tern cannot be leaked. +3.5.2 +Security Assumption. We assume that a block cipher (e.g., +AES) can be used as a pseudorandom generator (PRG) in practice. +We need this assumption because random numbers (i.e., the func- +tion 𝑅𝑛𝑑()) are essential for our protocols (see Algs. 1–3). This is a +well-accepted assumption in the literature of applied cryptography. +3.5.3 +Indistinguishability between Two Parties. Let 𝑚𝐿 and 𝑚𝑅 de- +note two distinct plaintext messages of the same length𝑙. According +to Alg. 1, when 𝑚𝐿 is encrypted, 𝑠0 ← 𝑅𝑛𝑑() and 𝑠1 ≔ 𝑚𝐿 −𝑠0. The +function can be similarly defined for 𝑚𝑅. Because we assume two +nodes 𝑁0 and 𝑁1 do not collude, only one share can be accessed. +Formally, we define the following function +𝑂𝐷𝐸𝑆2 +𝐿(𝑚𝐿,𝑚𝑅,𝑖 ∈ {0, 1}) def += +� +𝑠1 = 𝑅𝑛𝑑(), +𝑖 = 1 +𝑠0 = 𝑚𝐿 − 𝑠1, +𝑖 = 0 +and +𝑂𝐷𝐸𝑆2 +𝑅(𝑚𝐿,𝑚𝑅,𝑖 ∈ {0, 1}) def += +� +𝑠1 = 𝑅𝑛𝑑(), +𝑖 = 1 +𝑠0 = 𝑚𝑅 − 𝑠1. +𝑖 = 0 +The goal is to show that when an adversary A submits two +messages 𝑚𝐿 and 𝑚𝑅 to the ODES oracle, A cannot distinguish +whether a share 𝑠𝑖 is from 𝑂𝐷𝐸𝑆2 +𝐿 or 𝑂𝐷𝐸𝑆2 +𝑅, 𝑖 ∈ {0, 1}. If 𝑠1 is +accessed, then it is obvious that 𝑠1 cannot be distinguished because +both 𝑂𝐷𝐸𝑆2 +𝐿 and 𝑂𝐷𝐸𝑆2 +𝑅 return random numbers through 𝑅𝑛𝑑(). +If 𝑠0 is accessed, we need to show that 𝑚𝐿 − 𝑅𝑛𝑑() and 𝑚𝑅 − 𝑅𝑛𝑑() +are indistinguishable. This is indeed the case because 𝑅𝑛𝑑() is sup- +posed to be uniformly distributed in the message space and adding +its output to 𝑚𝐿 or 𝑚𝑅 would render it garbled. Technically, the +probability for A to distinguish two randomized numbers is 1 +2𝑙 , +which is a negligible function in 𝑙—the bit length of the plaintext. +In other words, the 𝑠0 shares generated by 𝑂𝐷𝐸𝑆2 +𝐿 and 𝑂𝐷𝐸𝑆2 +𝑅 are +interchangeable. +3.5.4 +Indistinguishability between Arbitrary Parties. In the general +case, 𝑡 nodes split the plaintexts, the secrets can be similarly defined +in extended functions: +𝑂𝐷𝐸𝑆𝑡 +𝐿(𝑚𝐿,𝑚𝑅,𝑖 ∈ [0,𝑡)) def += +� +𝑠𝑖 = 𝑅𝑛𝑑(), +𝑖 ≠ 0 +𝑠0 = 𝑚𝐿 − �𝑡−1 +𝑖=1 𝑠𝑖, +𝑖 = 0 +and +𝑂𝐷𝐸𝑆𝑡 +𝑅(𝑚𝐿,𝑚𝑅,𝑖 ∈ [0, 1)) def += +� +𝑠𝑖 = 𝑅𝑛𝑑(), +𝑖 ≠ 0 +𝑠0 = 𝑚𝑅 − �𝑡−1 +𝑖=1 𝑠𝑖. +𝑖 = 0 +When 𝑖 ≠ 0, the 𝑠𝑖 = 𝑅𝑛𝑑() values cannot be distinguished between +𝑂𝐷𝐸𝑆𝑡 +𝐿 and 𝑂𝐷𝐸𝑆𝑡 +𝑅. When 𝑖 = 0, repeated 𝑅𝑛𝑑()’s can only further +garble the𝑚𝐿 and𝑚𝑅 messages (recall that the arithmetic operation +over a set of negligible functions results in a negligible function as +well), making 𝑠0 interchangeable. +3.5.5 +IND-FAOCPA. To show that ODES is IND-FAOCPA, we need +to demonstrate two features: (i) releasing the access patterns does +not help the adversary to distinguish between two ciphertexts, and +(ii) the sequence of pairs of plaintext messages is always ordered +such that the adversary has no extra information to help the distin- +guishing process. We will show that both properties are satisfied +by ODES. A complete proof is beyond the scope of this conference +paper and we will keep our analysis at the descriptive level. +Frequency Analyzing. Every time a plaintext interacts with the +cluster of database servers, the plaintext is freshly decomposed +by the Share() primitive (§3.3). The probability that the shared +secrets are repeated is negligible, +1 +2𝑙·𝑚 , where 𝑙 denotes the bit- +string length of the message space and 𝑚 denotes the number of +database servers. Intuitively, this can be understood as the chance +of generating a repeated set of shares given the same plaintext is +extremely low. Therefore, the adversary cannot launch a successful +attack by studying the frequency or access patterns. +Ordered Plaintexts. Recall that one assumption of the proposed +ODES scheme is that they are non-colluding. This implies that the +database servers have no way to learn about the plaintext except for +the ordinal information of its local shares. However, as we discussed +in §3.5, those shares are indistinguishable from a random number +in the local table of the database server. For example, in Figure 2, +8 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +although the positive database server knows that 22-MAR has a +higher balance than that of 22-FEB (by collecting the 𝛿s), there is +no way for the server to reveal the real balance of either 22-MAR or +22-FEB (unless both servers send over the local share in its entirety, +which is not allowed in ODES). The order of local shares on an +individual database server does not help; for example, even the +servers themselves cannot reach a consensus (let alone whether +the order is consistent with the plaintext): the positive server says +the local share of 22-MAR ($11,000) is lower than that of 22-FEB +($14,000) and yet the negative server says the local share of 22-MAR +($2,000) is higher than that of 22-FEB ($-2,000). +4 +EVALUATION +4.1 +System Implementation +We have implemented the proposed ODES protocol with about +1,200 lines of Python code and Bash script, which will be released at +https://github.com/hpdic/odes. We choose the lightweight SQLite +as the local database instance. Note that SQLite is a file-based data- +base and does not support network access. We thus implement +a communication layer among remote SQLite instances through +the paramiko library for secure data transfer and remote query +invocation. Some of the most important libraries and dependencies +include: python 3.8.0, sqlite 3.31.1, numpy 2.21.0, paramiko 2.12.0, +scp 0.14.4, and cryptography 39.9.0. +4.2 +Experimental Setup +4.2.1 +Test Bed. We deploy the proposed ODES protocol and the +latest OPE scheme [22] with SQLite [15] on a 10-node cluster hosted +at CloudLab [13]. Each node is equipped with two 32-core Intel +Xeon Gold 6142 CPUs, 384 GB ECC DDR4-2666 memory, and two +1 TB SSDs. The operating system image is Ubuntu 20.04.3 LTS, +and the page size is 4 KB. All servers are connected via a 1 Gbps +control link (Dell D3048 switches) and a 10 Gbps experimental link +(Dell S5048 switches). We only use the experimental links for our +evaluation. +Specifically, we name the 10 nodes in the cluster as node0–node9. +The client runs on node0, the ODES server runs on node1–node8, +and the OPE server runs on node9. All 10 nodes are enabled with +password-less SSH connection for convenient communication since +our evaluation focuses on performance metrics rather than security +measurement. Figure 7 illustrates the topological structure of our +10-node cluster. +4.2.2 +Baseline Systems. The literature on order-preserving encryp- +tion (OPE) is mainly contributed by the security and database com- +munities. Not all proposed ideas had been implemented; as of the +writing of this paper, the state-of-the-art OPE system was presented +at [22], which adopts the symmetric encryption scheme 128-bit +AES [25] as a building block. Because the plaintext is not always +128-bit long, our implementation takes the PKCS7 [18] padding +scheme. We will use the term OPEA to refer to this particular sys- +tem. +The original OPEA system was implemented with a server user- +defined function on MySQL and a client script using Python. How- +ever, to make a fair comparison, we re-implement both the server +and client functions of OPEA for SQLite; in particular, we leverage +Test Bed: 10-node Cluster @ CloudLab +Client +(node0) +OPE DB Server +(node9) +ODES Servers (node1--node8) +... +Figure 4: 10-node cluster on CloudLab +the state-of-the-art secure communication framework paramiko +and the cryptographic package hazmat for security building blocks +in our implementation. +4.2.3 +Data Sets. The synthetic benchmark is TPC-H ver. 3.0.0 [31], +a standard database benchmark. There are overall eight tables in +TPC-H; we select four of them for evaluating the proposed work: +Supplier, Customer, Part, and Orders. The reason why we choose +these four tables is two-fold. First, they all exhibit a single-attribute +primary key that is straightforward to encode in the underlying data +structures. Second, they all have at least one numerical attribute that +is more interesting to encode than textual attributes. The attributes +we are interested in include: +Supplier.S_Acctbal, +Customer.C_Acctbal, +Part.P_Retailprice, +Orders.O_Totalprice. +In our TPC-H of scale 0.01, the above column includes 100, 1,500, +2,000, and 15,000 tuples, respectively. +In addition to the synthetic benchmark, we also evaluate the +proposed work with three real-world data sets. +• COVID-19. The first application is the U.S. national COVID- +19 statistics from April 2020 to March 2021 [12]. The data set +has 341 days of 16 metrics, such as death increase, positive +increase, and hospitalized increase. +• Bitcoin. The second application is the history of Bitcoin +trade volume [6] since it was first exchanged in the public in +February 2013. The data consists of the accumulated Bitcoin +exchange on a 3-day basis from February 2013 to January +2022, totaling 1,086 large numbers. +• Human Genome #38 (hg38). The third application is the hu- +man genome reference 38 [16], commonly known as hg38, +which includes 34,424 rows of singular attributes, e.g., tran- +scription positions, coding regions, and number of exons, last +updated in March 2020. +4.2.4 +Workloads. As discussed in [22], we are primarily interested +in two performance metrics in an order-preserving database en- +cryption scheme: insertion and query. +9 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +Supplier +Customer +Part +Orders +TPC-H Tables +101 +102 +103 +104 +Time (second) +End-to-End Performance +ODES +OPEA +Figure 5: Overall performance of ODES and OPEA +• The insertion workload works as follows. The client roughly +issues the following SQL to insert a series of records into the +SQLite database: +INSERT INTO VALUES ; +The insertion starts from scratch, i.e., the target table is +empty. During the insertion, some sort of indexing (depend- +ing on the scheme, e.g., ODES, OPEA) is carried out for future +queries touching on ordering/sorting operations. Essentially, +the insertion incurs not only the searching overhead of com- +paring the provided ciphertext and those existing (encrypted) +tuples already in the table but also the updating overhead +for maintaining the order information after the new record +is inserted. +• The query workload is simpler than the insertion. We ran- +domly make log(𝑛) point-wise queries and compare their +values, where 𝑛 denotes the total number of records in the ta- +ble, i.e., cardinality. Intuitively, if there are a large number of +records for insertion, the query time should be significantly +smaller than the insertion time. +We will also report the overhead incurred by the client, if applica- +ble. The client overhead includes the initialization and maintenance +of encoding the plaintext and optionally other auxiliary information +such as local table in OPEA or linear secret sharing in ODES. +We carry out all experiments at least three times and report the +average numbers. We do not plot the variances (i.e., the error bars) +because they are negligible. A complete log of the experiments is +available upon request. Unless otherwise stated, the default number +of ODES shares is two. +4.3 +End-to-end Performance on TPC-H +We start with reporting the end-to-end performance of the proposed +ODES protocol with the conventional OPEA. Figure 5 shows the +end-to-end execution time of inserting 𝑛 tuples into an empty table +followed by log(𝑛) queries, where 𝑛 denotes the total number of +records as discussed in §4.2.3. Four TPC-H tables are involved and +the time includes the overhead on the client side in addition to the +insertion and the query time. +According to the results of Figure 5, ODES outperforms OPEA by +orders of magnitude in all TPC-H tables. Specifically, for Customer +and Part tables, the improvement is about 100×; for Supplier and +Orders tables, the speedup is over 10×. Therefore, we claim that +ODES can improve the overall order-preserving encryption time +10×–100× faster on TPC-H. In the following three subsections, we +will break down the overall running time into three phases: client +overhead, insertion time, and query time. +4.4 +ODES for Real-World Applications +We report the ODES performance on three different real-world data +sets. We break down the execution time into client overhead, inser- +tion time, and query time to have a better idea of the distribution +of cost in real-world ODES applications. Since our goal is to gain +more insight into cost allocation in practice, we will not compare +the metric to the baseline, i.e., OPEA. +Figure 6 shows the time breakdown of those three real-world +data sets. There are a few interesting observations. First, the client +overhead is insignificant, if not negligible. This reaffirms our previ- +ous conclusion on the effectiveness of ODES for lightweight devices +in the Internet of Things (IoT) or edge computing applications. Sec- +ond, the cost allocation between insertion and query is dynamic. For +COVID-19, the insertion cost is almost 10× lower than the query +cost; however, a converse phenomenon occurs for hg38, where the +insertion cost is more than 10× higher than the query. This can be +best explained by the cardinality of the table: there are 341 rows in +COVID-19 and more than 30,000 records in hg38. +4.5 +Client Overhead +Recall that in OPEA, the client maintains a key-value store of plain- +text and ciphertext such that when inserting a new record the client +can locate the ciphertexts of the lower and upper encoding. This is +required because OPEA must synchronize the stateful client-side +table and the recording encoding on the server side. Each cipher- +text is essentially a 128-bit byte-string (because OPEA uses 128-bit +AES for encryption) regardless of the plaintext length1. Assuming +a plaintext numerical value is stored with 4 bytes (32 bits), the +expansion rate of the local table in OPEA is 128+32 +32 += 5×. +By contrast, the ODES client is stateless and lightweight: (i) the +client simply generates 𝑚 − 1 random numbers as shares and calcu- +late the the𝑚-th share by subtracting the plaintext with those𝑚 −1 +random numbers; (ii) the client does not maintain any data struc- +ture to store the intermediate values (technically, the expansion +rate is zero if we follow the same terminology as OPEA). Therefore, +it is reasonable to expect a much lower client overhead in ODES. +Figure 7 reports the overhead of ODES and OPEA on the client +side. Indeed, we observe that the ODES client overhead is about +3–4 orders of magnitude lower than OPEA for all TPC-H tables. In +addition to the lower overhead, the results also suggest that ODES +is a more practical solution for those lightweight devices such as +smartphones and smartwatches, which have limited computational +power and storage capacity. +1Obviously, we here assume the tuple is represented by less than 128 bits. This is indeed +the case because, practically speaking, the numerical values in database applications +are almost always less than 2128. +10 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +Client +Insertion +Query +10 +2 +10 +1 +100 +Time (second) +COVID-19 +Client +Insertion +Query +10 +2 +10 +1 +100 +Time (second) +Bitcoin +Client +Insertion +Query +100 +101 +102 +Time (second) +hg38 +Figure 6: ODES for real-world applications +Supplier +Customer +Part +Orders +TPC-H Tables +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +Time (second) +Client Overhead +ODES +OPEA +Figure 7: Client overhead of ODES and OPEA +4.6 +Insertion Performance +The main cost of inserting a record in OPEA lies in the interaction +between the client and the server. After the client locates the lower +and upper ciphertexts that are closest to the next plaintext, the client +needs to make two queries to the server to retrieve the encoding +of both ciphertexts. For simplicity, the encoding can be thought +of as the positions of the plaintexts. Roughly speaking, one OPEA +insertion incurs two additional database queries (one for the lower +neighbor and the other one for the upper neighbor). That is, the +number of queries is tripled at the time of insertions. In addition, +the client needs to adjust the local table to incorporate the newly +inserted plaintext and its associated ciphertexts, the entry position +in the local table, and the counter of possible repetition of duplicate +plaintexts. +In contrast, the main cost of ODES for inserting records lies in +the cost of uploading secret shares to multiple servers. However, +ODES is a stateless protocol, meaning that no extra information +needs to be maintained on either the client or the server side. That +is, the queries among distinct servers can be parallelized. As a result, +although the overall number of queries in ODES may exceed that +in OPEA (if there are more than three ODES servers), the running +time for executing these queries is about the same as a single query. +Combining all the above factors, we expect that ODES runs must +faster than OPEA for inserting records into the database. +Figure 9 reports the insertion performance of ODES and OPEA +on TPC-H tables. We observe that for small and medium scales +Supplier +Customer +Part +Orders +TPC-H Tables +10 +1 +100 +101 +102 +103 +104 +Time (second) +Insertion Performance +ODES +OPEA +Figure 8: Insertion performance of ODES and OPEA +(Supplier, Customer, Part), ODES outperforms OPEA by more than +two orders of magnitude. For the large-scale data set of the Orders +table, the speedup is also significant, exceeding 10×. +4.7 +Query Performance +The performance gap between ODES and OPEA for queries is signif- +icantly smaller than that for record insertion because the ordering +information of OPEA is stored as plaintext and therefore an effi- +cient binary sorting can be leveraged. However, as discussed above, +the expanded ciphertext (i.e., 5×) takes significantly more storage +space and network transmission than the plaintext. On the other +hand, ODES deals with 𝑚× plaintexts during the queries that can +be parallelized, where 𝑚 denotes the number of nodes (shares). +Figure 9 reports the query performance of ODES and OPEA. +We observe that ODES still outperforms OPEA in terms of query +performance, and yet the gap is only about 35%–55% rather than +orders of magnitude as we have seen for insertion. However, we +argue that 35%–55% is nonetheless a significant improvement in +query performance, demonstrating the superiority of the proposed +ODES protocol. +4.8 +Scalability of ODES +The last metric we are interested in learning about ODES perfor- +mance is the scalability regarding different numbers of shares. Thus +far, all experiments assume that there are two nodes, each of which +11 + +Conference’17, July 2017, Washington, DC, USA +Dongfang Zhao +Supplier +Customer +Part +Orders +TPC-H Tables +0 +1 +2 +3 +4 +5 +6 +7 +Time (second) +Query Performance +ODES +OPEA +Figure 9: Query Performance of ODES and OPEA +2 +4 +8 +Number of Nodes +0 +2 +4 +6 +8 +Time (second) +ODES@TPC-H Supplier +Overhead +Insertion +Query +2 +4 +8 +Number of Nodes +0 +5 +10 +15 +Time (second) +ODES@TPC-H Customer +Overhead +Insertion +Query +2 +4 +8 +Number of Nodes +0 +5 +10 +15 +Time (second) +ODES@TPC-H Part +Overhead +Insertion +Query +2 +4 +8 +Number of Nodes +0 +100 +200 +300 +Time (second) +ODES@TPC-H Orders +Overhead +Insertion +Query +Figure 10: Scalability of ODES on four TPC-H relations: Sup- +plier, Customer, Part, and Orders. +holds a secret share from the plaintext. There is no technical limita- +tion preventing the user from employing more nodes to increase +the security level, possibly by allowing more processing time. That +is, the adversary would need to compromise more nodes to launch +a successful attack. +Figure 10 reports the scalability of ODES when various numbers +of shares (i.e., nodes) are involved on four TPC-H tables. We ob- +serve that the client overhead is negligible at all scales, which is +consistent with what we have learned. The insertion cost mildly +increases when more shares are involved, which can be explained +by the higher cost of splitting the plaintext into a larger number +of shares. The query time exhibits a steeper trend concerning the +number of nodes; the cost looks proportional to the number of +Supplier +Customer +Part +Orders +TPC-H Tables +0 +100 +200 +300 +400 +500 +600 +700 +Database Size (Kilobytes) +Server Storage +ODES +OPEA +Figure 11: Server storage of ODES and OPEA +nodes. The main reason for this linear proportion is due to the se- +rial implementation of the paramiko library, which is synchronous +and therefore a remote query must complete before another one is +issued. We believe parallel processing from the client would greatly +mitigate this issue by utilizing more CPU cores and higher network +bandwidth, which will be explored in our future work. +4.9 +Storage Cost +Client Storage. Recall that ODES is a stateless protocol, meaning +that there is zero storage requirement for ODES. On the other hand, +the space overhead of OPEA is O(�𝑛), where �𝑛 denotes the number +of distinct values in the column. +Server Storage. Figure 11 reports the database size of ODES and +OPEA when various TPC-H tables are encrypted. The results sug- +gest that for small- and medium-scale data sets, the database size +is not significant. However, for larger data sets such as the orders +table, the space overhead is much higher in OPEA. This can be best +explained by the fact that OPEA involves the AES scheme, which +causes size expansion in the ciphertext. That is, no matter how +small the plaintext value is, the encrypted text will always be a +fixed length (e.g., 128 bits) for security reasons. Therefore, the more +plaintext we have, the larger space overhead is incurred. +5 +CONCLUSION AND FUTURE WORK +This paper proposes a new stateless order-preserving encryption +scheme, namely ODES (Ordered Database Encryption with Secret- +sharing), by incorporating secret-sharing primitives. ODES sup- +ports the latest IND-FAOCPA security. A series of database proto- +cols are designed based on ODES. The ODES scheme and database +protocols are implemented on top of a 10-node cluster of SQLite +databases. Experimental results show that ODES outperforms state- +of-the-art schemes by orders of magnitude on the TPC-H bench- +mark and multiple real-world applications. +Our future work will focus on more efficient secret-sharing prim- +itives that we hope will further help reduce the insertion and query +time in database applications. Another orthogonal research direc- +tion is to explore the feasibility of homomorphic encryption in +order-preserving encryption in database systems. +12 + +Order-Preserving Database Encryption with Secret Sharing +Conference’17, July 2017, Washington, DC, USA +REFERENCES +[1] Rakesh Agrawal, Jerry Kiernan, Ramakrishnan Srikant, and Yirong Xu. Order +preserving encryption for numeric data. In Proceedings of the 2004 ACM SIGMOD +International Conference on Management of Data, SIGMOD ’04, page 563–574, +New York, NY, USA, 2004. Association for Computing Machinery. +[2] Panagiotis Antonopoulos, Arvind Arasu, Kunal D. 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Springer, 2015. +13 + diff --git a/qtE3T4oBgHgl3EQfMAny/content/tmp_files/load_file.txt b/qtE3T4oBgHgl3EQfMAny/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b795472f20647b7d59b4fbaa3bbc2f096ef285f5 --- /dev/null +++ b/qtE3T4oBgHgl3EQfMAny/content/tmp_files/load_file.txt @@ -0,0 +1,957 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf,len=956 +page_content='Order-Preserving Database Encryption with Secret Sharing Dongfang Zhao University of Nevada, Reno United States ABSTRACT The order-preserving encryption (OPE) problem was initially for- mulated by the database community in 2004 soon after the para- digm database-as-a-service (DaaS) was coined in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Over the past two decades, OPE has drawn tremendous research interest from communities of databases, cryptography, and security;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' we have witnessed significant advances in OPE schemes both theoret- ically and systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' All existing OPE schemes assume that the outsourced database is modeled as a single semi-honest adver- sary who should learn nothing more than the order information of plaintext messages up to a negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This paper addresses the OPE problem from a new perspective: instead of mod- eling the outsourced database as a single semi-honest adversary, we assume the outsourced database service compromises a cluster of non-colluding servers, which is a practical assumption as all major cloud vendors support multiple database instances deployed to exclusive sub-networks or even to distinct data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This as- sumption allows us to design a new stateless OPE protocol, namely order-preserving database encryption with secret sharing (ODES), by employing secret-sharing schemes among those presumably non-colluding servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will demonstrate that ODES guarantees the latest security level, namely IND-FAOCPA, and outperforms the state-of-the-art scheme by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ACM Reference Format: Dongfang Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Order-Preserving Database Encryption with Secret Sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ACM, New York, NY, USA, 13 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Inception of Order-Preserving Encryption For two decades, we have witnessed the inception and prosperity of database as a service (DaaS) since the publication of the seminal pa- per [14] in ICDE’02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As of the writing of this paper, all major cloud computing vendors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', Amazon Web Services, Google Cloud Plat- form, Microsoft Azure) support DaaS with pay-as-you-go business models, which enables users to avoid the upfront cost of managing their in-house databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' From the user’s perspective, the DaaS can be thought of as an outsourced database maintained by cloud computing vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As with any outsourced service, the security of outsourced databases has been one of the top concerns for users: the threat in an outsourced database comes not only from outside Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='nnnnnnn attackers but also inside adversaries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', developers and adminis- trators of the cloud computing vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Among others, one avenue of research to address the above security issue is to encrypt the user’s sensitive data before uploading them to the outsourced database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' One key challenge of DaaS lies in the management of those en- crypted data, such as building indexes, because the index must be associated with the plaintext to speed up the query and modifica- tion requests from the user and yet all the server can learn about is the ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To that end, in SIGMOD’04, Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [1] proposed to encode the plaintext in the outsourced database while retaining the numerical order of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The paper demon- strated that it was possible to achieve the best of both worlds: the so-called order-preservation encryption (OPE) can ensure both the confidentiality and the ordinal of the outsourced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The work quickly drew a lot of research interest from the database and the security/cryptography communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Brief Timeline of OPE Security In EuroCrypt’09, Boldyreva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [8] presented the first security definition of OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Following the convention of cryptography, the definition is based on the canonical structure of an encryption scheme: (i) the security goal is computational indistinguishability, (ii) the threat model is to allow the adversary to obtain a polyno- mial number of ciphertexts of arbitrary plaintexts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the so-called chosen-plaintext attack (CPA), and (iii) a simulation-based reduc- tion to prove that distinguishability is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Unfortunately, it was shown that it is impossible to achieve indistinguishability under the standard CPA attack because the CPA definition is overly strong and can be violated if the adversary can learn about the ordinal of the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The good news was that a new security notion, namely indistinguishability under ordered chosen-plaintext attack (IND-OCPA), was proposed by [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' IND- OCPA is as strong as IND-CPA except for allowing the adversary to only learn about the ordinal of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Not long after IND-OCPA was proposed, Popa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [26] in Oakland’13 pointed out that IND-OCPA is insufficient for the well- known frequency attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The frequency attack is due to the deter- ministic ciphertexts, which are known to be insecure under the conventional CPA attack as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A stronger security notion was then proposed in CCS’15, namely indistinguishability under frequency-analyzing and ordered chosen- plaintext attack (IND-FAOCPA) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Multiple subsequent schemes claimed to meet IND-FAOCPA, such as [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As of the writing of this paper, IND-FAOCPA remains the strongest security notion for order-preserving encryption schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 System Research of OPE Schemes While the cryptography and security communities spent tremen- dous effort in properly defining and proving security from a theoret- ical perspective, the database and system communities are equally 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='04370v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='CR] 11 Jan 2023 Conference’17, July 2017, Washington, DC, USA Dongfang Zhao interested in other metrics such as performance and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Many leading cloud vendors are now supporting encrypted database ser- vices, such as Microsoft Azure [2] as reported in SIGMOD’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' An evaluation paper [7] in VLDB’19 summarized the pros and cons of major OPE schemes as of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The metrics include encryp- tion complexity, comparison complexity, ciphertext size, I/O cost, and communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There was no clear winner based on the reported numbers in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In VLDB’21, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [22] presented a new frequency-hiding OPE scheme with a 128-bit AES (OPEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' OPEA outperforms existing OPE schemes in almost all aspects: OPEA is IND-FAOCPA secure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' OPEA incurs a constant number of interactions between clients and servers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' OPEA completes both insertion and query requests signifi- cantly faster than the counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' OPEA incurs O(𝑁) client storage space, where 𝑁 denotes the number of distinct plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The only limitation of OPEA lies in the client storage: as a stateful scheme, OPEA requires the client to maintain a local table to keep track of the plaintext orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Although duplicate plaintexts only need to be stored one time, counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', POPE [27]) could take constant client storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' (However, POPE suffers the problem of possible incomparable elements) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 Motivation and Challenges of This Work The goal of this work is to eliminate the shortcoming of OPEA [22] while retaining its advantages compared with existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This implies that we want to achieve both strong security levels and high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' By high performance, we mean lower processing time, which is contributed by computational time, communication time, and I/O time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Clearly, the O(𝑁) client storage of OPEA is a performance bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' On the other hand, achieving both strong security guarantees and high performance is very challenging with currently available cryptographic primitives and our conventional wisdom of system optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As discussed above, client storage is necessitated by the stateful coordination between the client and the server, because a stateful mechanism is believed to achieve higher efficiency for maintaining the tuple orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As of the writing of this paper, we are only aware of one work [30] taking a stateless approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' but no experimental results were reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition, 128-AES is widely believed to be one of the most efficient and secure symmetric en- cryption schemes nowadays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' therefore, it is unlikely to improve the performance of OPEA by upgrading or optimizing the crypto- graphic subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Something more drastic is in need should we aim to further improve the performance without trading off the security level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 Proposed Solution This work proposes a new OPE scheme by employing secret-sharing primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' While secret-sharing can be thought of as an encryption scheme (in a broad sense) its internal machinery works quite differ- ently than single-node encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Instead of placing the encrypted test on a single machine, we now assume a cluster of machines that would not collude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The ciphertext is now distributed into multiple User Client Server Conventional Order-preserving Encryption Client Cluster of Servers Proposed Order Preservation w/ Multiparty Secrets Log on Figure 1: Proposed multiparty secrets vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' conventional single-node encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' database servers as secret shares and cannot be decrypted without the authorization of the data owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, confidentiality is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To achieve the ordinal of plaintexts, the secret-sharing primitives should allow us to compare the corresponding plaintext values by asking the server not to share its local secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' If we can achieve this comparison merely through some local computation of the secret shares, the resulting scheme would be stateless and save us some I/O costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As we will demonstrate in the latter sections, a specific type of secret-sharing scheme does allow us to achieve both confidentiality and ordinal of encoded data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To make matters more concrete, Fig- ure 1 illustrate the high-level difference between the conventional OPE schemes and the proposed scheme, which we coin as ordered database encryption with secret-sharing (ODES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='6 Contributions In summary, this work makes the following technical contributions: We propose the very first stateless order-preserving encryp- tion scheme for outsourced databases with secret sharing, namely ordered database encryption with secret-sharing (ODES);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We demonstrate that ODES guarantees a strong security level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', IND-FAOCPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We design various database protocols for leveraging the proposed ODES scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We implement the ODES scheme and database protocols on top of SQLite databases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' and We conduct a thorough evaluation of ODES by comparing it with state-of-the-art schemes on top of the TPC-H bench- mark and three real-world applications on a 10-node cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Order-Preserving Encryption The concept of order-preserving encryption (OPE) was originally proposed in the database community [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The motivation is evident: how could we achieve both the confidentiality and the ordinals of sensitive data in an outsourced database?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The confidentiality part is obvious and the ordinal requirement is also well justified: it is very common for database systems to build indexes to speed up the query and insertion requests and being able to sort or order the outsourced data sets is essential to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2 Order-Preserving Database Encryption with Secret Sharing Conference’17, July 2017, Washington, DC, USA The early-state solution to achieve the dual goals is somewhat straightforward: the plaintexts are encoded with the help of some statistical distribution such that the encoded values remain in the same order as the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There are a few issues with this ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' First, the encoded values are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This means the encoding cannot be secure if the adversary can somehow obtain the encoding of some chosen plaintexts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the so-called chosen- plaintext attack (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' It can be argued that in outsourced databases we do not need security as strong as CPA, but from a security point of view, a practical database system should always provide CPA security as the minimum [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We then run into a dilemma between CPA security and ordi- nal encoding: CPA security implies the randomness of ciphertexts which cannot retain the order of plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The solution is to intro- duce some function for the database to order the encrypted tuples without relying on the raw values of ciphertexts, which is called order-revealing encryption (ORE) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Accordingly, a new security notion was proposed to allow the adversary to learn about the orders of plaintexts, resulting in the so-called indistinguishabil- ity under ordered chosen-plaintext attack (IND-OCPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Obviously, there are many options to calculate the order values but they can be categorized into two categories: (i) a stateful scheme where the client and the server coordinate to maintain the order information of encrypted records in the database [26], and (ii) a stateless scheme where the order information can be retrieved on the fly [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Most OPE works in the literature focus on the stateful approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' as we will see in the latter sections, the proposed ODES is a stateless scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' It turned out that there were new issues for ORE and IND- OCPA: Many IND-OCPA schemes [8, 26] are vulnerable to attacks that leverage the access patterns of the queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To that end, a newer notion is defined, the indistinguishability under frequency- analyzing ordered chosen-plaintext attack (IND-FAOCPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Multiple IND-FAOCPA schemes have been proposed in the literature, such as [19, 22, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A relatively recent evaluation paper [7] reports the performance of some of the most popular OPE schemes, including what have not been mentioned in this paper: [10, 11, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As of the writing of this paper, the OPE scheme proposed in [22] achieves the best performance in almost all the metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', client storage, query rounds), and we will primarily compare the proposed ODES with the protocol proposed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Secret Sharing The idea of a secret sharing scheme (SSS) is straightforward: a given plaintext 𝑝𝑡 is converted into a set of encoded bytes 𝑐𝑡’s such that only a specific subset of 𝑐𝑡’s can reconstruct the original 𝑝𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The goal of SSS is to reduce the risk of disclosing the plaintext;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' instead of compromising the holder of the plaintext, the malicious adversary needs to compromise multiple entities before any of the shareholders detect the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Even for weaker attacks where only semi-honest adversaries are assumed, dispersing the secret shares to more parties raises the bar of a successful eavesdropping attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The SSS can be tuned by the subset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Formally, a 𝑡-out-of-𝑛 threshold SSS (TSSS) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Definition 1 (TSSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A TSSS is comprised of two algorithms: Share: a randomized algorithm that takes as input a plaintext 𝑝𝑡 and returns a sequence 𝑆 = (𝑠1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ,𝑠𝑛) of shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Reconstruct: a deterministic algorithm that takes a set of at least 𝑡 shares and returns the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The number 𝑡 is called the threshold of the TSSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Let 𝑈 of size 𝑡 be a subset of 𝑛 shares, |𝑈 | ≥ 𝑡 and 𝑈 ⊆ {𝑠1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ,𝑠𝑛}, we require that a TSSS holds the following property: 𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡(𝑈 ) = 𝑝𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As we will see in the next section §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3, the definition of TSSS leads to a slightly different security definition compared with the conventional encryption schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The canonical example of TSSS is due to Shamir [29], in which the secrets are revealed through a (𝑡 − 1)-degree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In essence, each share can reconstruct the coefficient of a specific de- gree of unknowns through the LaGrange polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition to Shamir’s construction, other schemes exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Ito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [17] pro- posed the replicated secret-sharing scheme, which is based on finite fields where each share is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' One nice property of replicated secret-sharing is its linearity: the addition and subtraction of local shares equal the addition and subtraction of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A simpler variant of replicated secret-sharing is additive secret sharing, where each share is a scalar value and the threshold 𝑡 is set to 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Indeed, this is a building block of our proposed ODES protocol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' we will see how the linearity of the additive secret-sharing allows us to preserve the orders of plaintexts in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' SSS has a tight connection with secure multiparty computation (MPC) [3], which has a long history [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The goal of MPC is more ambitious than SSS: in addition to keeping the plaintext confidential, we want to calculate an arbitrary function of the original plaintexts by touching on only the encoded data on multiple parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The original problem was solved by the so-called garbled circuits [32], whose idea was pretty simple: we can ask each party to encode the input with its private key, shuffle the encrypted ciphertexts, and then enumerate all the keys to decrypt the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Since we assume the encryption scheme is secure, the only way that the result can be revealed is that the correct combination of private keys is applied to one of the garbled outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is indeed a feasible solution, at least theoretically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' in practice, the circuits may grow exponentially and result in efficiency issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There are many more efficient MPC solutions, such as [4, 20, 24, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 Provable Security When employing an encryption scheme in an application, it is highly desirable to demonstrate its security provably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Formally, we need to identify the following three important pieces for the prov- able security of a given encryption scheme: security goal, threat model, and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The security goal spells out the desired effect when the application is under attack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' the threat model ar- ticulates what an adversary can do with the attack, such as what information of the plaintext/ciphertext can be collected and the resource/time limitation of the attack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' the assumption lists the presumed specifics of the subsystems or components of the crypto- graphic scheme, which is usually an important building block for security proof, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The security goal and threat model are usually called security definition collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3 Conference’17, July 2017, Washington, DC, USA Dongfang Zhao One well-accepted security definition with a good balance be- tween efficiency and security is that the adversary can launch a chosen-plaintext attack (CPA), defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Definition 2 (Chosen-Plaintext Attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Given a security parameter 𝑛, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the bitstring length of the key, an adversary can obtain up to 𝑝𝑜𝑙𝑦(𝑛) of plaintext-ciphertext pairs (𝑚,𝑐), where 𝑚 is arbitrarily chosen by the adversary and 𝑝𝑜𝑙𝑦(·) is a polynomial function on 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' With such information, the adversary tries to decrypt a 𝑐′ that is not included in the polynomial number of known ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The polynomial requirement is only for practical reasons, as we usually assume that the adversary should only be able to run a polynomial algorithm without unlimited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Accordingly, we want to design encryption schemes that are CPA secure: even if the adversary A can obtain those extra pieces of information, A should not be able to decode the ciphertext better than a random guess up to a very small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To quantify the degree of this small probability, negligible function is defined as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A function 𝜇(·) is called negligible if for all poly- nomials 𝑝𝑜𝑙𝑦(𝑛) the inequality 𝜇(𝑛) < 1 𝑝𝑜𝑙𝑦(𝑛) holds for sufficiently large 𝑛’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For completeness, we list the following lemmas for negligible functions that will be used in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We state them without the proofs, which can be found in introductory cryptography or complexity theory texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lemma 1 (Summation of two negligible functions is a neg- ligible function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Let 𝜇1(𝑛) and 𝜇2(𝑛) be both negligible func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Then 𝜇(𝑛) is a negligible function that is defined as 𝜇(𝑛) def= 𝜇1(𝑛) + 𝜇2(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lemma 2 (Quotient of a polynomial function over an ex- ponential function is a negligible function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑝𝑙𝑜𝑦(𝑛) 2𝑛 is a negligible function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, ∃𝑁 ∈ N, ∀𝑛 ≥ 𝑁 : 𝑝𝑙𝑜𝑦(𝑛) 2𝑛 < 1 𝑝𝑜𝑙𝑦(𝑛) , where N denotes natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The canonical method to prove the security of a proposed encryp- tion scheme, such as IND-CPA, is through reduction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Usually, breaking the scheme is reduced to a hard mathematical problem, which means that if an attack is possible for the scheme then the mathematical problem would be efficiently solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, the en- cryption scheme is not easier than the mathematical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The scheme is modeled as a subroutine, whose inputs are simulated such that the adversary cannot tell whether it is being involved in an attack or in a subroutine to help solve the math problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Although forward proof is possible, the more commonly used technique is the contradiction: by assuming that the adversary could distinguish some designed experiments with a non-negligible advantage, the re- duction would lead to a non-negligible probably to efficiently solve the hard mathematical problem that is believed to be intractable, thus leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Unfortunately, in the context of order-preserving encryption, it has been proven that the conventional IND-CPA is impossible [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the cryptography community proposed a relaxed nota- tion called indistinguishability under ordered chosen-plaintext attack (IND-OCPA) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, it was shown that [5] effective attacks can be launched on IND-OCPA security caused by the access pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The root cause of this issue lies in the deterministic cipher- text in early-stage order-preserving encryption schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Indeed, it is well known that a deterministic encryption scheme can be impossibly secure against CPA attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As a result, modern order- preserving encryption schemes are all randomized, which implies that the ciphertexts are not directly comparable and necessities indirect comparison between ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Such indirection com- parison is usually coined as order-revealing encryption (ORE) [9] that generalizes the original notion of OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In a more general sense, some so-called frequency-hiding order-preserving encryp- tion schemes [19, 27] were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Accordingly, a new security notion was proposed, namely indistinguishability under frequency- analyzing ordered chosen-plaintext attack (IND-FAOCPA) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' IND-FAOCPA is the latest security definition in this area and our proposed ODES scheme is IND-FAOCPA secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The above review of provable security assumes that the cipher- text is a single entity and does not consider the scenario where the ciphertext is a set, which is the case for secret sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The provable security of secret sharing takes a slightly different approach because of the additional assumption that not all shares will be accessible to the adversary per the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' While it is true that if we can prove the entire set of shares is secure then any subset of shares is also secure, a more common approach to proving the semantic security of secret shares is through interchangeable libraries [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The key idea is to model the scheme as a library with the input of either an 𝐿 or 𝑅 plaintext input, and then the proof will show that the library with 𝐿 input eventually looks identical to the library with the 𝑅 input through a series of interchangeable operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will see how this technique is used in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 3 ORDER-PRESERVING DATABASE ENCRYPTION WITH SECRET SHARING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Overview The intuitive idea behind the proposed secret-sharing-based order preservation is straightforward: we leverage the multiplicity of a cluster of database servers such that no plaintext is leaked while maintaining the comparative order among the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, we somehow break the original plaintext into multiple shares, each of which is allocated to a distinct server in a database cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The nodes in the cluster are assumed to be non-colluding, which can be implemented by deploying the servers into different sub-networks or different data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The order of the plaintext can be retrieved and updated by an aggregation of local functions on each server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To make matters more concrete, Figure 2 illustrate the idea in an oversimplified scenario where two servers are available to store encoded and ordered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Assume that the data owner wants to save the balance table into a remote database service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The table is as simple as a key-value store with the year-month as the key and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' dollar amount as the value for his business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Each of the two database servers stores some seemingly random numbers that will help keep the real amounts confidential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We must be able to reconstruct the original plaintext from the shares stored by the servers because otherwise, the user would lose the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In this example, the original balance amount can be reconstructed by simply adding up the shares from distinct database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' It 4 Order-Preserving Database Encryption with Secret Sharing Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' USA Balance Plaintext YY-MON Amount 22-JAN $15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-FEB $12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-MAR $13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-APR $14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 Balance Share0 YY-MON Amount 22-JAN $6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-FEB $14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-MAR $11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-APR $-6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 Balance Share1 YY-MON Amount 22-JAN $9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-FEB $-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-MAR $2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 22-APR $20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='000 No Collusion Balance Index YY-MON Order 22-JAN 4 22-FEB 1 22-MAR 2 22-APR 3 Cloud Services User Figure 2: Example of two shares with order preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' should be noted that the reconstruction should only happen on the client side as the outsourced databases are not fully trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, the database servers are not supposed to share their local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will formally define what we mean by “fully trusted” later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' but for now, we keep our discussion at a non-technical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Another important piece of information is the index metadata for keeping track of the order information of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The index table will be useful when a new record is inserted into the databases: it allows us to do a binary search to locate the correct order of the newly inserted record in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition, it would facilitate the effectiveness of order-related queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Here is another example for illustrating the plaintext comparison through two sets of secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Let’s say the user recently ob- tained a new record for (22-MAY, $10,000) and would like to know whether the balance is higher than the previous month, 22-APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Note that ODES is a stateless protocol, so the client cannot simply compare $10,000 to (22-APR, $14,000) since the latter does not exist after being secretly shared with the two database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' What the client would do is split $11,000 into two random numbers, say $3,000 and $8,000, which are combined with the key and sent to the two database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In our example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2, the plus server receives (22-MAY, $3,000) record and the minus server receives (22- MAY, $8,000) record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Both servers then carry out local computations of the 22-APR and 22-MAY records;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' for example, on the plus server, it calculates the following delta (𝛿0, 3000-(-6000) = 9000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Similarly, the minus server calculates (𝛿1, 8000-20000 = -12000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Both servers then broadcast their local 𝛿𝑗 to all other servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Note that this sharing would not reveal any information other than the orders and is therefore allowed (as opposed to the secret share itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Now, each database server has obtained all the 𝛿𝑗’s and then applies an aggregation over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For example, the plus server calculates 9000+(-12000) = -3000 < 0, which means the balance of 22-MAY is lower than 22-APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The server can also update the index metadata accordingly by conducting a binary search on the index file;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' in this Tuples SQLite 0 SQLite i SQLite m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Ordered Tuples Provider Customer Share Index Tuple 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Tuple n Outsourced Database Services Figure 3: The proposed architecture of preserving tuple or- ders with secret shares in outsourced SQLite databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' example, (22-MAY, 1) will be inserted into the index table and some existing orders will be incremented by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Architecture As shown in Figure 3, we envision a cluster of 𝑚 outsourced data- base servers that (i) do not share their local data and (ii) can access an index for the orders of the secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Since our system proto- type is implemented with SQLite, we assume there are 𝑚 SQLite instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As long as at least one of the 𝑚 SQLite instances does not collude with others, the outsourced data is secure, which overcomes the so-called dishonest majority problem in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This im- plies that more databases imply a higher security level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' however, this is at the cost of more computational and I/O overhead in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The architecture also differentiates two different user roles: a data provider (on the left) and a data customer (on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The key difference is that the provider may modify the secret shares in the database cluster and possibly update the index metadata as well, while the customer only makes read-only queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will see that both modification and read-only queries will be facilitated by the share index in the proposed protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 Primitives We are now ready to formally present the primitives of ODES to allow order preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' These primitives will be used as building blocks in various protocols later (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We assume that the plaintext is a string of 𝑙 bits and there are overall 𝑚 database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We start with the Share function that splits a given plain- text into 𝑚 secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 5 Conference’17, July 2017, Washington, DC, USA Dongfang Zhao Definition 4 (Share).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A Share() function is defined as 𝑆ℎ𝑎𝑟𝑒 : {0, 1}𝑙 → � {0, 1}𝑙 �𝑚 , 𝑝𝑡 ↦→ {𝑠𝑖}, 0 ≤ 𝑖 < 𝑚, where 𝑠𝑖 is a random number when 𝑖 ≠ 0 and 𝑠0 is calculated as 𝑠0 ≔ 𝑚−1 ∑︁ 𝑖=1 𝑠𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The Reconstruct function is the reverse of Share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Definition 5 (Reconstruct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A Reconstruct() function is defined as 𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡 : � {0, 1}𝑙 �𝑚 → {0, 1}𝑙, {𝑠0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ,𝑠𝑚−1} ↦→ 𝑚−1 ∑︁ 𝑖=0 𝑠𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A cluster of 𝑚 servers aims to compare two plaintexts by local computations of two sets of secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As the name suggests, a server is not allowed to disclose its local share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Let 𝑆𝐿 def = � 𝑠𝐿 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ,𝑠𝐿 𝑚−1 � denote the secret shares of the first plaintext 𝐿 and 𝑆𝑅 def = � 𝑠𝑅 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ,𝑠𝑅 𝑚−1 � denote the secret shares of the second plaintext 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Definition 6 (Compare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A Compare is a function: 𝐶𝑜𝑚𝑝𝑎𝑟𝑒 : � {0, 1}𝑙 �𝑚 × � {0, 1}𝑙 �𝑚 → {−1, 0, 1}, (𝑆𝐿,𝑆𝑅) ↦→ \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 −1, 𝐿 < 𝑅 0, 𝐿 = 𝑅 1, 𝐿 > 𝑅 , where Compare is assigned the same arithmetic sign as the following summation 𝑚−1 ∑︁ 𝑖=0 � 𝑠𝐿 𝑖 − 𝑠𝑅 𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Note that the terms in the above equation are calculated by each database server independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The servers broadcast their local differences to the entire cluster such that each server can decide the ordinal information of the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 Protocols We present three ODES protocols in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Various primitives will be used in the protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To make the protocols self-contained, we include some high-level implementation details of the primitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' when we do so, we make comments to the pseudocode to remind the readers that certain primitives are being called (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', Line 3 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We follow the cryptographic convention to use ≔ to denote a deterministic assignment and ← to denote a uniformly sampled value from a randomized algorithm or distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' When the system is deployed for the first time, we assume that the data provider has an initial list of data sets that will be encoded and uploaded to the remote database service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Several tasks must be completed before the proposed ODES system goes into operation, including: An initial index file is built by the data provider based on the local plaintexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Each plaintext field is decomposed into a list of shares;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The shares of the same plaintext should be randomized and distributed to distinct database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' All of those tasks are completed in the initialization phase, as illustrated in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For the 𝑖-th record 𝑟, the client (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the provider in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3) splits it into 𝑚 pieces (Lines 3–6): the last 𝑚 − 1 pieces 𝑠𝑖 [1 : 𝑗] are simply random values and the first piece is calculated as the difference between 𝑟 and �𝑠𝑖 [1 : 𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To add more randomness to the encoded ciphertext, we apply permutation to the 𝑚 shares as well (Line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In Line 9, the client sends each of the 𝑚 shares to a distinct database server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' in Line 10, the server receives the share and inserts it into its local share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' After processing the record 𝑟, the index metadata is updated in Line 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' After completing all the 𝑛 records, the client broadcasts the index table to all the database servers in Lines 14–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Algorithm 1: ODES Init Input: A relation 𝑅 of cardinality 𝑛, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', there are |𝑅| = 𝑛 plaintext tuples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' a set of 𝑚 non-colluding nodes 𝑁;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' function 𝑅𝑛𝑑() returning a random number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Output: Each node 𝑁𝑗 holds a list of ciphertext shares 𝑅𝑗, |𝑅𝑗 | = 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' the index 𝑖𝑑𝑥 holding the orders of 𝑅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1 for i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' i < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' i++ do 2 r = R[i] 3 for j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do // Share() 4 𝑠𝑖 [𝑗] ← 𝑅𝑛𝑑() 5 𝑠𝑖 [0] � 𝑟 − 𝑠𝑖 [𝑗] 6 end 7 Permute elements in 𝑠𝑖 8 for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do 9 Send 𝑠𝑖 [𝑗] to 𝑁𝑗 10 𝑅𝑗 [𝑖] � 𝑠𝑖 [𝑗] // On server 𝑁𝑗 11 end 12 Update 𝑖𝑑𝑥 13 end 14 for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do 15 Send 𝑖𝑑𝑥 to 𝑁𝑗 16 end The complexity of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 3–6 take O(𝑚) steps, Line 7 takes O(𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=') steps, and Lines 8–11 take O(𝑚) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, Lines 1–13 take O(𝑛𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 14–16 trivially take O(𝑚) steps and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The total complexity of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1 is therefore O(𝑛𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Although the O(𝑚) factor in the asymptotic complexity seems costly, it is usually not an issue in practice because𝑚 is taken at a relatively small value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', 2, 4, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The insertion protocol assumes that the cluster of database servers already holds secret shares and will take in a new record from the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In this context, the protocol comprises two phases: (i) the client prepares the secret shares of the new record 6 Order-Preserving Database Encryption with Secret Sharing Conference’17, July 2017, Washington, DC, USA and sends them to the cluster of servers, and (ii) the servers update their local shares and the index metadata for ordering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We depict both phases in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Client protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Similarly to the initialization protocol, the client splits the given record 𝑟 into 𝑚 shares in Lines 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The client then permutes the shares and then sends each of them to a distinct database server (Lines 5–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Server protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A server always inserts the received share into its local table, as shown in Line 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 10–25 work on updating the index through a binary search for the correct order of the new record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Line 13 computes the difference between the received share and the share whose corresponding plaintext is the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Line 14 allows all database servers to learn about the differences so that they can decide whether to move to the smaller or the larger half of the sorted shares (Lines 16–19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' If there exists a duplicate value in the database, then the protocol will end up at Line 21 (𝛿 = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' otherwise, Line 24 sets the order of the new record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The complexity of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 2 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 1–4 take O(𝑚) steps, Line 5 takes O(𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=') steps, and Lines 6–8 take O(𝑚) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the client takes overall O(𝑚!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=') steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 11–12 and 16–21 imply O(log𝑛) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Line 14 takes O(𝑚) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 21 and 24 take constant steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the server takes overall O(𝑚 log𝑛) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We represent the order-related predicate as𝑂𝑃, which is part of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There are three phases in the query protocol: In the first phase of the query protocol, the client (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the provider in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3) simply broadcasts the predicate to all database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The second phase happens on each database server, which works on the local shares that qualify for the 𝑂𝑃 predicate and send them back to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In the third phase, the client (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the customer in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3) col- lects all the qualified shares and reconstructs the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' These three phases are summarized in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 5–8 check every indexed order to see whether the local share qualifies for 𝑂𝑃 and if so, the server 𝑁𝑗 replies its local share 𝑅𝑗 [𝑘] to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 13–19 reconstruct the list of plaintexts by aggregating the shares into each of the tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The complexity of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For the server, the overall complexity is O(𝑛) (Lines 5–10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For the client, Lines 1–3 take O(𝑚) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Line 11 itself takes O(𝑛𝑚) steps because 𝐾 ≤ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Lines 13–19 take O(𝑛𝑚) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the overall complexity of the client is O(𝑛𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 Deletion and Modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The deletion of ODES is more straightforward than insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The first phase is asking every data- base server to remove the corresponding local shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The second phase is to remove the order information of the to-be-removed record in the index file and update the index file by decrement- ing (by one) the order values of those records whose orders are larger than the order of the deleted record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We omit the detailed description of this protocol in this conference paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The modification of ODES can be trivially implemented by first deleting the record and then inserting the new value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We skip the formal protocol description of the modification in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Algorithm 2: ODES Insert Input: A new record 𝑟 of record identifier 𝑟𝑖𝑑 to be inserted into a relation 𝑅, |𝑅| = 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑅 is not directly accessible and can only be reconstructed from 𝑅𝑗, each of which is stored at node 𝑁𝑗, 1 ≤ 𝑗 ≤ 𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' a global index 𝑖𝑑𝑥 holding the order information of 𝑅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Output: 𝑅𝑗 is updated with an additional record;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝐼𝑛𝑥 is updated to reflect the new order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' /* On client / 1 for j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do // Share() 2 𝑠[𝑗] ← 𝑅𝑛𝑑() 3 𝑠[0] � 𝑟 − 𝑠[𝑗] 4 end 5 Permute elements in 𝑠 6 for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do 7 Send 𝑠[𝑗] to 𝑁𝑗 8 end /* On server 𝑁𝑗 / 9 𝑅𝑗 � 𝑅𝑗 ∪ {𝑠[𝑗]} // Update the data share 10 𝑙𝑜 � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ℎ𝑖 � 𝑛 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝛿 ≔ 0 11 while 𝑙𝑜 < ℎ𝑖 do // Binary search for order position of 𝑟 12 𝑚𝑖𝑑 � ⌊ 𝑙𝑜+ℎ𝑖 2 ⌋ 13 𝛿𝑗 � 𝑠[𝑗] − 𝑅𝑗 [𝑖𝑑𝑥[𝑚𝑖𝑑]] 14 Broadcast 𝛿𝑗 to 𝑁’s // Sharing 𝛿 is fine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' but not 𝑠 [𝑗 ] 15 𝛿 � �𝑚 𝑗=0 𝛿𝑗 16 if 𝛿 > 0 then // Compare() 17 𝑙𝑜 � 𝑚𝑖𝑑 + 1 18 else if 𝛿 < 0 then // Compare() 19 ℎ𝑖 � 𝑚𝑖𝑑 − 1 20 else 21 insert(𝑖𝑑𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑚𝑖𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑟𝑖𝑑) // Found duplicate values in 𝑅 22 end 23 if 𝛿 ≠ 0 then 24 insert(𝑖𝑑𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑙𝑜𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑟𝑖𝑑) // No existing values found in 𝑅 25 end 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 Security Analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Threat Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The database servers are assumed to be semi- honest, meaning that they are not interested in launching active attacks such as compromising the secret shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, those servers may eavesdrop on the messages and share passively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' for example, the servers may manage to collect secret shares of arbi- trary plaintexts, which essentially implies a chosen-plaintext attack (CPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In the context of order-preserving data confidentiality, however, achieving the conventional CPA indistinguishability (IND-CPA) is impossible because the ordering information does allow the adver- sary to make a guess much better than a random one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, we assume the adversary server is interested in learning information other than the orders of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is also called IND-OCPA in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A stronger security notion was proposed called 7 Conference’17, July 2017, Washington, DC, USA Dongfang Zhao Algorithm 3: ODES Query Input: An order-predicate 𝑂𝑃(·) based on which a set of records are returned from 𝑅, |𝑅| = 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' each of share of 𝑅𝑗 is stored at node 𝑁𝑗, 1 ≤ 𝑗 ≤ 𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Output: The client receives a set of records 𝑟’s whose order 𝑟𝑜𝑟𝑑 satisfying 𝑂𝑃(𝑟𝑜𝑟𝑑) = 𝑇𝑟𝑢𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' /* On client / 1 for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do 2 Send 𝑂𝑃(·) to 𝑅𝑗 3 end /* On server 𝑁𝑗 / 4 Receive 𝑂𝑃(·) from client 5 for i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' i < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' i++ do 6 if 𝑂𝑃(𝑖) then 7 𝑘 ≔ 𝑖𝑑𝑥[𝑖] 8 Send 𝑅𝑗 [𝑘] to the client 9 end 10 end /* On client / 11 Collect 𝐾 values from each of 𝑚 nodes 12 res = [] 13 for k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' k < K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' k++ do 14 𝑣𝑎𝑙 ≔ 0 15 for j = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j < m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' j++ do // Reconstruct() 16 𝑣𝑎𝑙 += 𝑅𝑗 [𝑘] 17 end 18 res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='append(𝑣𝑎𝑙) 19 end 20 return res indistinguishability under frequency-analyzing ordered chosen- plaintext attack (IND-FAOCPA) [19], implying that the access pat- tern cannot be leaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Security Assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We assume that a block cipher (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', AES) can be used as a pseudorandom generator (PRG) in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We need this assumption because random numbers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the func- tion 𝑅𝑛𝑑()) are essential for our protocols (see Algs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is a well-accepted assumption in the literature of applied cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 Indistinguishability between Two Parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Let 𝑚𝐿 and 𝑚𝑅 de- note two distinct plaintext messages of the same length𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' According to Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1, when 𝑚𝐿 is encrypted, 𝑠0 ← 𝑅𝑛𝑑() and 𝑠1 ≔ 𝑚𝐿 −𝑠0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The function can be similarly defined for 𝑚𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Because we assume two nodes 𝑁0 and 𝑁1 do not collude, only one share can be accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Formally, we define the following function 𝑂𝐷𝐸𝑆2 𝐿(𝑚𝐿,𝑚𝑅,𝑖 ∈ {0, 1}) def = � 𝑠1 = 𝑅𝑛𝑑(), 𝑖 = 1 𝑠0 = 𝑚𝐿 − 𝑠1, 𝑖 = 0 and 𝑂𝐷𝐸𝑆2 𝑅(𝑚𝐿,𝑚𝑅,𝑖 ∈ {0, 1}) def = � 𝑠1 = 𝑅𝑛𝑑(), 𝑖 = 1 𝑠0 = 𝑚𝑅 − 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑖 = 0 The goal is to show that when an adversary A submits two messages 𝑚𝐿 and 𝑚𝑅 to the ODES oracle, A cannot distinguish whether a share 𝑠𝑖 is from 𝑂𝐷𝐸𝑆2 𝐿 or 𝑂𝐷𝐸𝑆2 𝑅, 𝑖 ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' If 𝑠1 is accessed, then it is obvious that 𝑠1 cannot be distinguished because both 𝑂𝐷𝐸𝑆2 𝐿 and 𝑂𝐷𝐸𝑆2 𝑅 return random numbers through 𝑅𝑛𝑑().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' If 𝑠0 is accessed, we need to show that 𝑚𝐿 − 𝑅𝑛𝑑() and 𝑚𝑅 − 𝑅𝑛𝑑() are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is indeed the case because 𝑅𝑛𝑑() is sup- posed to be uniformly distributed in the message space and adding its output to 𝑚𝐿 or 𝑚𝑅 would render it garbled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Technically, the probability for A to distinguish two randomized numbers is 1 2𝑙 , which is a negligible function in 𝑙—the bit length of the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In other words, the 𝑠0 shares generated by 𝑂𝐷𝐸𝑆2 𝐿 and 𝑂𝐷𝐸𝑆2 𝑅 are interchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 Indistinguishability between Arbitrary Parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In the general case, 𝑡 nodes split the plaintexts, the secrets can be similarly defined in extended functions: 𝑂𝐷𝐸𝑆𝑡 𝐿(𝑚𝐿,𝑚𝑅,𝑖 ∈ [0,𝑡)) def = � 𝑠𝑖 = 𝑅𝑛𝑑(), 𝑖 ≠ 0 𝑠0 = 𝑚𝐿 − �𝑡−1 𝑖=1 𝑠𝑖, 𝑖 = 0 and 𝑂𝐷𝐸𝑆𝑡 𝑅(𝑚𝐿,𝑚𝑅,𝑖 ∈ [0, 1)) def = � 𝑠𝑖 = 𝑅𝑛𝑑(), 𝑖 ≠ 0 𝑠0 = 𝑚𝑅 − �𝑡−1 𝑖=1 𝑠𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 𝑖 = 0 When 𝑖 ≠ 0, the 𝑠𝑖 = 𝑅𝑛𝑑() values cannot be distinguished between 𝑂𝐷𝐸𝑆𝑡 𝐿 and 𝑂𝐷𝐸𝑆𝑡 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' When 𝑖 = 0, repeated 𝑅𝑛𝑑()’s can only further garble the𝑚𝐿 and𝑚𝑅 messages (recall that the arithmetic operation over a set of negligible functions results in a negligible function as well), making 𝑠0 interchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 IND-FAOCPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' To show that ODES is IND-FAOCPA, we need to demonstrate two features: (i) releasing the access patterns does not help the adversary to distinguish between two ciphertexts, and (ii) the sequence of pairs of plaintext messages is always ordered such that the adversary has no extra information to help the distin- guishing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will show that both properties are satisfied by ODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A complete proof is beyond the scope of this conference paper and we will keep our analysis at the descriptive level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Frequency Analyzing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Every time a plaintext interacts with the cluster of database servers, the plaintext is freshly decomposed by the Share() primitive (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The probability that the shared secrets are repeated is negligible, 1 2𝑙·𝑚 , where 𝑙 denotes the bit- string length of the message space and 𝑚 denotes the number of database servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Intuitively, this can be understood as the chance of generating a repeated set of shares given the same plaintext is extremely low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the adversary cannot launch a successful attack by studying the frequency or access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Ordered Plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Recall that one assumption of the proposed ODES scheme is that they are non-colluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This implies that the database servers have no way to learn about the plaintext except for the ordinal information of its local shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, as we discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5, those shares are indistinguishable from a random number in the local table of the database server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For example, in Figure 2, 8 Order-Preserving Database Encryption with Secret Sharing Conference’17, July 2017, Washington, DC, USA although the positive database server knows that 22-MAR has a higher balance than that of 22-FEB (by collecting the 𝛿s), there is no way for the server to reveal the real balance of either 22-MAR or 22-FEB (unless both servers send over the local share in its entirety, which is not allowed in ODES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The order of local shares on an individual database server does not help;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' for example, even the servers themselves cannot reach a consensus (let alone whether the order is consistent with the plaintext): the positive server says the local share of 22-MAR ($11,000) is lower than that of 22-FEB ($14,000) and yet the negative server says the local share of 22-MAR ($2,000) is higher than that of 22-FEB ($-2,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4 EVALUATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 System Implementation We have implemented the proposed ODES protocol with about 1,200 lines of Python code and Bash script, which will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='com/hpdic/odes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We choose the lightweight SQLite as the local database instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Note that SQLite is a file-based data- base and does not support network access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We thus implement a communication layer among remote SQLite instances through the paramiko library for secure data transfer and remote query invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Some of the most important libraries and dependencies include: python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0, sqlite 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1, numpy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0, paramiko 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0, scp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4, and cryptography 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Experimental Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 Test Bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We deploy the proposed ODES protocol and the latest OPE scheme [22] with SQLite [15] on a 10-node cluster hosted at CloudLab [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Each node is equipped with two 32-core Intel Xeon Gold 6142 CPUs, 384 GB ECC DDR4-2666 memory, and two 1 TB SSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The operating system image is Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 LTS, and the page size is 4 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' All servers are connected via a 1 Gbps control link (Dell D3048 switches) and a 10 Gbps experimental link (Dell S5048 switches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We only use the experimental links for our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Specifically, we name the 10 nodes in the cluster as node0–node9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The client runs on node0, the ODES server runs on node1–node8, and the OPE server runs on node9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' All 10 nodes are enabled with password-less SSH connection for convenient communication since our evaluation focuses on performance metrics rather than security measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 7 illustrates the topological structure of our 10-node cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 Baseline Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The literature on order-preserving encryp- tion (OPE) is mainly contributed by the security and database com- munities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Not all proposed ideas had been implemented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' as of the writing of this paper, the state-of-the-art OPE system was presented at [22], which adopts the symmetric encryption scheme 128-bit AES [25] as a building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Because the plaintext is not always 128-bit long, our implementation takes the PKCS7 [18] padding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will use the term OPEA to refer to this particular sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The original OPEA system was implemented with a server user- defined function on MySQL and a client script using Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' How- ever, to make a fair comparison, we re-implement both the server and client functions of OPEA for SQLite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' in particular, we leverage Test Bed: 10-node Cluster @ CloudLab Client (node0) OPE DB Server (node9) ODES Servers (node1--node8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 4: 10-node cluster on CloudLab the state-of-the-art secure communication framework paramiko and the cryptographic package hazmat for security building blocks in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 Data Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The synthetic benchmark is TPC-H ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 [31], a standard database benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There are overall eight tables in TPC-H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' we select four of them for evaluating the proposed work: Supplier, Customer, Part, and Orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The reason why we choose these four tables is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' First, they all exhibit a single-attribute primary key that is straightforward to encode in the underlying data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Second, they all have at least one numerical attribute that is more interesting to encode than textual attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The attributes we are interested in include: Supplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='S_Acctbal, Customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='C_Acctbal, Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='P_Retailprice, Orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='O_Totalprice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In our TPC-H of scale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='01, the above column includes 100, 1,500, 2,000, and 15,000 tuples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition to the synthetic benchmark, we also evaluate the proposed work with three real-world data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The first application is the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' national COVID- 19 statistics from April 2020 to March 2021 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The data set has 341 days of 16 metrics, such as death increase, positive increase, and hospitalized increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Bitcoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The second application is the history of Bitcoin trade volume [6] since it was first exchanged in the public in February 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The data consists of the accumulated Bitcoin exchange on a 3-day basis from February 2013 to January 2022, totaling 1,086 large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Human Genome #38 (hg38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The third application is the hu- man genome reference 38 [16], commonly known as hg38, which includes 34,424 rows of singular attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', tran- scription positions, coding regions, and number of exons, last updated in March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 Workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As discussed in [22], we are primarily interested in two performance metrics in an order-preserving database en- cryption scheme: insertion and query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 9 Conference’17, July 2017, Washington, DC, USA Dongfang Zhao Supplier Customer Part Orders TPC-H Tables 101 102 103 104 Time (second) End-to-End Performance ODES OPEA Figure 5: Overall performance of ODES and OPEA The insertion workload works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The client roughly issues the following SQL to insert a series of records into the SQLite database: INSERT INTO
VALUES ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The insertion starts from scratch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the target table is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' During the insertion, some sort of indexing (depend- ing on the scheme, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', ODES, OPEA) is carried out for future queries touching on ordering/sorting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Essentially, the insertion incurs not only the searching overhead of com- paring the provided ciphertext and those existing (encrypted) tuples already in the table but also the updating overhead for maintaining the order information after the new record is inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The query workload is simpler than the insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We ran- domly make log(𝑛) point-wise queries and compare their values, where 𝑛 denotes the total number of records in the ta- ble, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Intuitively, if there are a large number of records for insertion, the query time should be significantly smaller than the insertion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We will also report the overhead incurred by the client, if applica- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The client overhead includes the initialization and maintenance of encoding the plaintext and optionally other auxiliary information such as local table in OPEA or linear secret sharing in ODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We carry out all experiments at least three times and report the average numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We do not plot the variances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', the error bars) because they are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A complete log of the experiments is available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Unless otherwise stated, the default number of ODES shares is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 End-to-end Performance on TPC-H We start with reporting the end-to-end performance of the proposed ODES protocol with the conventional OPEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 5 shows the end-to-end execution time of inserting 𝑛 tuples into an empty table followed by log(𝑛) queries, where 𝑛 denotes the total number of records as discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Four TPC-H tables are involved and the time includes the overhead on the client side in addition to the insertion and the query time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' According to the results of Figure 5, ODES outperforms OPEA by orders of magnitude in all TPC-H tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Specifically, for Customer and Part tables, the improvement is about 100×;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' for Supplier and Orders tables, the speedup is over 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, we claim that ODES can improve the overall order-preserving encryption time 10×–100× faster on TPC-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In the following three subsections, we will break down the overall running time into three phases: client overhead, insertion time, and query time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ODES for Real-World Applications We report the ODES performance on three different real-world data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We break down the execution time into client overhead, inser- tion time, and query time to have a better idea of the distribution of cost in real-world ODES applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Since our goal is to gain more insight into cost allocation in practice, we will not compare the metric to the baseline, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', OPEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 6 shows the time breakdown of those three real-world data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There are a few interesting observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' First, the client overhead is insignificant, if not negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This reaffirms our previ- ous conclusion on the effectiveness of ODES for lightweight devices in the Internet of Things (IoT) or edge computing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Sec- ond, the cost allocation between insertion and query is dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For COVID-19, the insertion cost is almost 10× lower than the query cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' however, a converse phenomenon occurs for hg38, where the insertion cost is more than 10× higher than the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This can be best explained by the cardinality of the table: there are 341 rows in COVID-19 and more than 30,000 records in hg38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 Client Overhead Recall that in OPEA, the client maintains a key-value store of plain- text and ciphertext such that when inserting a new record the client can locate the ciphertexts of the lower and upper encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is required because OPEA must synchronize the stateful client-side table and the recording encoding on the server side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Each cipher- text is essentially a 128-bit byte-string (because OPEA uses 128-bit AES for encryption) regardless of the plaintext length1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Assuming a plaintext numerical value is stored with 4 bytes (32 bits), the expansion rate of the local table in OPEA is 128+32 32 = 5×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' By contrast, the ODES client is stateless and lightweight: (i) the client simply generates 𝑚 − 1 random numbers as shares and calcu- late the the𝑚-th share by subtracting the plaintext with those𝑚 −1 random numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' (ii) the client does not maintain any data struc- ture to store the intermediate values (technically, the expansion rate is zero if we follow the same terminology as OPEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, it is reasonable to expect a much lower client overhead in ODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 7 reports the overhead of ODES and OPEA on the client side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Indeed, we observe that the ODES client overhead is about 3–4 orders of magnitude lower than OPEA for all TPC-H tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition to the lower overhead, the results also suggest that ODES is a more practical solution for those lightweight devices such as smartphones and smartwatches, which have limited computational power and storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 1Obviously, we here assume the tuple is represented by less than 128 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This is indeed the case because, practically speaking, the numerical values in database applications are almost always less than 2128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 10 Order-Preserving Database Encryption with Secret Sharing Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' USA Client Insertion Query 10 2 10 1 100 Time (second) COVID-19 Client Insertion Query 10 2 10 1 100 Time (second) Bitcoin Client Insertion Query 100 101 102 Time (second) hg38 Figure 6: ODES for real-world applications Supplier Customer Part Orders TPC-H Tables 10 3 10 2 10 1 100 101 102 103 Time (second) Client Overhead ODES OPEA Figure 7: Client overhead of ODES and OPEA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='6 Insertion Performance The main cost of inserting a record in OPEA lies in the interaction between the client and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' After the client locates the lower and upper ciphertexts that are closest to the next plaintext, the client needs to make two queries to the server to retrieve the encoding of both ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For simplicity, the encoding can be thought of as the positions of the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Roughly speaking, one OPEA insertion incurs two additional database queries (one for the lower neighbor and the other one for the upper neighbor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, the number of queries is tripled at the time of insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In addition, the client needs to adjust the local table to incorporate the newly inserted plaintext and its associated ciphertexts, the entry position in the local table, and the counter of possible repetition of duplicate plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In contrast, the main cost of ODES for inserting records lies in the cost of uploading secret shares to multiple servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, ODES is a stateless protocol, meaning that no extra information needs to be maintained on either the client or the server side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, the queries among distinct servers can be parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' As a result, although the overall number of queries in ODES may exceed that in OPEA (if there are more than three ODES servers), the running time for executing these queries is about the same as a single query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Combining all the above factors, we expect that ODES runs must faster than OPEA for inserting records into the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 9 reports the insertion performance of ODES and OPEA on TPC-H tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We observe that for small and medium scales Supplier Customer Part Orders TPC-H Tables 10 1 100 101 102 103 104 Time (second) Insertion Performance ODES OPEA Figure 8: Insertion performance of ODES and OPEA (Supplier, Customer, Part), ODES outperforms OPEA by more than two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' For the large-scale data set of the Orders table, the speedup is also significant, exceeding 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='7 Query Performance The performance gap between ODES and OPEA for queries is signif- icantly smaller than that for record insertion because the ordering information of OPEA is stored as plaintext and therefore an effi- cient binary sorting can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, as discussed above, the expanded ciphertext (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', 5×) takes significantly more storage space and network transmission than the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' On the other hand, ODES deals with 𝑚× plaintexts during the queries that can be parallelized, where 𝑚 denotes the number of nodes (shares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 9 reports the query performance of ODES and OPEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We observe that ODES still outperforms OPEA in terms of query performance, and yet the gap is only about 35%–55% rather than orders of magnitude as we have seen for insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, we argue that 35%–55% is nonetheless a significant improvement in query performance, demonstrating the superiority of the proposed ODES protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 Scalability of ODES The last metric we are interested in learning about ODES perfor- mance is the scalability regarding different numbers of shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Thus far,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' all experiments assume that there are two nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' each of which 11 Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' USA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Dongfang Zhao ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Supplier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Customer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Orders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='TPC-H Tables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Query Performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='ODES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='OPEA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Figure 9: Query Performance of ODES and OPEA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Number of Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='ODES@TPC-H Supplier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Insertion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Number of Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='ODES@TPC-H Customer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Insertion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Number of Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='ODES@TPC-H Part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Insertion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Number of Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='ODES@TPC-H Orders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Overhead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Insertion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='Figure 10: Scalability of ODES on four TPC-H relations: Sup- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='plier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Customer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' and Orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' holds a secret share from the plaintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' There is no technical limita- tion preventing the user from employing more nodes to increase the security level, possibly by allowing more processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, the adversary would need to compromise more nodes to launch a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 10 reports the scalability of ODES when various numbers of shares (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', nodes) are involved on four TPC-H tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We ob- serve that the client overhead is negligible at all scales, which is consistent with what we have learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The insertion cost mildly increases when more shares are involved, which can be explained by the higher cost of splitting the plaintext into a larger number of shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The query time exhibits a steeper trend concerning the number of nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' the cost looks proportional to the number of Supplier Customer Part Orders TPC-H Tables 0 100 200 300 400 500 600 700 Database Size (Kilobytes) Server Storage ODES OPEA Figure 11: Server storage of ODES and OPEA nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The main reason for this linear proportion is due to the se- rial implementation of the paramiko library, which is synchronous and therefore a remote query must complete before another one is issued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' We believe parallel processing from the client would greatly mitigate this issue by utilizing more CPU cores and higher network bandwidth, which will be explored in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='9 Storage Cost Client Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Recall that ODES is a stateless protocol, meaning that there is zero storage requirement for ODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' On the other hand, the space overhead of OPEA is O(�𝑛), where �𝑛 denotes the number of distinct values in the column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Server Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Figure 11 reports the database size of ODES and OPEA when various TPC-H tables are encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The results sug- gest that for small- and medium-scale data sets, the database size is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' However, for larger data sets such as the orders table, the space overhead is much higher in OPEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' This can be best explained by the fact that OPEA involves the AES scheme, which causes size expansion in the ciphertext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' That is, no matter how small the plaintext value is, the encrypted text will always be a fixed length (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=', 128 bits) for security reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Therefore, the more plaintext we have, the larger space overhead is incurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK This paper proposes a new stateless order-preserving encryption scheme, namely ODES (Ordered Database Encryption with Secret- sharing), by incorporating secret-sharing primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' ODES sup- ports the latest IND-FAOCPA security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' A series of database proto- cols are designed based on ODES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' The ODES scheme and database protocols are implemented on top of a 10-node cluster of SQLite databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Experimental results show that ODES outperforms state- of-the-art schemes by orders of magnitude on the TPC-H bench- mark and multiple real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Our future work will focus on more efficient secret-sharing prim- itives that we hope will further help reduce the insertion and query time in database applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Another orthogonal research direc- tion is to explore the feasibility of homomorphic encryption in order-preserving encryption in database systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' 12 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12(8):933–947, apr 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [8] Alexandra Boldyreva, Nathan Chenette, Younho Lee, and Adam O’Neill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Order- preserving symmetric encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In Proceedings of the 28th Annual International Conference on Advances in Cryptology - EUROCRYPT 2009 - Volume 5479, page 224–241, Berlin, Heidelberg, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [9] Dan Boneh, Kevin Lewi, Mariana Raykova, Amit Sahai, Mark Zhandry, and Joe Zimmerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Semantically secure order-revealing encryption: Multi-input functional encryption without obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In Elisabeth Oswald and Marc Fis- chlin, editors, Advances in Cryptology - EUROCRYPT 2015, pages 563–594, Berlin, Heidelberg, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Springer Berlin Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [10] David Cash, Feng-Hao Liu, Adam O’Neill, Mark Zhandry, and Cong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Parameter-hiding order revealing encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' In Thomas Peyrin and Steven Galbraith, editors, Advances in Cryptology – ASIACRYPT 2018, pages 181–210, Cham, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' [11] Nathan Chenette, Kevin Lewi, Stephen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE3T4oBgHgl3EQfMAny/content/2301.04370v1.pdf'} +page_content=' Weis, and David J.' metadata={'source': 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[math.CA] 27 Jan 2023 +REGULARITY OF ALMOST-SURELY INJECTIVE PROJECTIONS IN +EUCLIDEAN SPACES +KRZYSZTOF BARAŃSKI∗, YONATAN GUTMAN†, AND ADAM ŚPIEWAK§ +Abstract. It is known that if a finite Borel measure µ in a Euclidean space has Hausdorff dimension +smaller than a positive integer k, then the orthogonal projection onto almost every k-dimensional +linear subspace is injective on a set of full µ-measure. We study the regularity of the inverses of +such projections. We prove that if µ has a compact support X and (respectively) the Hausdorff, +upper box-counting or Assouad dimension of X is smaller than k, then the inverse is (respectively) +continuous, pointwise Hölder for some α ∈ (0, 1) or pointwise Hölder for every α ∈ (0, 1). The result +generalizes to the case of typical linear perturbations of Lipschitz maps. Additionally, we construct a +non-trivial measure on the plane which admits almost-surely injective projections in every direction, +and show that no homogeneous self-similar measure has this property. +1. Introduction and main results +1.1. Projections of sets and measures in Euclidean spaces. The study of geometric and +dimensional properties of the images of a set X ⊂ RN, N ∈ N, under orthogonal projections +PV : RN → V +onto k-dimensional linear spaces V ⊂ RN, is a well-known subject of interest in geometric mea- +sure theory. The space of all k-dimensional linear subspaces of RN (or, equivalently, the space of +corresponding orthogonal projections) forms the Grassmannian Gr(k, N), which has a structure of +a k(N − k)-dimensional manifold, equipped with the standard rotation-invariant (Haar) measure. +Throughout the paper, the terms ‘almost every linear space’ or ‘almost every projection’ will be +used in relation to this measure. +A classical result in this area is the celebrated Marstrand–Mattila theorem, proved in [Mar54, +Mat75]. +Theorem 1.1 (Marstrand–Mattila Projection Theorem). Let X be a Borel set in RN. Then +(i) dimH PV (X) = min{k, dimH X} for almost every k-dimensional linear subspace V of RN. +(ii) If dimH X > k, then PV (X) has positive k-dimensional Hausdorff measure for almost every +k-dimensional linear subspace V of RN. +Here and in the sequel dimH denotes the Hausdorff dimension. There have been a number of +further results related to Marstrand–Mattila projection theorem, including versions valid for various +types of dimension and estimates on the size of the set of exceptional projections, see e.g. [FH97, +PS00, Mat04, Bou10, FO14, FO17, Orp21] and the references therein, as well as [FFJ15] for a +comprehensive survey. +∗Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland +†Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, +Poland +§Department of Mathematics, Bar-Ilan University, Ramat Gan, 5290002, Israel +E-mail addresses: baranski@mimuw.edu.pl, gutman@impan.pl, ad.spiewak@gmail.com. +Date: January 30, 2023. +1 + +In particular, there has been established a version of Marstrand–Mattila’s projection theorem for +measures (see [HT94, HK97]). +Theorem 1.2 (Marstrand–Matilla Projection Theorem for measures). Let µ be a finite +Borel measure in RN. Then +(i) dimH PV µ = min{k, dimH µ} for almost every k-dimensional linear subspace V of RN. +(ii) If dimH µ > k, then PV µ is absolutely continuous with respect to k-dimensional Hausdorff +measure for almost every k-dimensional linear subspace V of RN. +Here PV µ denotes the image of µ under PV . +1.2. Injective and almost-surely injective projections. Apart from considering the dimension +of the images of X under orthogonal projections PV , another line of research is to study under which +conditions the projections PV are injective on X, at least for typical V . Note that if this occur, +then PV provides a topological embedding of X into a k-dimensional linear space V , and X can be +considered as the graph of a function from PV (X) ⊂ V ≃ Rk to V ⊥ ≃ RN−k. +It is known that if X is a compact subset of RN and 2dimB X < k, where dimB denotes the +upper box-counting (Minkowski) dimension, then PV is injective for a typical k-dimensional linear +space V ⊂ RN. This fact is commonly referred to as the Mañé projection theorem. Indeed, Mañé +proved this result for topologically generic projections in [Mn81], while a version valid for almost +every projection (or, equivalently, for almost every linear map L: RN → Rk) was presented in +[SYC91, Rob11]. Notice that this agrees with the well-known Menger–Nöbeling embedding theorem +(see e.g. [HW41, Theorem 5.2], which states that for a compact metric space X with Lebesgue +covering dimension at most k, a generic continuous transformation φ: X → R2k+1 is injective. +Given a projection PV which is injective on a set X, it is natural to ask what is the regularity of +its inverse +(PV |X)−1 : PV (X) → X. +In [HK99, Theorem 3.1], Hunt and Kaloshin proved that if X ⊂ RN is compact and 2dimB X < k, +then for almost every k-dimensional linear space V ⊂ RN, the projection PV restricted to X has +an α-Hölder inverse for 0 < α < 1 − 2dimB X +k +(see also [BAEFN93, EFNT94] for earlier results in +this direction and [Rob11] for a detailed exposition). In [RS19, Theorem 2.1], Rossi and Shmerkin +gave upper bounds on the Hausdorff dimension of the set of exceptional projections. Furthermore, +the regularity of the inverses can be improved, if the assumption 2dimB X < k is replaced by +2 dimA X < k, where dimA is the Assouad dimension (see Definition 2.2). More precisely, in this +case almost all projections onto k-dimensional linear subspaces of RN have inverses which are α- +Hölder for any α ∈ (0, 1), see [Ols02, Theorem 5.2] and [Rob11, Theorem 9.18]. The problem of +existence of linear embeddings and regularity of their inverses was also studied for finite-dimensional +subsets of Banach spaces, see [Rob11, Chapters 5-9] and references there. +It is known that in general, the bound 2dimB X < k in the Mañé projection theorem cannot be +diminished (see [HW41, Example V.3]), and dimB cannot be replaced by dimH (see [SYC91, Ap- +pendix])1. However, the situation changes if instead of the injectivity of PV on X, one is interested +in almost sure injectivity of PV , i.e. the injectivity of PV on a full µ-measure Borel subset of X, +according to a given Borel measure µ on X. In our previous paper [BGŚ20, Corollary 3.4], strength- +ening a result by Alberti, Bölcskei, De Lellis, Koliander and Riegler [ABDL+19], we showed that if +X is a Borel subset of RN equipped with a Borel σ-finite measure µ and the k-Hausdorff measure of +1In fact, these examples show that even the existence of an injective orthogonal projection does not hold under +weaker assumptions +2 + +X is zero (in particular, if dimH X < k ≤ N), then for almost every k-dimensional linear subspace V +of RN, the orthogonal projection PV is injective on a full µ-measure Borel set XV ⊂ X (see Theorem +1.11 for a more general statement). In this way, for a given set X, the minimal embedding dimension +k for a typical projection can be reduced by half when considered in a ‘probabilistic’ setup instead +of a ‘deterministic’ one. +We note that the condition dimH µ ≤ k is necessary for a typical almost sure injectivity of PV . +Indeed, we have the following. +Proposition 1.3. Let µ be a finite Borel measure in RN with dimH µ > k. Then for almost every +k-dimensional linear space V ⊂ RN and every Borel set Y ⊂ RN of full µ-measure, the orthogonal +projection PV onto V is not injective on Y . +The proof of Proposition 1.3 is contained in Section 3. +In the border case dimH µ = k, if µ is not singular with respect to the k-dimensional Hausdorff +measure, then different types of behaviour can occur, e.g. for k = 1 and 1-dimensional Hausdorff +measure on an interval in R2, a projection onto a typical line is injective on supp µ, while for 1- +dimensional Hausdorff measure on a circle in R2, no Lipschitz map φ : R2 → R is injective on a set +of full measure, see [BGŚ20, Example 3.5]. +In this paper, assuming suitable bounds on the dimension of a compact set X ⊂ RN, we study +the question of the regularity of the almost sure inverse map +(PV |XV )−1 : PV (XV ) → XV +for a typical k-dimensional linear space V ⊂ RN, where PV is injective on a full µ-measure Borel set +XV ⊂ X. +Definition 1.4. A map φ: A → RN, where A ⊂ Rk, is pointwise α-Hölder, if for every x ∈ A there +exists cx > 0 such that +∥φ(x) − φ(y)∥ ≤ cx∥x − y∥α +for every y ∈ A. +The basic result of this paper is the following. +Theorem 1.5 (Regularity of the inverse of almost-surely injective projections). Let X be +a compact subset of RN and let µ be a finite Borel measure supported on X. Consider orthogonal +projections PV : RN → V onto k-dimensional linear spaces V ⊂ RN. Then the following hold. +(i) If dimH X < k, then for almost every V there exists a full µ-measure Borel set XV ⊂ X +such that the restriction of PV to XV is injective with continuous inverse. +(ii) If dimB X < k, then for almost every V there exists a full µ-measure Borel set XV ⊂ X +such that the restriction of PV to XV is injective with inverse which is pointwise α-Hölder +for every α ∈ +� +0, 1 − dimB X +k +� +. +(iii) If dimA X < k, then for almost every V there exists a full µ-measure Borel set XV ⊂ X +such that the restriction of PV to XV is injective with inverse which is pointwise α-Hölder +for every α ∈ (0, 1). +Theorem 1.5 follows from a more general Theorem 1.12, presented in Subsection 1.4. +Remark 1.6. It is important to note that in general we cannot obtain Hölder continuity (instead +of pointwise Hölder continuity) of the inverse of PV |XV in Theorem 1.5, even under the assumption +dimA X < k. +Indeed, if X ⊂ RN is a compact set with dimA X < k, which does not embed +3 + +topologically into Rk (for example, there exist simplicial complexes of dimension n which do not +embed topologically into R2n, see [HW41, Example V.3]) and µ is any measure with suppµ = X, +then a projection PV onto a k-dimensional linear space V ⊂ RN, which is injective on a set of full +µ-measure with a Hölder inverse is actually a homeomorphism on X, which is impossible as X does +not embed topologically into Rk. See [BGŚ20, Remark 3.10] for the details of this argument. +Remark 1.7. It is easy to see that in general, Theorem 1.5 does not hold if we assume only +dimH µ < k. Indeed, let X be a dense countable subset of [0, 1]N and let µ be any finite Borel +measure on X satisfying µ({x}) > 0 for every x ∈ X. Then dimH µ = 0, but no projection onto a +k-dimensional linear space V ⊂ RN for k ≤ N − 1 can be injective on a full µ-measure subset of X +with a continuous inverse, since X cannot be a graph of a continuous function defined on a subset +of V . +1.3. Measures with almost-surely injective projections in every direction. An important +question in the geometric measure theory, which has gained an increasing interest in recent years, +is finding conditions under which suitable projection theorems hold for every (rather than almost +every) projection. This question is particularly interesting in the context of self-similar measures +for iterated function systems, i.e. Borel probability measures µ in RN satisfying +µ = +� +i∈I +pi ϕiµ, +where I is a finite set, (pi)i∈I is a strictly positive probability vector and the iterated function system +(IFS) is given by contracting similarities +ϕi : RN → RN, +ϕi(x) = riOi(x) + ti +with scales ri ∈ (0, 1), orthogonal matrices Oi ∈ RN×N and translation vectors ti ∈ RN. +In [HS12, Theorem 1.6], Hochman and Shmerkin proved that if the iterated function system +{ϕi : i ∈ I} satisfies the strong separation condition and the semigroup generated by {Oi : i ∈ I} +acts minimally on Gr(k, N), then the first assertion of Marstrand–Mattila’s projection theorem for +measures (Theorem 1.2) holds for all orthogonal projections onto k-dimensional linear subspaces of +RN. On the other hand, Rapaport [Rap17] constructed a self-similar measure in the same class, which +does not satisfy the absolute continuity part of Theorem 1.2 for a Baire-residual set of projections. +Similarly, he showed that for such measures, the Slicing theorem (Theorem 3.2) can fail for a residual +set of projections. Nevertheless, in [Rap20] he proved that typical homogeneous self-similar measures +(i.e. the ones satisfying riOi = rjOj for all i, j) in the plane with the strong separation condition +will satisfy both assertions of Theorem 1.2 for every projection. In the case of random constructions, +Simon and Rams [RS14, RS15] showed that almost every fractal percolation satisfies the assertions +of Theorem 1.1 for every projection. +Considering the question of the injectivity of projections, we note that obviously, a non-singleton +set in RN cannot be projected injectively onto all k-dimensional linear subspaces of RN, for any +1 ≤ k ≤ N − 1. This naturally leads us to the setup of almost-surely injective projections. The first +question appearing in this context is whether there exist non-trivial (with non-singleton support) +measures which project almost-surely injectively in every direction. The following result shows that +in fact such measures exist. +Theorem 1.8 (Existence of measure with almost-surely injective projections in every +direction). There exists a Borel probability measure µ in R2 with non-singleton compact support +4 + +and positive Hausdorff dimension, such that every orthogonal projection PV : R2 → V onto a line +V ⊂ R2 is injective on a Borel set XV of full µ-measure. +The proof of Theorem 1.8 is presented in Section 5. +Furthermore, we show that, unlike for the Marstrand–Mattila’s and Slicing Theorems, non- +degenerated homogeneous self-similar measures cannot satisfy this property, even generically. +Proposition 1.9. Let µ be a self-similar measure in RN corresponding to a homogeneous IFS +ϕi(x) = rO(x) + ti, i ∈ I such that ϕi are not all equal. Then for every k ∈ {1, . . . , N − 1} there +exists a k-dimensional linear space V ⊂ RN such that for every Borel set Y ⊂ RN of full µ-measure, +the orthogonal projection PV onto V is not injective on Y . +Proposition 1.9 follows from a more general fact (Proposition 5.4), which is proved in Section 5. +Remark 1.10. Note that we cannot extend Proposition 1.3 to projections onto all k-dimensional +linear subspaces V ⊂ RN. An obvious example is the lift of 1-dimensional Lebesgue measure on +the x-axes on the plane to a graph of a function R → R with Hausdorff dimension greater than +1 (e.g. the graph of a non-differentiable Weierstrass-type function), which projects injectively onto +the x-axis. On the other hand, if µ is s-analytic for s > k, then no projection onto a k-dimensional +linear space V ⊂ RN is injective on a set of positive µ-measure, see [ABDL+19, Corollary IV.2]. +1.4. Regularity of the inverse for typical linear perturbations of Lipschitz maps. The +result described in Theorem 1.5 can be generalized to the setup of almost-surely injective linear +perturbations of Lipschitz maps on compact sets in Euclidean spaces. +In a previous paper, we +proved the following. +Theorem 1.11 ([BGŚ20, Theorem 3.1]). Let µ be a finite or σ-finite Borel measure in RN with +support X = suppµ, such that µ is singular with respect to the k-dimensional Hausdorff measure for +some k ∈ N (in particular, it is enough to assume dimH µ < k), and let φ: X → Rk be a Lipschitz +map. Then for almost every linear transformation L: RN → Rk there exists a Borel set XL ⊂ X of +full µ-measure, such that the map φL = φ + L is injective on XL. +Here and in the sequel, ‘almost every linear map’ refers to the Lebesgue measure in the space +Lin(RN, Rk) ≃ RNk. The result can be generalized to the case of Hölder maps φ: X → Rk, see +[BGŚ20] for details. +The conclusion of Theorem 1.11 holds also for prevalent sets in spaces of +Lipschitz and Cr maps φ: X → Rk (see Definition 2.3 and Remark 2.4). +Extending the setup of Subsection 1.2, we study regularity properties of the inverse maps +(φL|XL)−1 : φL(XL) → XL. +The following is our main result. +Theorem 1.12 (Regularity of the inverse for almost-surely injective linear perturbations +of Lipschitz maps). Let µ be a finite Borel measure in RN with a compact support X = supp µ +and let φ: X → Rk be a Lipschitz map. We write φL = φ + L for linear maps L ∈ Lin(RN, Rk). +Then the following hold. +(i) If dimH(supp µ) < k, then for almost every linear map L there exists a Borel set XL ⊂ X +of full µ-measure such that φL is injective on XL with continuous inverse. +(ii) If dimB (supp µ) < k, then for almost every linear map L there exists a Borel set XL ⊂ X +of full µ-measure such that φL is injective on XL with inverse which is pointwise α-Hölder +for every α ∈ +� +0, 1 − dimB (supp µ) +k +� +. +5 + +(iii) If dimA(suppµ) < k, then for almost every linear map L there exists a Borel set XL ⊂ X of +full µ-measure on which φL is injective with inverse which is pointwise α-Hölder for every +α ∈ (0, 1). +Note that orthogonal projections PV : RN → V for V ∈ Gr(k, N) with the rotation-invariant +measure can be treated as linear maps from Lin(RN, Rk) ≃ RNk with the Lebesgue measure (see +Remark 3.1). Therefore, Theorem 1.12 implies immediately Theorem 1.5 by setting φ = 0. On the +other hand, considering all Lipschitz or Cr-maps φ gives a result for almost every map in the given +class in the sense of prevalence, as defined in [HSY92]. See Definition 2.3 and Remark 2.4 for more +details. +1.5. Relation to the theory of compressed sensing. The field of compressed sensing grew +out of the work of Candès, Donoho, Romberg and Tao ([Can06, CRT06b, Don06, FR13]). The +fundamental problem of the theory is to give conditions enabling to recover an input vector x ∈ RN +from its linear measurement y = Ax ∈ Rm, where A ∈ Rm×N even though m << N. +A key +theorem ([FR13, Theorem 9.12], see also [CT06, CRT06a]) states that with high probability one +may recover x with ∥x∥0 := |{j : xj ̸= 0}| ≤ s (the s-sparsity condition) from y when A ∈ Rm×N +is a random Gaussian matrix with m ≈ s ln(N +s ) via an ℓ1-minimization basis pursuit algorithm +([Mal99, §1.4.3], see also [CDS01]). Capitalizing on sparsity, compressed sensing has found many +applications ([LDP07, DDT+08, BS07, HS09]). +In [ABDL+18] Alberti, Bölcskei, De Lellis, Koliander and Riegler studied the above-mentioned +problem in a setting where both the input vector x and the sensing matrix A are random: x ∈ RN +is given according to a probability measure µ in Rn, A ∈ Rm×N is given according to the Lebesgue +measure on Rm×N and one seeks to recover x from y = Ax µ-almost surely. In particular, they +proved a version of Theorem 1.11 for probability measures, with φ = 0 and the Hausdorff dimension +replaced with the lower modified Minkowski dimension, see [ABDL+18, Theorem II.1]. +In recent years there has been a surge in interest in a compressed sensing framework for analog +signals modeled by continuous-alphabet discrete-time stochastic processes2 with general (not nec- +essarily sparse) distributions ([WV10, DT10, DMM11, JP17, RJEP17, GK19, GŚ19, GŚ20]). One +of the tenants of the theory, introduced in the pioneering work of Wu and Verdú [WV10], is that +the regularity class of the decompressor is of foremost importance, as it introduces robustness to +noise. Translated to the setting of [ABDL+18] and Theorem 1.11, this corresponds to investigating +the regularity of the inverse map L−1 : L(XL) → XL. Thus one may interpret our main result as +giving almost sure regularity guarantees for decompression under various dimension assumptions on +the measure generating the input vector. +1.6. Further related topics. Let us also mention that almost sure injectivity plays an important +role in some other nonlinear projection schemes. For instance, in the context of natural projection +maps from the symbolic space for iterated function systems, this property is called weak almost +unique coding (see [KS19, Definition 1.9]) and for self-similar systems it is known to be equivalent +to the no dimension drop condition (i.e. the equality of the dimension of given ergodic invariant +measure to the ratio of its entropy and Lyapunov exponent), see [KS19, Appendix] and [Fen19, +Corollary 4.7]. This observation can be successfully utilized for obtaining dimension results for cer- +tain classes of fractal attractors, see e.g. [KS19]. Moreover, basic techniques developed for studying +2The rigorous passage between continuous-time signals and discrete-time signals is justified by the Shannon sam- +pling theorem ([Hig96, Chapter 1]). +6 + +typical properties of orthogonal projections can be often transferred to parametrized families of it- +erated functions systems satisfying the transversality condition (an analogue of Lemma 4.1). It was +first used by Pollicott and Simon [PS95] and led, for example, to results analogous to Marstrand’s +projection theorem [SSU01a, SSU01b] (see also [BSSŚ22] for a more detailed overview). +Finally, techniques originating from the study of orthogonal projections can also be used to analyse +so-called delay-coordinate maps, i.e. maps of the form φ(x) = (h(x), h(Tx), . . . , h(T k−1x)), where +T : X → X is a discrete-time dynamical system and h: X → R is a real function (observable). +The injectivity or almost sure injectivity of φ implies that the original dynamics can be faithfully +modelled based only on the values of the observable. Related embedding results, know as Takens- +type theorems, serve as a framework for applications in natural sciences see e.g. [Tak81, SM90, +SYC91, Aba96]. Recently, a probabilistic counterpart of this theory has been developed [SSOY98, +BGŚ20, BGŚ22a, BGŚ22b], where the regularity of almost-surely defined mappings related to φ +plays a crucial role. See [BGŚ22b] for a more detailed discussion and further references. +Structure of the paper. Section 2 contains basic definitions and a description of technical tools +used in subsequent parts of the paper. In Section 3 we prove Proposition 1.3, while Section 4 contains +the proof of Theorem 1.12. Theorem 1.8 and Proposition 1.9 are proved in Section 5. +2. Preliminaries +We consider the Euclidean space RN with the standard inner product ⟨·, ·⟩ and denote, respec- +tively, by ∥·∥ and |·| the corresponding norm and diameter. The symbol BN(x, ε) denotes an ε-ball +centred at x in the Euclidean norm in RN. We often write B(x, ε) when the dimension is clear from +the context. +2.1. Dimensions. +Definition 2.1. For s > 0, the s-dimensional (outer) Hausdorff measure of a set X ⊂ RN is defined +as +Hs(X) = lim +δ→0 inf +� ∞ +� +i=1 +|Ui|s : X ⊂ +∞ +� +i=1 +Ui, |Ui| ≤ δ +� +. +The Hausdorff dimension of X is given as +dimH X = inf{s > 0 : Hs(X) = 0} = sup{s > 0 : Hs(X) = ∞}. +For a bounded set X ⊂ RN and δ > 0, let N(X, δ) denote the minimal number of balls of radius δ +required to cover X. The lower and upper box-counting (Minkowski) dimensions of X are defined, +respectively, as +dimBX = lim inf +δ→0 +log N(X, δ) +− log δ, +and +dimB X = lim sup +δ→0 +log N(X, δ) +− log δ +. +For a finite Borel measure µ in RN, we define its (upper) Hausdorff dimension as +dimH µ = inf +� +dimH X : X ⊂ RN Borel with µ(RN \ X) = 0 +� +and the lower Hausdorff dimension as +dimHµ = inf +� +dimH X : X ⊂ RN Borel with µ(X) > 0 +� +. +7 + +Definition 2.2. A bounded set X ⊂ RN is said to be (M, s)-homogeneous if N(X ∩ B(x, r), ρ) ≤ +M(r/ρ)s for every x ∈ X, 0 < ρ < r, i.e. the intersection B(x, r) ∩ X can be covered by at most +M(r/ρ)s balls of radius ρ. The Assouad dimension of X is defined as +dimA X = inf{s > 0 : X is (M, s)-homogeneous for some M > 0}. +It is easy to see that in the definitions of box-counting and Assouad dimensions it is enough to +consider covers by balls centred in X. For a bounded set X ⊂ RN and a finite Borel measure µ on +X, we have the following inequalities (see e.g. [Rob11, (9.1)]): +(1) +dimHµ ≤ dimH µ ≤ dimH X ≤ dimBX ≤ dimB X ≤ dimA X. +2.2. Prevalence. A notion of prevalence was introduced by Hunt, Shroer and Yorke in [HSY92] +and is regarded to be an analogue of ‘Lebesgue almost sure’ condition in infinite dimensional linear +spaces. +Definition 2.3. Let V be a complete linear metric space (i.e. a linear space with a complete metric +which makes addition and scalar multiplication continuous). A Borel set S ⊂ V is called prevalent if +there exists a Borel measure ν in V, which is positive and finite on some compact set in V, such that +for every v ∈ V, we have v + e ∈ S for ν-almost every e ∈ V. A non-Borel subset of V is prevalent if +it contains a prevalent Borel subset. +We focus mainly on the prevalence in the space Lip(X, Rk) of all Lipschitz functions h: X → Rk +on a compact set X ⊂ RN, endowed with the Lipschitz norm ∥h∥Lip = ∥h∥∞+Lip(h), where ∥h∥∞ is +the supremum norm and Lip(h) is the Lipschitz constant of h. Note however, that in Theorem 1.12 +we can consider prevalence in other spaces, as explained in the following remark. +Remark 2.4. Let V be any of the spaces of Lipschitz or Cr, r = 1, 2, . . . , ∞ maps from (a bounded +open neighbourhood of) a compact set X ⊂ RN into Rk endowed with the natural complete linear +metric. Note that in order to show prevalence of a set S ⊂ V via linear maps, it is enough to prove +that for every φ ∈ V and Lebesgue-almost every L ∈ Lin(RN, Rk) ≃ RNk we have φ + L ∈ S. As +all the Cr-spaces listed above are contained in the space of Lipschitz maps, it is enough to consider +Lipschitz maps φ. +2.3. Conditional measures. It will be useful to work with a system of conditional measures for +considered projections. +Definition 2.5. Given a continuous map φ: X → Rk on a compact set X ⊂ RN there exists a +system of conditional measures of a probability measure µ on X with respect to φ, i.e. a family +{µy : y ∈ Rk}, such that +(i) for every y ∈ Rk, µy is a (possibly zero) Borel measure on φ−1({y}), +(ii) for φµ-almost every y ∈ Rk, µy is a Borel probability measure, +(iii) for every µ-measurable set A ⊂ X, the function Rk ∋ y �→ µy(A) is φµ-measurable and +µ(A) = +ˆ +Rk +µy(A)dφµ(y). +The existence and φµ-almost sure uniqueness of the system of conditional measures follows from +the Rokhlin’s Disintegration Theorem [Roh52]. See also [Sim12] for a more direct approach. +The following lemma characterizes almost sure injectivity in terms of conditional measures. +8 + +Lemma 2.6. Let φ: X → Rk be a continuous map on a compact set X ⊂ Rk and let µ be a Borel +probability measure on X. Then Xφ is injective on a Borel set Xφ ⊂ X of full µ-measure if and only +if µφ(x) = δx for µ-almost every x ∈ X. +Proof. If φ is injective on Xφ, then setting µφ(x) = δx for x ∈ Xφ and µy = 0 for y /∈ φ(Xφ) +gives a system of conditional measures of µ with respect to φ (see [BGŚ22a, p. 620] for a detailed +argument). +Hence, the first implication follows by the almost sure uniqueness of the system of +conditional measures. For the other implication, assume µφ(x) = δx for µ-almost every x ∈ X. Then +Xφ = {x ∈ X : µφ(x) = δx} is the required set of injectivity. Indeed, µ(Xφ) = 1 by assumption, and +if x, y ∈ Xφ and φ(x) = φ(y), then δx = δy, so x = y. +□ +3. Orthogonal projections and slices +As noted in introduction, we denote by Gr(k, N) the Grassmannian of k-dimensional linear sub- +spaces of RN and by γk,N the unique rotation-invariant measure on Gr(k, N) (see [Mat95, Section +3.9] for details and [FR02] for an alternative construction). Recall that for V ∈ Gr(k, N) we denote +by PV : RN → V ≃ Rk the orthogonal projection onto V . +Remark 3.1. As we switch between Lebesgue-almost sure statements for linear transformations L ∈ +Lin(RN, Rk) ≃ RNk and γk,N-almost sure statements for orthogonal projections PV , V ∈ Gr(k, N), +it is useful to note that the two statements are equivalent if one is interested in almost sure injectivity +of L and PV . Namely, any linear map L: RN → Rk of full rank can be represented uniquely as +L = ψ ◦ PV , where PV is the orthogonal projection onto the k-dimensional orthogonal complement +of Ker L and ψ: V → Rk is a linear isomorphism, hence (almost sure) injectivity of L is equivalent +to the almost sure injectivity of PV . It is easy to see that this identification preserves sets of full +measure. +Following the notation from [Mat95], for a compactly supported finite measure µ in RN, a linear +space V ∈ Gr(k, N) and the orthogonal complement W = V ⊥ ∈ Gr(N − k, N), we denote by +{µW,a : a ∈ V } the system of conditional measures of µ with respect to PV . The measures µW,a +are concentrated on P −1 +V (a) = W + a and are called the sliced measures of µ (in direction W). +Järvenpää and Mattila [JM98, Theorem 3.3] proved a general ‘slicing’ theorem for measures, which +in our notation reads as follows: +Theorem 3.2 (Slicing Theorem). Let µ be a compactly supported finite Borel measure on RN. +For γk,N-almost every V ∈ Gr(k, N) and W = V ⊥ we have +dimHµW,a ≥ dimHµ − k for Hk-almost every a ∈ V with µW,a(RN) > 0. +By Lemma 2.6 and Remark 3.1, Theorem 1.11 implies the following corollary. +Corollary 3.3. Let µ be a compactly supported finite Borel measure on RN with dimH µ < k. Then +for γk,N-almost every V ∈ Gr(k, N) and W = V ⊥, the sliced measure µW,a is a point mass for +PV µ-almost every a ∈ V . +Now we can give a proof of Proposition 1.3. +Proof of Proposition 1.3. We claim that under the assumptions of the proposition, there exists a +compact set X ⊂ RN such that µ(X) > 0 and ν = µ|X satisfies dimHν > k. Indeed, recall (see +e.g. [Fal97, Proposition 10.3]) that +dimH µ = esssup +x∼µ +lim inf +r→0 +log µ(B(x, r)) +log r +. +9 + +Now choose X to be a compact subset of positive µ-measure of the set +� +x ∈ RN : lim inf +r→0 +log µ(B(x, r)) +log r +≥ s +� +for a fixed s > 0 with k < s < dimH µ. Then applying Frostman’s lemma (see e.g. [PU10, Theorem +8.6.3]) we obtain dimHν ≥ s > k. +By Theorem 3.2, for γk,N-almost every V ∈ Gr(k, N) and +W = V ⊥, +dimHνW,a ≥ dimHν − k > 0 for Hk-almost every a ∈ V such that νW,a(RN) > 0. +On the other hand, Theorem 1.2 implies that PV ν is absolutely continuous with respect to Hk for +γk,N-almost every V ∈ Gr(k, N), hence we can conclude +dimHνW,PV x > 0 for ν-almost every x ∈ RN. +As Dirac’s delta has dimension zero, this means that for γk,N-almost every V ∈ Gr(k, N), almost +every conditional measure of ν with respect to PV is not a Dirac’s delta, hence by Lemma 2.6 there +cannot exist a set of full ν-measure on which PV is injective. As ν is absolutely continuous with +respect to µ, the same is true for µ. By Remark 3.1, this holds also for almost every linear map +L: RN → Rk. +□ +4. Proof of Theorem 1.12 +Let E = E(N, k) denote the set of linear maps L: RN → Rk of the form +Lx = +� +⟨l1, x⟩, . . . , ⟨lk, x⟩ +� +, +where l1, . . . , lk ∈ RN satisfy ∥l1∥, . . . , ∥lk∥ ≤ 1. As E may be identified with (BN(0, 1))k, we will +denote by Leb the normalized k-fold product of Lebesgue measures on BN(0, 1), considered as a +probability measure on E. Note that it is enough to prove the assertion of Theorem 1.12 for Leb- +almost every L ∈ E, as then rescaling gives the result for almost every linear mapping L: RN → Rk. +By the Cauchy–Schwarz inequality, for all x ∈ RN +(2) +∥Lx∥ ≤ +√ +N ∥x∥. +The following lemma is the key technical ingredient of the proof. +Lemma 4.1. Let L: RN → Rk be a linear transformation. Then for every x ∈ RN \ {0}, z ∈ Rk +and ε > 0, +Leb({L ∈ E : ∥Lx + z∥ ≤ ε}) ≤ C εk +∥x∥k , +where C = C(N, k) > 0 is a constant depending only on N and k. +For the proof see [Rob11, Lemma 4.1]. We will prove each of the items of Theorem 1.12 separately. +4.1. Proof of assertion (i) of Theorem 1.12. We actually prove the following, slightly stronger +version of the theorem. +Theorem 4.2. Let µ be a finite Borel measure on RN with a compact support X = suppµ satisfying +Hk(X) = 0 and let φ: X → Rk be a Lipschitz map. Then for almost every linear map L: RN → Rk +there exists a Borel set XL ⊂ X of full µ-measure such that for every x ∈ XL and every ε > 0, there +exists δ > 0 for which the map φL = φ + L satisfies +(3) +for every y ∈ X, if ∥φL(x) − φL(y)∥ ≤ δ, then ∥x − y∥ ≤ ε. +10 + +Proof. The first part of the argument is obtained directly from the proof of [BGŚ20, Theorem 3.1]. +We include the arguments for the convenience of the reader. First, we will prove that for every +x ∈ X we have +(4) +Leb +�� +L ∈ E : +∃ +y∈X\{x} φL(x) = φL(y) +�� += 0. +Note that the above set, as well as all similar sets we consider in this section, are Borel measurable as +a consequence of standard considerations (see [BGŚ20, Lemma 2.4]). As φ: X → Rk is a Lipschitz +map, there exists H > 0 so that for all x, y ∈ X, +(5) +∥φ(x) − φ(y)∥ ≤ H ∥x − y∥. +Fix x ∈ X, ε > 0 and let +Kn = +� +y ∈ X : ∥x − y∥ ≥ 1 +n +� +for n ∈ N. Define +Bn = +� +L ∈ E : +∃ +y∈Kn φL(x) = φL(y) +� +. +and note that for (4) it suffices to prove Leb(Bn) = 0 for each n. As dimH Kn ≤ dimH X < k, there +exists a collection of balls BN(yi, εi), i ∈ N, for some yi ∈ Kn and εi > 0, such that +(6) +Kn ⊂ +� +i∈N +BN(yi, εi) +and +∞ +� +i=1 +εk +i ≤ ε. +Let L ∈ Bn. Then there is y ∈ Kn such that φL(x) = φL(y). Clearly, y ∈ BN(yi, εi) for some i ∈ N. +We calculate +∥φL(x) − φL(yi)∥ ≤ ∥φL(x) − φL(y)∥ + ∥φL(y) − φL(yi)∥ += ∥φL(y) − φL(yi)∥ +≤ ∥φ(yi) − φ(y)∥ + ∥L(yi − y)∥ +≤ H∥yi − y∥ + +√ +N∥yi − y∥ +≤ Mεi +for M = H + +√ +N, by (2) and (5). This shows +Bn ⊂ +� +i∈N +{L ∈ E : ∥φL(x) − φL(yi)∥ ≤ Mεi}. +Thus, using Lemma 4.1 and the fact ∥x − yi∥ ≥ 1 +n, we obtain +Leb(Bn) ≤ +∞ +� +i=1 +Leb({L ∈ E : ∥L(x − yi) + φ(x) − φ(yi)∥ ≤ Mεi}) +≤ CMk +1/nk +∞ +� +i=1 +εk +i ≤ CMknkε. +As ε > 0 is arbitrary, we obtain Leb(Bn) = 0, and thus (4) is established. Combining (4) with +Fubini’s theorem (see e.g. [Rud87, Thm. 8.8]), we obtain +(7) +µ +�� +x ∈ X : +∃ +y∈X\{x} φL(x) = φL(y) +�� += 0 +11 + +for Leb-almost every L ∈ E. Hence, the set +XL = X \ +� +x ∈ X : +∃ +y∈X\{x}φL(x) = φL(y) +� +is a full µ-measure set on which φL is injective (this proves Theorem 1.11). To obtain additionally +the continuity of φ−1 +L +on XL, fix L ∈ E satisfying (7). We claim that every x ∈ XL satisfies (3). If +not, then there exists ε > 0 such that for every n ≥ 1 there exists yn ∈ X satisfying +∥φL(x) − φL(yn)∥ ≤ 1 +n +and +∥x − yn∥ > ε. +As X is compact, there is a converging subsequence ynk → y for some y = y(L, x) ∈ X. By the +continuity of φL, we have φL(x) = φL(y) and ∥x − y∥ ≥ ε, in particular x ̸= y, contradicting +x ∈ XL. +□ +4.2. Proof of assertion (ii) of Theorem 1.12. The proof combines the techniques of [HK99, +Theorem 3.1] and [BGŚ20, Theorem 3.1]. +Lemma 4.3. Let X be a compact subset of RN with dimB (X) < k. Fix θ ∈ +� +0, k − dimB (X) +� +and +a Lipschitz map φ : X → Rk. Then there exists a constant D > 0 such that inequality +Leb +�� +L ∈ E : +∃ +y∈X ∥φL(x) − φL(y)∥ ≤ ε and ∥x − y∥ ≥ δ +�� +≤ Dδ−kεk−dimB X−θ +holds for every x ∈ X and 0 < 2ε ≤ δ, where φL = φ + L. +Proof. By the definition of dimB , there exists a constant D = D(X, θ) such that for every ε > 0 +there exists a cover +(8) +X ⊂ +Nε +� +i=1 +B(yi, ε) with Nε ≤ Dε−(d+θ). +Consider x, y ∈ X and L ∈ E such that ∥φL(x) − φL(y)∥ ≤ ε and ∥x − y∥ ≥ δ. Let yi be such that +y ∈ B(yi, ε). Then, recalling that we assume 2ε ≤ δ, +∥L(x − yi) − (φ(yi) − φ(x)) ∥ = ∥φL(x) − φL(yi)∥ ≤ ∥φL(x) − φL(y)∥ + ∥φL(y) − φL(yi)∥ +≤ ε + +� +Lip(φ) + sup +L∈E +∥L∥ +� +ε ≤ Mε +, +where M = 1 + Lip(φ) + sup +L∈E +∥L∥ < ∞. Moreover, +∥x − yi∥ ≥ ∥x − y∥ − ∥y − yi∥ ≥ δ − ε ≥ δ +2. +Therefore, +� +L ∈ E : +∃ +y∈X ∥φL(x) − φL(y)∥ ≤ ε and ∥x − y∥ ≥ δ +� +⊂ +Nε +� +i=1 +{L ∈ E : ∥L(x − yi) − (φ(yi) − φ(x)) ∥ ≤ Mε and ∥x − yi∥ ≥ δ/2} . +Hence, by Lemma 4.1 and (8), +Leb +�� +L ∈ E : +∃ +y∈X ∥φL(x) − φL(y)∥ ≤ ε and ∥x − y∥ ≥ δ +�� +≤ C2kNεδ−kMkεk +≤ C2kDMkδ−kεk−d−θ. +□ +12 + +Proof of Theorem 1.12.(ii). Set d = dimB (supp µ). Fix α ∈ +� +0, 1 − d +k +� +and let θ ∈ (0, k − d) be such +that α < 1 − d+θ +k . Let H = sup +L∈E +diam (φL(X)). For a fixed x ∈ X, by (4) (which we can apply as +dimH µ ≤ d < k) and Lemma 4.3, +Leb +�� +L ∈ E : +∀ +M>0 +∃ +y∈X ∥x − y∥ > M∥φL(x) − φL(y)∥α�� += lim +M→∞ Leb +�� +L ∈ E : +∃ +y∈X ∥x − y∥ > M∥φL(x) − φL(y)∥α�� += lim +M→∞ +∞ +� +m=0 +Leb +�� +L ∈ E : +∃ +y∈X 2−(m+1)H < ∥φL(x) − φL(y)∥ ≤ 2−mH +and ∥x − y∥ > M∥φL(x) − φL(y)∥α�� ++ Leb +�� +L ∈ E : +∃ +y∈X φL(x) = φL(y) and ∥x − y∥ > 0 +�� +(4) +≤ +lim +M→∞ +∞ +� +m=0 +Leb +�� +L ∈ E : +∃ +y∈X ∥φL(x) − φL(y)∥ ≤ 2−mH and ∥x − y∥ > MHα2−α(m+1)�� +Lemma 4.3 +≤ +lim +M→∞ +∞ +� +m=0 +DM−kH−αk2αk(m+1)2−m(k−d−θ)Hk−d−θ += lim +M→∞ DM−kHk(1−α)−d−θ2αk +∞ +� +m=0 +2m(αk−k+d+θ) += 0, +as the series +∞ +� +m=0 +2m(αk−k+d+θ) converges since we assume α < 1 − d+θ +k . This proves that for every +x ∈ X that the condition +(9) +∥x − y∥ ≤ M∥φLx − φLy∥α +for some M = M(x, L) and every y ∈ X +is satisfied for almost every L ∈ E. Therefore, by Fubini’s theorem, for almost every L ∈ E condition +(9) holds for µ-almost every x ∈ X. Finally, note that by taking a countable intersection of full +Lebesgue measure sets, we can assume that for almost every L ∈ E, the condition (9) holds for every +α < 1 − d+θ +k . +□ +4.3. Proof of assertion (iii) of Theorem 1.12. Again, we prove a stronger result. +Theorem 4.4. Let R > 0 and η > 1. +Let µ be a probability measure in RN with a compact +support X = suppµ satisfying |X| ≤ R and dimA X < k. Fix a Lipschitz map φ: X → Rk and +θ ∈ (0, k − dimA(supp µ)). Then for almost every linear mapping L : RN → Rk there exists a Borel +set XL ⊂ X of full µ-measure such that for each point x ∈ XL there is a constant C > 0 for which +the map φL = φ + L satisfies +(10) +∥φLx − φLy∥ ≥ Cf(∥x − y∥) +for every y ∈ X, +where +f(x) = x +� +1 +log2(2R/x) +�η/θ +. +Proof. Once more, by Fubini’s theorem it is enough to prove that for every x ∈ X, The condition +(10) holds for almost every L ∈ E. The rest of the proof is a combination of the methods set forth +13 + +in [Ols02, Theorem 5.2] and [BGŚ20, Theorem 3.1]. As φ: X → Rk is a Lipschitz map, there exists +H > 0 so that for all x, y ∈ X, +(11) +∥φ(x) − φ(y)∥ ≤ H ∥x − y∥. +Fix x ∈ X. +Define rn = +R +2n and ρn = f(rn−1) > 0. +Note that for 0 < x ≤ R, +f(x) +x +≤ 1 as +log2(2R +x ) ≥ 1. Moreover, a simple calculation shows that the function f is monotone increasing on +(0, R]. For n ≥ 1, define +Zn = {y ∈ X : rn ≤ ∥y − x∥ ≤ rn−1}. +By the definition of the Assouad dimension, there exists K > 0 such that for every 0 < s < rn−1 +and a ball B of radius rn−1, the set X ∩ B may be covered by K(rn−1 +s )k−θ balls of radius s. Let +c > 2 satisfy cθ > K. We conclude that the set Zn, which is contained in a ball B of radius rn−1 +around x, may be covered by at most ℓn,i ≤ Kci(k−θ)(rn−1 +ρn )k−θ balls {B(an,i,j, ρn +ci )}ℓn,i +j=1 of radius ρn +ci +(with centers in Zn) for i ≥ 1 (recall ρn ≤ rn−1). Thus, Zn ⊂ +ℓn,i +� +j=1 +B(an,i,j, ρn +ci ). For i ≥ 2, define +Ui = +∞ +� +n=1 +ℓn,i +� +j=1 +B +� +an,i,j, 2ρn +ci +� +. +Every center an,i,j satisfies ∥an,i,j − x∥ ≥ rn, so a ball of radius 2ρn +ci +< ρn +2 ≤ rn−1 +2 +≤ rn does not +contain x. Thus, for i ≥ 2, +X \ {x} ⊂ Ui. +In order to establish the condition (10) for a fixed x ∈ X and Leb-almost every L ∈ E, it is enough +to show +lim +i→∞ Leb +� +L ∈ E : +∃ +y∈X +∃ +n≥1 ∥φLx − φLy∥ < ρn +ci and ∥x − y∥ ≥ rn +� += 0 +Indeed, this implies that for almost every L ∈ E there exists i = i(L) ≥ 2, such that for every +y ∈ X, ∥x − y∥ ≥ rn implies ∥φLx − φLy∥ ≥ ρn +ci . As every y ∈ X \ {x} is contained in some Zn +(recall |X| ≤ R), this implies that for every y ∈ X \ {x} (using monotonicity of f on (0, R]), +∥φLx − φLy∥ ≥ ρn +ci = f(rn−1) +ci +≥ 1 +ci f(∥x − y∥). +Denote Ai = +� +L ∈ E : +∃ +y∈X +∃ +n≥1 ∥φLx − φLy∥ < ρn +ci and ∥x − y∥ ≥ rn +� +. Clearly, Ai is a Borel set. +Let L ∈ Ai. Thus one may find y ∈ X and n ≥ 1 so that ∥φLx − φLy∥ < ρn +ci and ∥x − y∥ ≥ rn. +Consequently, y ∈ �n +m=1 Zm. Therefore, one may find a center am,i,j such that y ∈ B(am,i,j, ρm +ci ). +Note that +∥φLx − φLam,i,j∥ ≤ ∥φLx − φLy∥ + ∥φLy − φLam,i,j∥. +By (2) and (11), +∥φLy − φLam,i,j∥ ≤ ∥Ly − Lam,i,j∥ + ∥φ(y) − φ(am,i,j)∥ ≤ +√ +N ρm +ci + H ρm +ci . +Thus, +∥φLx − φLam,i,j∥ ≤ ρn +ci + ( +√ +N + H)ρm +ci ≤ Qρm +ci , +where Q = 1 + +√ +N + H. Set Am,i,j = +� +L ∈ E : ∥φL(x − am,i,j)∥ ≤ Q ρm +ci +� +. We conclude +Ai ⊂ +∞ +� +m=1 +ℓm,i +� +j=1 +Am,i,j. +14 + +By Lemma 4.1 (recall that am,i,j ∈ Zm), +Leb(Am,i,j) ≤ C (Q ρm +ci )k +rkm +. +Thus, +Leb(Ai) ≤ +∞ +� +m=1 +ℓm,i +� +j=1 +Leb(Am,i,j) ≤ +∞ +� +m=1 +Kci(k−θ)�rm−1 +ρm +�k−θ +C (Q ρm +ci )k +rkm +≤ KC(2Q)k +ciθ +∞ +� +m=1 +�f(rm−1) +rm−1 +�θ +. +We notice +�f(rm−1) +rm−1 +�θ += +� +1 +log2(2m) +�η += +1 +mη . +Thus +Leb(Ai) ≤ KC(2Q)k +ciθ +∞ +� +m=1 +1 +mη . +As +∞ +� +m=1 +1 +mη < ∞, this implies lim +i→∞ Leb(Ai) = 0 as desired. +□ +5. Measure with all almost-surely injective projections +5.1. Proof of Theorem 1.8. The measure µ will be defined in two steps. First, we will define a +measure ν on the unit interval by randomizing digits in dyadic expansions and then we will push +ν to the graph of the function x �→ x2. For the first step, it is convenient to work in the symbolic +space {0, 1}N. +Partition N = {1, 2, . . .} into blocks Bn = {2n − 1, 2n}, n ≥ 1 and denote Ln = 2n − 1, Rn = 2n, +so that the block Bn consists of the left bit Ln and the right bit Rn. For elements ω ∈ {0, 1}N we +use the notation ω = (ω1, ω2, . . .). Set +Σ = {ω ∈ {0, 1}N : ωLn = 0 for each n ≥ 1}. +Fix p ∈ (0, 1/2). Define a probability measure p on {0, 1}2 as +p = pδ(0,1) + (1 − p)δ(0,0) +and let P be a probability measure on {0, 1}N given by +P = p⊗N +(we use below the identification {0, 1}N = +� +{0, 1}2�N). Clearly, P(Σ) = 1. Now we transport P to +the unit interval by setting +ν = ΠP and X = Π(Σ), +where Π is given by +(12) +Π(ω) = +∞ +� +j=1 +ωj2−j. +The last step is to lift ν to the graph of a non-linear function. Let f : [0, 1] → [0, 1] be given by +f(x) = x2 and Ψ: [0, 1] → [0, 1]2 by Ψ(x) = (x, f(x)). Finally, we set +µ = Ψν +and claim that the measure µ satisfies the property from Theorem 1.8. First, note that dimH ν = +−p log p−(1−p) log(1−p) +log 4 +, see [BP17, Example 1.5.6]. As Ψ is bi-Lipschitz, we have dimH µ = dimH ν > 0. +For the injectivity part of Theorem 1.8, let us fix a non-zero linear map L: R2 → R, L(x, y) = +15 + +αx + βy. As f is a bijection on [0, 1], we can assume that α ̸= 0 and β ̸= 0, as otherwise L is +injective on the whole graph Ψ([0, 1]) and the claim of Theorem 1.8 follows trivially. Note that for +x, y ∈ [0, 1] with x ̸= y we have +(13) +L(x, f(x)) = L(y, f(y)) ⇔ f(y) − f(x) +x − y += α +β ⇔ x + y = −α +β . +Therefore, in order to show the injectivity of L, we need to study solutions of the equation x+y = z +for fixed z and x, y taken from X. We will do so in terms of the dyadic expansions. +Note that every z ∈ [0, 1) has a unique dyadic expansion z = �∞ +j=1 zj2−j such that the sequence +(zj)∞ +j=1 ∈ {0, 1}N is not eventually equal to 1 (we say that the dyadic expansions does not terminate +with 1’s). Moreover, the only points z ∈ [0, 1) which have a non-unique dyadic expansion are the +dyadic rationals from (0, 1). For them, there are exactly two expansions, one terminating with 1’s +and one terminating with 0’s. As Σ does not contain sequences terminating with 1’s, we see that Π +is injective on Σ. Therefore, for x, y ∈ X we will write x = +∞ +� +j=1 +xj2−j, y = +∞ +� +j=1 +yj2−j for its unique +dyadic expansion which does not terminate with 1’s, so that Π((x1, x2, . . .)) = x, Π((y1, y2, . . .)) = y +and (x1, x2, . . .), (y1, y2, . . .) ∈ Σ. First, we need a technical lemma. +Lemma 5.1. Let x, y, z ∈ [0, 1) have dyadic expansions +x = +∞ +� +j=1 +xj2−j, y = +∞ +� +j=1 +yj2−j, z = +∞ +� +j=1 +zj2−j +which do not terminate with 1’s. Assume x+y = z. Then for every k ∈ N, the condition xk = yk = 0 +implies � +j